Planck's Half-Quanta: A History of the Concept of Zero-Point Energy

Foundations of Physics

Volumn 29, Issue 1, 1999, Pages 91-132

  Mehra, Jagdish ^a   Rechenberg, Helmut ^b  

^a UNIVERSITY OF HOUSTON   (United States)

^b MAX PLANCK INSTITUT FÜR PHYSIK   (Germany)

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EID: 0033449722     PISSN: 00159018     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1018869221019     Document Type: Article

References (241)

1. The first Solvay Conference on Physics was held in Brussels, Belgium, for 5 days of lectures and discussions, from 30 October to 3 November 1911, on the general theme of " The Theory of Radiation and the Quanta." Its scientific promoter had been Walther Nernst, who became interested in the quantum theory around 1910 when experiments on he low-temperature behavior of the specific heat of solids showed that this property goes to zero, in agreement with the quantum-theoretical treatment of Albert Einstein on specific heats. In order to further advance quantum theory and to solve its problems, Nernst thought that the best way would be to hold an international conference (similar to the Karlsruhe Chemical Congress of 1860, at which the participants had successfully dealt with various questions connected with the chemical nomenclature). The Belgian industrialist Ernest Solvay, a great enthusiast of science (who had himself developed an "amateurish" unified theory of the gravitational and the electromagnetic field, and wished to bring it to the notice of the great physicists), provided the funds to hold the first "Conseil Solvay," which became so successful in its outcome that Solvay himself endowed the institution of holding periodic "Solvay Conferences" in physics and chemistry, a tradition which continues till today. At the first Solvay Conference, lectures were given by H. A. Lorentz - who was also appointed the President of the "Conseil" - (On the Application of the Equipartition Theorem to Radiation), J. H. Jeans (The Kinetic Theory of Specific Heat According to Maxwell and Boltzmann), E. Warburg (The Experimental Test of Planck's Formula for Blackbody Radiation), H. Rubens (The Test of Planck's Radiation Formula for Long Waves), M. Planck (The Laws of Heat Radiation and the Hypothesis of the Elementary Quantum of Action), M. Knudsen (Kinetic Theory and the Observable Properties of Ideal Gases), J. Perrin ( Proofs of the Real Existence of Molecules), W. Nernst (Application of the Quantum Theory to a Series of Physico-Chemical Problems), H. Kamerlingh Onnes (On Electrical Resistance), A. Sommerfeld (The Meaning of the Quantum of Action for Aperiodic Molecular Processes in Physics), P. Langevin (The Kinetic Theory of Magnetism and Magnetons), and A. Einstein (The Present Status of the Problem of Specific Heat). Among the other participants were H. Rubens and W. Wien from Germany, E. Rutherford from England, M. Brillouin, Marie Curie and H. Poincaré from France, and F. Hasenöhrl from Austria. Hendrik Lorentz, who presided over the first "Conseil Solvay," continued as President until his death in 1928 (the last Solvay Conference on which he presided was the fifth one in 1927, at which quantum mechanics was first presented publicly by its protagonists. M. de Broglie, R. Goldschmidt and F. A. Lindemann acted as the Scientific Secretaries of the Conference, and two personal guests of Ernest Solvay (E. Herzen and G. Hostelet) were also included. The proceedings of the 1911 Solvay Conference were published as: Theorie du Rayonnement et les Quanta (P. Langevin and M. de Broglie, eds.), published by Gauthier-Villars, Paris, 1912; German edition, Die Theorie der Strahlung und der Quanten, (A. Eucken, ed. Abhandlungen der Deutschen Bunsen-Gesellschaft für Angewandte Chemie, No. 7 (Halle, 1914). For a detailed historical backgrand of the Solvay Conferences, see: Jagdish Mehra, The Solvay Conferences on Physics: Aspects of the Development of Physics Since 1911 (D. Reidel Publishing Co., Dordrecht, Holland, 1975).
2. The first Solvay Conference on Physics was held in Brussels, Belgium, for 5 days of lectures and discussions, from 30 October to 3 November 1911, on the general theme of " The Theory of Radiation and the Quanta." Its scientific promoter had been Walther Nernst, who became interested in the quantum theory around 1910 when experiments on he low-temperature behavior of the specific heat of solids showed that this property goes to zero, in agreement with the quantum-theoretical treatment of Albert Einstein on specific heats. In order to further advance quantum theory and to solve its problems, Nernst thought that the best way would be to hold an international conference (similar to the Karlsruhe Chemical Congress of 1860, at which the participants had successfully dealt with various questions connected with the chemical nomenclature). The Belgian industrialist Ernest Solvay, a great enthusiast of science (who had himself developed an "amateurish" unified theory of the gravitational and the electromagnetic field, and wished to bring it to the notice of the great physicists), provided the funds to hold the first "Conseil Solvay," which became so successful in its outcome that Solvay himself endowed the institution of holding periodic "Solvay Conferences" in physics and chemistry, a tradition which continues till today. At the first Solvay Conference, lectures were given by H. A. Lorentz - who was also appointed the President of the "Conseil" - (On the Application of the Equipartition Theorem to Radiation), J. H. Jeans (The Kinetic Theory of Specific Heat According to Maxwell and Boltzmann), E. Warburg (The Experimental Test of Planck's Formula for Blackbody Radiation), H. Rubens (The Test of Planck's Radiation Formula for Long Waves), M. Planck (The Laws of Heat Radiation and the Hypothesis of the Elementary Quantum of Action), M. Knudsen (Kinetic Theory and the Observable Properties of Ideal Gases), J. Perrin ( Proofs of the Real Existence of Molecules), W. Nernst (Application of the Quantum Theory to a Series of Physico-Chemical Problems), H. Kamerlingh Onnes (On Electrical Resistance), A. Sommerfeld (The Meaning of the Quantum of Action for Aperiodic Molecular Processes in Physics), P. Langevin (The Kinetic Theory of Magnetism and Magnetons), and A. Einstein (The Present Status of the Problem of Specific Heat). Among the other participants were H. Rubens and W. Wien from Germany, E. Rutherford from England, M. Brillouin, Marie Curie and H. Poincaré from France, and F. Hasenöhrl from Austria. Hendrik Lorentz, who presided over the first "Conseil Solvay," continued as President until his death in 1928 (the last Solvay Conference on which he presided was the fifth one in 1927, at which quantum mechanics was first presented publicly by its protagonists. M. de Broglie, R. Goldschmidt and F. A. Lindemann acted as the Scientific Secretaries of the Conference, and two personal guests of Ernest Solvay (E. Herzen and G. Hostelet) were also included. The proceedings of the 1911 Solvay Conference were published as: Theorie du Rayonnement et les Quanta (P. Langevin and M. de Broglie, eds.), published by Gauthier-Villars, Paris, 1912; German edition, Die Theorie der Strahlung und der Quanten, (A. Eucken, ed. Abhandlungen der Deutschen Bunsen-Gesellschaft für Angewandte Chemie, No. 7 (Halle, 1914). For a detailed historical backgrand of the Solvay Conferences, see: Jagdish Mehra, The Solvay Conferences on Physics: Aspects of the Development of Physics Since 1911 (D. Reidel Publishing Co., Dordrecht, Holland, 1975).
3. The first Solvay Conference on Physics was held in Brussels, Belgium, for 5 days of lectures and discussions, from 30 October to 3 November 1911, on the general theme of " The Theory of Radiation and the Quanta." Its scientific promoter had been Walther Nernst, who became interested in the quantum theory around 1910 when experiments on he low-temperature behavior of the specific heat of solids showed that this property goes to zero, in agreement with the quantum-theoretical treatment of Albert Einstein on specific heats. In order to further advance quantum theory and to solve its problems, Nernst thought that the best way would be to hold an international conference (similar to the Karlsruhe Chemical Congress of 1860, at which the participants had successfully dealt with various questions connected with the chemical nomenclature). The Belgian industrialist Ernest Solvay, a great enthusiast of science (who had himself developed an "amateurish" unified theory of the gravitational and the electromagnetic field, and wished to bring it to the notice of the great physicists), provided the funds to hold the first "Conseil Solvay," which became so successful in its outcome that Solvay himself endowed the institution of holding periodic "Solvay Conferences" in physics and chemistry, a tradition which continues till today. At the first Solvay Conference, lectures were given by H. A. Lorentz - who was also appointed the President of the "Conseil" - (On the Application of the Equipartition Theorem to Radiation), J. H. Jeans (The Kinetic Theory of Specific Heat According to Maxwell and Boltzmann), E. Warburg (The Experimental Test of Planck's Formula for Blackbody Radiation), H. Rubens (The Test of Planck's Radiation Formula for Long Waves), M. Planck (The Laws of Heat Radiation and the Hypothesis of the Elementary Quantum of Action), M. Knudsen (Kinetic Theory and the Observable Properties of Ideal Gases), J. Perrin ( Proofs of the Real Existence of Molecules), W. Nernst (Application of the Quantum Theory to a Series of Physico-Chemical Problems), H. Kamerlingh Onnes (On Electrical Resistance), A. Sommerfeld (The Meaning of the Quantum of Action for Aperiodic Molecular Processes in Physics), P. Langevin (The Kinetic Theory of Magnetism and Magnetons), and A. Einstein (The Present Status of the Problem of Specific Heat). Among the other participants were H. Rubens and W. Wien from Germany, E. Rutherford from England, M. Brillouin, Marie Curie and H. Poincaré from France, and F. Hasenöhrl from Austria. Hendrik Lorentz, who presided over the first "Conseil Solvay," continued as President until his death in 1928 (the last Solvay Conference on which he presided was the fifth one in 1927, at which quantum mechanics was first presented publicly by its protagonists. M. de Broglie, R. Goldschmidt and F. A. Lindemann acted as the Scientific Secretaries of the Conference, and two personal guests of Ernest Solvay (E. Herzen and G. Hostelet) were also included. The proceedings of the 1911 Solvay Conference were published as: Theorie du Rayonnement et les Quanta (P. Langevin and M. de Broglie, eds.), published by Gauthier-Villars, Paris, 1912; German edition, Die Theorie der Strahlung und der Quanten, (A. Eucken, ed. Abhandlungen der Deutschen Bunsen-Gesellschaft für Angewandte Chemie, No. 7 (Halle, 1914). For a detailed historical backgrand of the Solvay Conferences, see: Jagdish Mehra, The Solvay Conferences on Physics: Aspects of the Development of Physics Since 1911 (D. Reidel Publishing Co., Dordrecht, Holland, 1975).
4. M. Planck, 1911 Solvay Report (German edition, p. 91).
5. M. Planck, Verh. Deutsch. Phys. Ges. 13, 138 (1911); reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Vol. II (F. Vieweg & Sohn, Braunschweig, 1958), pp. 249-259. We shall quote from Max Planck's papers throughout from his Collected Papers, abbreviated as Phys. Abh. There exist three volumes of these, the first two containing his papers in various journals; the third contains a selection of Planck's lectures given on various occasions, as well as the obituaries of his esteemed colleagues. See also: Sitz. Ber. Preuss. Akad. Wiss. (Berlin, 1911), p. 723; reprinted in Phys. Abh. II, pp. 260-268; Ann. Phys. (Leipzig) 37, 642 (1912); reprinted in Phys. Abh. II 287-301.
6. M. Planck, Verh. Deutsch. Phys. Ges. 13, 138 (1911); reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Vol. II (F. Vieweg & Sohn, Braunschweig, 1958), pp. 249-259. We shall quote from Max Planck's papers throughout from his Collected Papers, abbreviated as Phys. Abh. There exist three volumes of these, the first two containing his papers in various journals; the third contains a selection of Planck's lectures given on various occasions, as well as the obituaries of his esteemed colleagues. See also: Sitz. Ber. Preuss. Akad. Wiss. (Berlin, 1911), p. 723; reprinted in Phys. Abh. II, pp. 260-268; Ann. Phys. (Leipzig) 37, 642 (1912); reprinted in Phys. Abh. II 287-301.
7. M. Planck, Verh. Deutsch. Phys. Ges. 13, 138 (1911); reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Vol. II (F. Vieweg & Sohn, Braunschweig, 1958), pp. 249-259. We shall quote from Max Planck's papers throughout from his Collected Papers, abbreviated as Phys. Abh. There exist three volumes of these, the first two containing his papers in various journals; the third contains a selection of Planck's lectures given on various occasions, as well as the obituaries of his esteemed colleagues. See also: Sitz. Ber. Preuss. Akad. Wiss. (Berlin, 1911), p. 723; reprinted in Phys. Abh. II, pp. 260-268; Ann. Phys. (Leipzig) 37, 642 (1912); reprinted in Phys. Abh. II 287-301.
8. M. Planck, Verh. Deutsch. Phys. Ges. 13, 138 (1911); reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Vol. II (F. Vieweg & Sohn, Braunschweig, 1958), pp. 249-259. We shall quote from Max Planck's papers throughout from his Collected Papers, abbreviated as Phys. Abh. There exist three volumes of these, the first two containing his papers in various journals; the third contains a selection of Planck's lectures given on various occasions, as well as the obituaries of his esteemed colleagues. See also: Sitz. Ber. Preuss. Akad. Wiss. (Berlin, 1911), p. 723; reprinted in Phys. Abh. II, pp. 260-268; Ann. Phys. (Leipzig) 37, 642 (1912); reprinted in Phys. Abh. II 287-301.
9. M. Planck, Verh. Deutsch. Phys. Ges. 13, 138 (1911); reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Vol. II (F. Vieweg & Sohn, Braunschweig, 1958), pp. 249-259. We shall quote from Max Planck's papers throughout from his Collected Papers, abbreviated as Phys. Abh. There exist three volumes of these, the first two containing his papers in various journals; the third contains a selection of Planck's lectures given on various occasions, as well as the obituaries of his esteemed colleagues. See also: Sitz. Ber. Preuss. Akad. Wiss. (Berlin, 1911), p. 723; reprinted in Phys. Abh. II, pp. 260-268; Ann. Phys. (Leipzig) 37, 642 (1912); reprinted in Phys. Abh. II 287-301.
10. M. Planck, Verh. Deutsch. Phys. Ges. 13, 138 (1911); reprinted in Max Planck, Physikalische Abhandlungen und Vorträge, Vol. II (F. Vieweg & Sohn, Braunschweig, 1958), pp. 249-259. We shall quote from Max Planck's papers throughout from his Collected Papers, abbreviated as Phys. Abh. There exist three volumes of these, the first two containing his papers in various journals; the third contains a selection of Planck's lectures given on various occasions, as well as the obituaries of his esteemed colleagues. See also: Sitz. Ber. Preuss. Akad. Wiss. (Berlin, 1911), p. 723; reprinted in Phys. Abh. II, pp. 260-268; Ann. Phys. (Leipzig) 37, 642 (1912); reprinted in Phys. Abh. II 287-301.
11. There exist quite a few reports and reviews on the history of the early quantum theory, in particular: F. Reiche, Die Quantentheorie: Ihr Ursprung und ihre Entwicklung (J. Springer, Berlin, 1921), English translation, The Quantum Theory (E. P. Dutton and Co., New York, 1922) (our quotations are from the English version); A. Hermann, Frühgeschichte der Quantentheorie (Physik Verlag, Mosbach, 1969; Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, Vol. I (Springer-Verlag New York. 1982) (in two parts). A rather detailed account is also given in the treatise of E. Whittaker, A History of the Theories of Aether and Electricity, Vol. II: The Modern Theories 1900-1925 (T. Nelson & Sons, London, 1953); reprinted in Harper Torchbooks, New York, 1960. James Hopwood Jeans, a participant in the development of the radiation law, himself gave two fair reports on the early development: Report on Radiation and the Quantum Theory, (The Physical Society of London, London, 1914, 1924). The important concepts are carefully reviewed: Max Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966). Finally, we refer to various articles on the subject of the old quantum theory by M. J. Klein, Arch. Hist. Exact Sci. 1, 459 (1962); The Natural Philosopher 1, 83 (1963); L. Rosenfeld, Osiris 2, 149 (1936); Max Planck et la définition de l'entropie, in Max Planck-Festschrift, 1958 (Berlin, 1958), p. 203. Martin J. Klein has also discussed certain aspects of the early quantum theory in his book Paul Ehrenfest, The Making of a Theoretical Physicist, Vol. I (North-Holland, Amsterdam, 1970), Chap. 10.
12. There exist quite a few reports and reviews on the history of the early quantum theory, in particular: F. Reiche, Die Quantentheorie: Ihr Ursprung und ihre Entwicklung (J. Springer, Berlin, 1921), English translation, The Quantum Theory (E. P. Dutton and Co., New York, 1922) (our quotations are from the English version); A. Hermann, Frühgeschichte der Quantentheorie (Physik Verlag, Mosbach, 1969; Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, Vol. I (Springer- Verlag New York. 1982) (in two parts). A rather detailed account is also given in the treatise of E. Whittaker, A History of the Theories of Aether and Electricity, Vol. II: The Modern Theories 1900-1925 (T. Nelson & Sons, London, 1953); reprinted in Harper Torchbooks, New York, 1960. James Hopwood Jeans, a participant in the development of the radiation law, himself gave two fair reports on the early development: Report on Radiation and the Quantum Theory, (The Physical Society of London, London, 1914, 1924). The important concepts are carefully reviewed: Max Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966). Finally, we refer to various articles on the subject of the old quantum theory by M. J. Klein, Arch. Hist. Exact Sci. 1, 459 (1962); The Natural Philosopher 1, 83 (1963); L. Rosenfeld, Osiris 2, 149 (1936); Max Planck et la définition de l'entropie, in Max Planck-Festschrift, 1958 (Berlin, 1958), p. 203. Martin J. Klein has also discussed certain aspects of the early quantum theory in his book Paul Ehrenfest, The Making of a Theoretical Physicist, Vol. I (North-Holland, Amsterdam, 1970), Chap. 10.
13. There exist quite a few reports and reviews on the history of the early quantum theory, in particular: F. Reiche, Die Quantentheorie: Ihr Ursprung und ihre Entwicklung (J. Springer, Berlin, 1921), English translation, The Quantum Theory (E. P. Dutton and Co., New York, 1922) (our quotations are from the English version); A. Hermann, Frühgeschichte der Quantentheorie (Physik Verlag, Mosbach, 1969; Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, Vol. I (Springer- Verlag New York. 1982) (in two parts). A rather detailed account is also given in the treatise of E. Whittaker, A History of the Theories of Aether and Electricity, Vol. II: The Modern Theories 1900-1925 (T. Nelson & Sons, London, 1953); reprinted in Harper Torchbooks, New York, 1960. James Hopwood Jeans, a participant in the development of the radiation law, himself gave two fair reports on the early development: Report on Radiation and the Quantum Theory, (The Physical Society of London, London, 1914, 1924). The important concepts are carefully reviewed: Max Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966). Finally, we refer to various articles on the subject of the old quantum theory by M. J. Klein, Arch. Hist. Exact Sci. 1, 459 (1962); The Natural Philosopher 1, 83 (1963); L. Rosenfeld, Osiris 2, 149 (1936); Max Planck et la définition de l'entropie, in Max Planck-Festschrift, 1958 (Berlin, 1958), p. 203. Martin J. Klein has also discussed certain aspects of the early quantum theory in his book Paul Ehrenfest, The Making of a Theoretical Physicist, Vol. I (North-Holland, Amsterdam, 1970), Chap. 10.
14. There exist quite a few reports and reviews on the history of the early quantum theory, in particular: F. Reiche, Die Quantentheorie: Ihr Ursprung und ihre Entwicklung (J. Springer, Berlin, 1921), English translation, The Quantum Theory (E. P. Dutton and Co., New York, 1922) (our quotations are from the English version); A. Hermann, Frühgeschichte der Quantentheorie (Physik Verlag, Mosbach, 1969; Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, Vol. I (Springer- Verlag New York. 1982) (in two parts). A rather detailed account is also given in the treatise of E. Whittaker, A History of the Theories of Aether and Electricity, Vol. II: The Modern Theories 1900-1925 (T. Nelson & Sons, London, 1953); reprinted in Harper Torchbooks, New York, 1960. James Hopwood Jeans, a participant in the development of the radiation law, himself gave two fair reports on the early development: Report on Radiation and the Quantum Theory, (The Physical Society of London, London, 1914, 1924). The important concepts are carefully reviewed: Max Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966). Finally, we refer to various articles on the subject of the old quantum theory by M. J. Klein, Arch. Hist. Exact Sci. 1, 459 (1962); The Natural Philosopher 1, 83 (1963); L. Rosenfeld, Osiris 2, 149 (1936); Max Planck et la définition de l'entropie, in Max Planck-Festschrift, 1958 (Berlin, 1958), p. 203. Martin J. Klein has also discussed certain aspects of the early quantum theory in his book Paul Ehrenfest, The Making of a Theoretical Physicist, Vol. I (North-Holland, Amsterdam, 1970), Chap. 10.
15. There exist quite a few reports and reviews on the history of the early quantum theory, in particular: F. Reiche, Die Quantentheorie: Ihr Ursprung und ihre Entwicklung (J. Springer, Berlin, 1921), English translation, The Quantum Theory (E. P. Dutton and Co., New York, 1922) (our quotations are from the English version); A. Hermann, Frühgeschichte der Quantentheorie (Physik Verlag, Mosbach, 1969; Jagdish Mehra and Helmut Rechenberg, The Historical Development of Quantum Theory, Vol. I (Springer- Verlag New York. 1982) (in two parts). A rather detailed account is also given in the treatise of E. Whittaker, A History of the Theories of Aether and Electricity, Vol. II: The Modern Theories 1900-1925 (T. Nelson & Sons, London, 1953); reprinted in Harper Torchbooks, New York, 1960. James Hopwood Jeans, a participant in the development of the radiation law, himself gave two fair reports on the early development: Report on Radiation and the Quantum Theory, (The Physical Society of London, London, 1914, 1924). The important concepts are carefully reviewed: Max Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966). Finally, we refer to various articles on the subject of the old quantum theory by M. J. Klein, Arch. Hist. Exact Sci. 1, 459 (1962); The Natural Philosopher 1, 83 (1963); L. Rosenfeld, Osiris 2, 149 (1936); Max Planck et la définition de l'entropie, in Max Planck-Festschrift, 1958 (Berlin, 1958), p. 203. Martin J. Klein has also discussed certain aspects of the early quantum theory in his book Paul Ehrenfest, The Making of a Theoretical Physicist, Vol. I (North-Holland, Amsterdam, 1970), Chap. 10.
16. M. Planck, Verh. Deutsch. Phys. Ges. 2, 202 (1900); Phys. Abh. I, 687- 689.
17. M. Planck, Verh. Deutsch. Phys. Ges. 2, 202 (1900); Phys. Abh. I, 687- 689.
18. F. Kurlbaum and H. Rubens, Verh. Deutsch. Phys. Ges. 2, 181 (1900); Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 929 (1900); Ann. Phys. (Leipzig) 4, 277 (1901).
19. F. Kurlbaum and H. Rubens, Verh. Deutsch. Phys. Ges. 2, 181 (1900); Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 929 (1900); Ann. Phys. (Leipzig) 4, 277 (1901).
20. F. Kurlbaum and H. Rubens, Verh. Deutsch. Phys. Ges. 2, 181 (1900); Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 929 (1900); Ann. Phys. (Leipzig) 4, 277 (1901).
21. An extensive and detailed report about Planck's theory of blackbody radiation law is presented in H. Kangro, Vorgeschichte des Planckschen Strahlungsgesetzes (F. Steiner, Wiesbaden, 1970).
22. -27(erg.sec). The constant b, introduced already in 1899, is the so-called "quantum of action" and was later denoted by h. According to Hermann, one should therefore date the birth of quantum theory on the day when Planck presented his derivation (18 May 1899). The constant a is equal to b/k, where k is now universally known as Boltzmann's constant; k was also first introduced by Planck in 1900.
23. -27(erg.sec). The constant b, introduced already in 1899, is the so-called "quantum of action" and was later denoted by h. According to Hermann, one should therefore date the birth of quantum theory on the day when Planck presented his derivation (18 May 1899). The constant a is equal to b/k, where k is now universally known as Boltzmann's constant; k was also first introduced by Planck in 1900.
24. -27(erg.sec). The constant b, introduced already in 1899, is the so-called "quantum of action" and was later denoted by h. According to Hermann, one should therefore date the birth of quantum theory on the day when Planck presented his derivation (18 May 1899). The constant a is equal to b/k, where k is now universally known as Boltzmann's constant; k was also first introduced by Planck in 1900.
25. -27(erg.sec). The constant b, introduced already in 1899, is the so-called "quantum of action" and was later denoted by h. According to Hermann, one should therefore date the birth of quantum theory on the day when Planck presented his derivation (18 May 1899). The constant a is equal to b/k, where k is now universally known as Boltzmann's constant; k was also first introduced by Planck in 1900.
26. -27(erg.sec). The constant b, introduced already in 1899, is the so-called "quantum of action" and was later denoted by h. According to Hermann, one should therefore date the birth of quantum theory on the day when Planck presented his derivation (18 May 1899). The constant a is equal to b/k, where k is now universally known as Boltzmann's constant; k was also first introduced by Planck in 1900.
27. At about the same time other, more complicated, improvements of Wien's formula were proposed and discussed by: M. Thiessen, Verh. Deutsch. Phys. Ges. 2, 67 (1900); O. Lummer and E. Jahnke, Ann. Physik (Leipzig) 3, 288 (1900); O. Lummer and E. Pringsheim, Verh. Deutsch. Phys. Ges. (Berlin) 2, 174 (1900).
28. At about the same time other, more complicated, improvements of Wien's formula were proposed and discussed by: M. Thiessen, Verh. Deutsch. Phys. Ges. 2, 67 (1900); O. Lummer and E. Jahnke, Ann. Physik (Leipzig) 3, 288 (1900); O. Lummer and E. Pringsheim, Verh. Deutsch. Phys. Ges. (Berlin) 2, 174 (1900).
29. At about the same time other, more complicated, improvements of Wien's formula were proposed and discussed by: M. Thiessen, Verh. Deutsch. Phys. Ges. 2, 67 (1900); O. Lummer and E. Jahnke, Ann. Physik (Leipzig) 3, 288 (1900); O. Lummer and E. Pringsheim, Verh. Deutsch. Phys. Ges. (Berlin) 2, 174 (1900).
30. M. Planck, "Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum,quot; Verh. Deutsch. Phys. Ges. 2, 237(1900); reprinted in Phys. Abh. I, 698-706.
31. M. Planck, "Zur Theorie des Gesetzes der Energieverteilung im Normalspectrum,quot; Verh. Deutsch. Phys. Ges. 2, 237(1900); reprinted in Phys. Abh. I, 698-706.
32. M. Planck, Phys. Abh. I, 700-701. By this procedure, Planck used a method proposed by Ludwig Boltzmann in dealing with the energy distribution among the molecules: L. Boltzmann, Sitz. Ber. Kai. Akad. Wiss. (Wien), Abh. II 79, 373(1977). Before that time, Planck had not favored Boltzmann's statistical approach to thermodynamics, in particular his use of the concept of entropy. Now, under the pressure to explain the new situation he changed his attitude and employed the statistical method with a more far- reaching consequence than Boltzmann himself. Later on, Max Planck described this derivation of the radiation formula (by employing the statistical method) "as an act of despair." Letter to R. W. Wood, dated 7 October 1931, reprinted in Hermann's Frühgeschichte 31-32. Hermann came to the conclusion that Planck regarded the application of the statistical method as a revolutionary deed, much more so than the existence of the energy-quantum (loc.cit., pp. 31-35).
33. M. Planck, Phys. Abh. I, 700-701. By this procedure, Planck used a method proposed by Ludwig Boltzmann in dealing with the energy distribution among the molecules: L. Boltzmann, Sitz. Ber. Kai. Akad. Wiss. (Wien), Abh. II 79, 373(1977). Before that time, Planck had not favored Boltzmann's statistical approach to thermodynamics, in particular his use of the concept of entropy. Now, under the pressure to explain the new situation he changed his attitude and employed the statistical method with a more far-reaching consequence than Boltzmann himself. Later on, Max Planck described this derivation of the radiation formula (by employing the statistical method) "as an act of despair." Letter to R. W. Wood, dated 7 October 1931, reprinted in Hermann's Frühgeschichte 31-32. Hermann came to the conclusion that Planck regarded the application of the statistical method as a revolutionary deed, much more so than the existence of the energy-quantum (loc.cit., pp. 31-35).
34. Planck found this equation by equating the emission and absorption energies of a weakly damped Hertzian oscillator: M. Planck's, Phys. Abh. I, 575 (1899).
35. See, e.g., Planck's remarks in his Solvay Conference lecture, reprinted in Phys. Abh. II 275; see also the summary of his lecture in Mehra, cited in Ref. 1.
36. Scientists like Lord Rayleigh and James Jeans preferred to wonder about, and even doubt the results of well-known experiments such as those of Paschen, Lummer, Pringsheim, Rubens, and Kurbaum, as late as 1911. See the letter of Lord Rayleigh to the Solvay Conference in 1911, German edition, pp. 40-41. J. H. Jeans, Solvay Report, loc.cit., pp. 45-64.
37. Scientists like Lord Rayleigh and James Jeans preferred to wonder about, and even doubt the results of well-known experiments such as those of Paschen, Lummer, Pringsheim, Rubens, and Kurbaum, as late as 1911. See the letter of Lord Rayleigh to the Solvay Conference in 1911, German edition, pp. 40-41. J. H. Jeans, Solvay Report, loc.cit., pp. 45-64.
38. Lorentz had long reflected on the quantum radiation law of Planck and the classical radiation law of Rayleigh and Jeans: Lord Rayleigh. Phil. Mag. 49, 539 (1900);
39. Nature 72, 54(1905);
40. 2R, where R is the radius of the electron, v the average velocity of the electron, c the velocity of light, and a is a numerical constant. In the past year I have pondered endlessly on this problem, and finally seen that I cannot arrive at the goal in this manner." Quoted in Hermann, Frühgeschichte, pp. 45-46. The method which Lorentz pursued for a long time was to derive the experimental deviations from the Rayleigh-Jeans law from a study of the motion of electrons in metals. At a congress of mathematicians in April 1908, Lorentz gave a report on the radiation problem where he clearly favored the Rayleigh-Jeans theory, although it obviously contradicted experiments. To explain the deviation, Lorentz assumed with Jeans "that the maximum [of the radiation curve] is an illusion; if one believes in observing it, it is because one cannot really obtain a blackbody for short wavelengths":
41. H. A. Lorentz, Collected Papers, VII (The Hague, 1934), pp. 317-343, especially p. 339. Soon afterwards, however, Lorentz convinced himself that Jeans' "physical" argument was wrong and would lead to conclusions that contradict experiment at low temperatures:
42. O. Lummer and E. Pringsheim. Phys. Zs. 9, 449(1908); Letter from Lorentz to W. Wien, dated 6 June 1908, partially quoted in Hermann's Frühgeschichte, p. 51.
43. A. Einstein, Ann. Phys. (Leipzig) 17, 132 (1905).
44. M. Planck, Vorlesungen über die Theorie der Wärmestrahlung (J. A. Barth, Leipzig, 1908); English translation, M. Masius, The Theory of Heat Radiation (P. Blackiston's & Co., Philadelphia, 1914). F. Reiche recalled that these lectures were given in winter 1904 at the University of Berlin. See the interview with F. Reiche in "Sources for the History of Quantum Physics." Max Planck, in the Preface to his book, referred to his lectures of winter 1905/06. In the English translation, this date is misprinted as 1906/07.
45. M. Planck, Vorlesungen über die Theorie der Wärmestrahlung (J. A. Barth, Leipzig, 1908); English translation, M. Masius, The Theory of Heat Radiation (P. Blackiston's & Co., Philadelphia, 1914). F. Reiche recalled that these lectures were given in winter 1904 at the University of Berlin. See the interview with F. Reiche in "Sources for the History of Quantum Physics." Max Planck, in the Preface to his book, referred to his lectures of winter 1905/06. In the English translation, this date is misprinted as 1906/07.
46. M. Planck, Ann. Phys. (Leipzig) 31, 758 (1910); reprinted in Phys. Abh. II 237-247.
47. M. Planck, Ann. Phys. (Leipzig) 31, 758 (1910); reprinted in Phys. Abh. II 237-247.
48. M. Planck, loc. cit., p. 247.
49. We shall return to the point as to how the classical theory had to be modified later on.
50. See, e.g., the discussion by H. A. Lorentz in "Alte und neue Fragen der Physik," on the occasion of his Wolfskehl lectures held in Göttingen 24-29 October 1910. Notes of these lectures were published by Max Born, Phys. Zs. 11, 1234 (1910).
51. M. Planck, Introduction of the Paper cited in Ref. 18.
52. M. Planck, quoted in Ref. 3.
53. M. Planck, Phys. Abh. II 253-254.
54. See Planck's lecture at the 1911 Solvay Conference; reprinted in Phys. Abh. II, especially p. 286.
55. M. Planck, Phys. Abh. II 256-257.
56. Even as late as 1921, F. Reiche, in his book quoted in Ref. 4, remarked: "... many facts undoubtedly support the conception that at absolute zero by no means all motion has ceased. We need only draw attention to the fact that, according to the view of F. Richarz, P. Langevin, and according to the experiments of A. Einstein, W. J. de Haas, and E. Back, para- and dia-magnetism are produced by rotating electrons and that this magnetism remains in existence down to the lowest temperatures" (see p. 25).
57. A. Einstein, Ann. Phys. (Leipzig) 22, 180 (1907).
58. W. Nernst, Sitz. Ber. Preuss. Akad. Wiss. 247, 262 (1914); 306, 494 (1911).
59. W. Nernst, Sitz. Ber. Preuss. Akad. Wiss. 247, 262 (1914); 306, 494 (1911).
60. M. Planck, Phys. Abh. II 258.
61. M. Planck, Sitz. Ber. Preuss. Akad. Wiss. (1914); reprinted in PPhys. Abh. II 330-335.
62. M. Planck, Sitz. Ber. Preuss. Akad. Wiss. (1914); reprinted in PPhys. Abh. II 330-335.
63. M. Planck, Phys. Abh. II 331. It is remarkable that at this time Niels Bohr had already formulated the quantum theory of the hydrogen atom: Phil. Mag. 26, 1, 476, 857 (1913).
64. M. Planck, Phys. Abh. II 331. It is remarkable that at this time Niels Bohr had already formulated the quantum theory of the hydrogen atom: Phil. Mag. 26, 1, 476, 857 (1913).
65. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
66. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
67. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
68. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
69. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
70. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
71. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent
72. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
73. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
74. M. Planck, Über die Energieverteilung in einem System rotierender Dipole (Elster und Geitel-Festschrift, Braunschweig, 1915); reprinted in Phys. Abh. II 334-340. See also Planck's later papers, e.g., in Ver. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Phys. (Leipzig) 50, 385 (1916), Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 324 (1917); reprinted in Phys. Ahh. II 349-360, 362-375, 386-419, 435-452. Planck's work on the quantum theory in the second decade of the 20th century might be regarded as partly related to the questions which H. Poincaré had raised at the 1911 Solvay Conference. Poincaré could be considered, together with James Jeans, as an opponent of the quantum concept, and he very actively raised his voice to point out the differences between the new ideas and the accepted form of classical theory. With his analytical mind he pinpointed many of the critical questions during the discussions of the lectures presented by Planck, Sommerfeld, and Einstein. For Poincaré the central problem was that the new researches called into question not only the fundamental principles of mechanics, but also seemed to demolish the essential notion that the laws of nature could be expressed in terms of differential equations: Die Theorie der Strahlung und der Quanten, p. 365. Immediately after the Conference Poincaré turned to this question in one of his last papers: J. Phys. (Paris) 2, 5 (1912) . His answer was "no" because the probability for an oscillator to take a definite energy value was a discontinuous function. He therefore concluded that Planck's second hypothesis was inconsistent with the quantum idea. Certainly this fundamental criticism was in Planck's mind when he considered the energy probability distribution and its relation to his 1911 and 1914 quantum hypotheses later on. In an article on "Henri Poincaré and the Quantum Theory" he summarized his achievements, noting that Poincaré's calculations could also be made on the basis of the second and third quantum hypotheses: M. Planck, Acta math. 38, 387 (1919); reprinted in Phys. Abh. II 516-526.
75. W. Gerlach and O. Stern, Zeit. f. Phys. 8, 110 (1922).
76. M. Planck, Naturwiss. 11, 535 (1923); reprinted in Phys. Abh. II 543-545, especially p. 545.
77. M. Planck, Naturwiss. 11, 535 (1923); reprinted in Phys. Abh. II 543-545, especially p. 545.
78. A. Einstein, quoted in Ref. 16.
79. A. Haas, Sitz. Ber. Kais. Akad. Wiss. (Wien), Abt. IIa 119, 119 (1910). Haas' ideas were pursued further by A. Schidlof, Ann. Phys. (Leipzig) 35, 90 (1911); F. Hasenöhrl, Phys. Zs. 12, 931 (1911). At the first Solvay Conference, Planck made a rather negative comment on the efforts of Haas (German version, pp. 89-90). On the other hand, Lorentz (see Ref. 15) and Sommerfeld found those considerations useful in explaining the meaning of the notion of the quantum. Arthur Haas had used in his considerations the atomic model of J. J. Thomson, which assumed the existence of discrete electrons in continuously (within a sphere) distributed positive charge. Later on, Niels Bohr, who did not know about Haas' attempt, used the more realistic model of Rutherford and constructed a successful quantum theory of the atom: Mehra and Rechenberg, Vol. 1, Part 1 (1982).
80. A. Haas, Sitz. Ber. Kais. Akad. Wiss. (Wien), Abt. IIa 119, 119 (1910). Haas' ideas were pursued further by A. Schidlof, Ann. Phys. (Leipzig) 35, 90 (1911); F. Hasenöhrl, Phys. Zs. 12, 931 (1911). At the first Solvay Conference, Planck made a rather negative comment on the efforts of Haas (German version, pp. 89-90). On the other hand, Lorentz (see Ref. 15) and Sommerfeld found those considerations useful in explaining the meaning of the notion of the quantum. Arthur Haas had used in his considerations the atomic model of J. J. Thomson, which assumed the existence of discrete electrons in continuously (within a sphere) distributed positive charge. Later on, Niels Bohr, who did not know about Haas' attempt, used the more realistic model of Rutherford and constructed a successful quantum theory of the atom: Mehra and Rechenberg, Vol. 1, Part 1 (1982).
81. A. Haas, Sitz. Ber. Kais. Akad. Wiss. (Wien), Abt. IIa 119, 119 (1910). Haas' ideas were pursued further by A. Schidlof, Ann. Phys. (Leipzig) 35, 90 (1911); F. Hasenöhrl, Phys. Zs. 12, 931 (1911). At the first Solvay Conference, Planck made a rather negative comment on the efforts of Haas (German version, pp. 89-90). On the other hand, Lorentz (see Ref. 15) and Sommerfeld found those considerations useful in explaining the meaning of the notion of the quantum. Arthur Haas had used in his considerations the atomic model of J. J. Thomson, which assumed the existence of discrete electrons in continuously (within a sphere) distributed positive charge. Later on, Niels Bohr, who did not know about Haas' attempt, used the more realistic model of Rutherford and constructed a successful quantum theory of the atom: Mehra and Rechenberg, Vol. 1, Part 1 (1982).
82. A. Haas, Sitz. Ber. Kais. Akad. Wiss. (Wien), Abt. IIa 119, 119 (1910). Haas' ideas were pursued further by A. Schidlof, Ann. Phys. (Leipzig) 35, 90 (1911); F. Hasenöhrl, Phys. Zs. 12, 931 (1911). At the first Solvay Conference, Planck made a rather negative comment on the efforts of Haas (German version, pp. 89-90). On the other hand, Lorentz (see Ref. 15) and Sommerfeld found those considerations useful in explaining the meaning of the notion of the quantum. Arthur Haas had used in his considerations the atomic model of J. J. Thomson, which assumed the existence of discrete electrons in continuously (within a sphere) distributed positive charge. Later on, Niels Bohr, who did not know about Haas' attempt, used the more realistic model of Rutherford and constructed a successful quantum theory of the atom: Mehra and Rechenberg, Vol. 1, Part 1 (1982).
83. Sommerfeld's interest in quantum theory had started with a quarrel which he had with Johannes Stark concerning the bremsstrahlung's spectrum. Sommerfeld could explain the dependence of the intensity on direction on the basis of classical theory: Phys. Zs. 10, 69 (1909) which was denied by Stark, Phys. Zs. 10, 579, 902 (1909). Nevertheless Sommerfeld himself applied the quantum concept later to this problem and discovered the "dynamical" quantum hypothesis: Sitz. Ber. Bayer. Akad. Wiss. (München), Math. Phys. Kl. 41, (1911).
84. Sommerfeld's interest in quantum theory had started with a quarrel which he had with Johannes Stark concerning the bremsstrahlung's spectrum. Sommerfeld could explain the dependence of the intensity on direction on the basis of classical theory: Phys. Zs. 10, 69 (1909) which was denied by Stark, Phys. Zs. 10, 579, 902 (1909). Nevertheless Sommerfeld himself applied the quantum concept later to this problem and discovered the "dynamical" quantum hypothesis: Sitz. Ber. Bayer. Akad. Wiss. (München), Math. Phys. Kl. 41, (1911).
85. Sommerfeld's interest in quantum theory had started with a quarrel which he had with Johannes Stark concerning the bremsstrahlung's spectrum. Sommerfeld could explain the dependence of the intensity on direction on the basis of classical theory: Phys. Zs. 10, 69 (1909) which was denied by Stark, Phys. Zs. 10, 579, 902 (1909). Nevertheless Sommerfeld himself applied the quantum concept later to this problem and discovered the "dynamical" quantum hypothesis: Sitz. Ber. Bayer. Akad. Wiss. (München), Math. Phys. Kl. 41, (1911).
86. The last two papers were published in Ann. Phys. (Leipzig) 17, 549 (1905); 17, 891 (1905).
87. The last two papers were published in Ann. Phys. (Leipzig) 17, 549 (1905); 17, 891 (1905).
88. It is important to recall that Einstein had explored the kinetic theory of matter in detail in three earlier papers, which constituted an independent creation of the field of statistical mechanics, besides the work J. W. Gibbs with which he was not familiar: A. Einstein, Ann. Phys. (Leipzig) 9, 417 (1902): 11, 170 (1903); 14, 354 (1904). In the last paper of this series, Einstein explained Wien's displacement law with the help of molecular kinetic theory of fluctuations. Pursuing the logical consequences of this result he arrived at his "heuristic point of view" concerning the nature of light: Ann. Phys. 17, 132 (1905).
89. It is important to recall that Einstein had explored the kinetic theory of matter in detail in three earlier papers, which constituted an independent creation of the field of statistical mechanics, besides the work J. W. Gibbs with which he was not familiar: A. Einstein, Ann. Phys. (Leipzig) 9, 417 (1902): 11, 170 (1903); 14, 354 (1904). In the last paper of this series, Einstein explained Wien's displacement law with the help of molecular kinetic theory of fluctuations. Pursuing the logical consequences of this result he arrived at his "heuristic point of view" concerning the nature of light: Ann. Phys. 17, 132 (1905).
90. It is important to recall that Einstein had explored the kinetic theory of matter in detail in three earlier papers, which constituted an independent creation of the field of statistical mechanics, besides the work J. W. Gibbs with which he was not familiar: A. Einstein, Ann. Phys. (Leipzig) 9, 417 (1902): 11, 170 (1903); 14, 354 (1904). In the last paper of this series, Einstein explained Wien's displacement law with the help of molecular kinetic theory of fluctuations. Pursuing the logical consequences of this result he arrived at his "heuristic point of view" concerning the nature of light: Ann. Phys. 17, 132 (1905).
91. It is important to recall that Einstein had explored the kinetic theory of matter in detail in three earlier papers, which constituted an independent creation of the field of statistical mechanics, besides the work J. W. Gibbs with which he was not familiar: A. Einstein, Ann. Phys. (Leipzig) 9, 417 (1902): 11, 170 (1903); 14, 354 (1904). In the last paper of this series, Einstein explained Wien's displacement law with the help of molecular kinetic theory of fluctuations. Pursuing the logical consequences of this result he arrived at his "heuristic point of view" concerning the nature of light: Ann. Phys. 17, 132 (1905).
92. A. Einstein, loc. cit., p. 133. Einstein assumed that light exhibits a corpuscular structure under certain extreme conditions, e.g., at low temperature and low "density" and high frequency of radiation; the "fluctuation" phenomena which then arise cannot be calculated from the classical wave theory. But it is the fluctuations which lead to Wien's distribution law. At high density and the high temperature range of the radiation law, the fluctuations become very small and the "classical" law derived from the wave theory (Einstein independently derived the Rayleigh-Jeans distribution law in his 1905 paper) seems to apply.
93. The photoelectric effect might well have been observed first by Heinrich Hertz, Wiedemann's Ann. Phys. 31, 982 (1887); but W. Hallwachs made the clear experimental demonstration in 1888. Later studies by Hallwachs and Righi in 1888 and Philip Lenard served to clarify the details of photoelectric phenomena. For further references, see the article by P. Lenard and A. Becker, "Lichtelektrische Wirkung," in W. Wien and F. Harms, eds. Handbuch der Experimental physik, Vol. 23/2, Akad. Verl. Ges. (Leipzig) (1928). If the surface of a metal is illuminated by light, especially ultraviolet light or X-rays, electrons are emitted having the properties: (a) the velocity of the emitted electrons depends only on the frequency of light; (b) the number of emitted electrons is proportional to the intensity of light. The first attempt at an explanation of the photoelectric effect was made by Lenard, who, like many experimentalists of those days was used to developing his own theoretical concepts: P. Lenard, Ann. Phys. (Leipzig) 2, 359 (1900); 8, 149 (1902). His "Auslösungstheorie" said that the incident light induces larger oscillations of those electrons in atoms which oscillate synchronously with the light, and the electrons finally separate from the atom. The "first" or "initial" energy (Erstenergie) of the electrons stems from the oscillatory motion of the electrons within the atom, and not from the incident light which only induces separation. According to Einstein, however, all the energy comes from the light-quantum or the photon, and it is used partly in overcoming the metallic potential barrier (Austrittsarbeit) and the rest is imparted to the electron as its kinetic energy. It is remarkable that all the consequences of Einstein's theory of the photoelectric effect were not accurately verified experimentally until the work of Robert A. Millikan in 1916, and the other theories of the photoelectric effect continued to be proposed even after Einstein's successful quantum explanation of this phenomenon. Among the contributors can be found the names of well-known scientists like P. Lenard and C. Ramsauer (1911), F. A. Lindemann (1911), and O. W. Richardson (1912). For the details of references, see the Handbuch article of Lenard and Becker. J. J. Thomson invented another "Auslösungstheorie" in which he assumed an atom to consist of a dipole surrounded by electrons: Thomson, Phil. Mag. 20, 243, 544 (1910). The incident light strikes the axis of the dipole and changes its direction; the electrons then become free and preserve their kinetic energy, electrons with higher primary energy being more stable than others. In the following we shall mention another approach by Peter Debye and Arnold Sommerfeld.
94. The photoelectric effect might well have been observed first by Heinrich Hertz, Wiedemann's Ann. Phys. 31, 982 (1887); but W. Hallwachs made the clear experimental demonstration in 1888. Later studies by Hallwachs and Righi in 1888 and Philip Lenard served to clarify the details of photoelectric phenomena. For further references, see the article by P. Lenard and A. Becker, "Lichtelektrische Wirkung," in W. Wien and F. Harms, eds. Handbuch der Experimental physik, Vol. 23/2, Akad. Verl. Ges. (Leipzig) (1928). If the surface of a metal is illuminated by light, especially ultraviolet light or X-rays, electrons are emitted having the properties: (a) the velocity of the emitted electrons depends only on the frequency of light; (b) the number of emitted electrons is proportional to the intensity of light. The first attempt at an explanation of the photoelectric effect was made by Lenard, who, like many experimentalists of those days was used to developing his own theoretical concepts: P. Lenard, Ann. Phys. (Leipzig) 2, 359 (1900); 8, 149 (1902). His "Auslösungstheorie" said that the incident light induces larger oscillations of those electrons in atoms which oscillate synchronously with the light, and the electrons finally separate from the atom. The "first" or "initial" energy (Erstenergie) of the electrons stems from the oscillatory motion of the electrons within the atom, and not from the incident light which only induces separation. According to Einstein, however, all the energy comes from the light-quantum or the photon, and it is used partly in overcoming the metallic potential barrier (Austrittsarbeit) and the rest is imparted to the electron as its kinetic energy. It is remarkable that all the consequences of Einstein's theory of the photoelectric effect were not accurately verified experimentally until the work of Robert A. Millikan in 1916, and the other theories of the photoelectric effect continued to be proposed even after Einstein's successful quantum explanation of this phenomenon. Among the contributors can be found the names of well-known scientists like P. Lenard and C. Ramsauer (1911), F. A. Lindemann (1911), and O. W. Richardson (1912). For the details of references, see the Handbuch article of Lenard and Becker. J. J. Thomson invented another "Auslösungstheorie" in which he assumed an atom to consist of a dipole surrounded by electrons: Thomson, Phil. Mag. 20, 243, 544 (1910). The incident light strikes the axis of the dipole and changes its direction; the electrons then become free and preserve their kinetic energy, electrons with higher primary energy being more stable than others. In the following we shall mention another approach by Peter Debye and Arnold Sommerfeld.
95. The photoelectric effect might well have been observed first by Heinrich Hertz, Wiedemann's Ann. Phys. 31, 982 (1887); but W. Hallwachs made the clear experimental demonstration in 1888. Later studies by Hallwachs and Righi in 1888 and Philip Lenard served to clarify the details of photoelectric phenomena. For further references, see the article by P. Lenard and A. Becker, "Lichtelektrische Wirkung," in W. Wien and F. Harms, eds. Handbuch der Experimental physik, Vol. 23/2, Akad. Verl. Ges. (Leipzig) (1928). If the surface of a metal is illuminated by light, especially ultraviolet light or X-rays, electrons are emitted having the properties: (a) the velocity of the emitted electrons depends only on the frequency of light; (b) the number of emitted electrons is proportional to the intensity of light. The first attempt at an explanation of the photoelectric effect was made by Lenard, who, like many experimentalists of those days was used to developing his own theoretical concepts: P. Lenard, Ann. Phys. (Leipzig) 2, 359 (1900); 8, 149 (1902). His "Auslösungstheorie" said that the incident light induces larger oscillations of those electrons in atoms which oscillate synchronously with the light, and the electrons finally separate from the atom. The "first" or "initial" energy (Erstenergie) of the electrons stems from the oscillatory motion of the electrons within the atom, and not from the incident light which only induces separation. According to Einstein, however, all the energy comes from the light-quantum or the photon, and it is used partly in overcoming the metallic potential barrier (Austrittsarbeit) and the rest is imparted to the electron as its kinetic energy. It is remarkable that all the consequences of Einstein's theory of the photoelectric effect were not accurately verified experimentally until the work of Robert A. Millikan in 1916, and the other theories of the photoelectric effect continued to be proposed even after Einstein's successful quantum explanation of this phenomenon. Among the contributors can be found the names of well-known scientists like P. Lenard and C. Ramsauer (1911), F. A. Lindemann (1911), and O. W. Richardson (1912). For the details of references, see the Handbuch article of Lenard and Becker. J. J. Thomson invented another "Auslösungstheorie" in which he assumed an atom to consist of a dipole surrounded by electrons: Thomson, Phil. Mag. 20, 243, 544 (1910). The incident light strikes the axis of the dipole and changes its direction; the electrons then become free and preserve their kinetic energy, electrons with higher primary energy being more stable than others. In the following we shall mention another approach by Peter Debye and Arnold Sommerfeld.
96. The photoelectric effect might well have been observed first by Heinrich Hertz, Wiedemann's Ann. Phys. 31, 982 (1887); but W. Hallwachs made the clear experimental demonstration in 1888. Later studies by Hallwachs and Righi in 1888 and Philip Lenard served to clarify the details of photoelectric phenomena. For further references, see the article by P. Lenard and A. Becker, "Lichtelektrische Wirkung," in W. Wien and F. Harms, eds. Handbuch der Experimental physik, Vol. 23/2, Akad. Verl. Ges. (Leipzig) (1928). If the surface of a metal is illuminated by light, especially ultraviolet light or X-rays, electrons are emitted having the properties: (a) the velocity of the emitted electrons depends only on the frequency of light; (b) the number of emitted electrons is proportional to the intensity of light. The first attempt at an explanation of the photoelectric effect was made by Lenard, who, like many experimentalists of those days was used to developing his own theoretical concepts: P. Lenard, Ann. Phys. (Leipzig) 2, 359 (1900); 8, 149 (1902). His "Auslösungstheorie" said that the incident light induces larger oscillations of those electrons in atoms which oscillate synchronously with the light, and the electrons finally separate from the atom. The "first" or "initial" energy (Erstenergie) of the electrons stems from the oscillatory motion of the electrons within the atom, and not from the incident light which only induces separation. According to Einstein, however, all the energy comes from the light-quantum or the photon, and it is used partly in overcoming the metallic potential barrier (Austrittsarbeit) and the rest is imparted to the electron as its kinetic energy. It is remarkable that all the consequences of Einstein's theory of the photoelectric effect were not accurately verified experimentally until the work of Robert A. Millikan in 1916, and the other theories of the photoelectric effect continued to be proposed even after Einstein's successful quantum explanation of this phenomenon. Among the contributors can be found the names of well-known scientists like P. Lenard and C. Ramsauer (1911), F. A. Lindemann (1911), and O. W. Richardson (1912). For the details of references, see the Handbuch article of Lenard and Becker. J. J. Thomson invented another "Auslösungstheorie" in which he assumed an atom to consist of a dipole surrounded by electrons: Thomson, Phil. Mag. 20, 243, 544 (1910). The incident light strikes the axis of the dipole and changes its direction; the electrons then become free and preserve their kinetic energy, electrons with higher primary energy being more stable than others. In the following we shall mention another approach by Peter Debye and Arnold Sommerfeld.
97. The notion of the light-quantum as "photon" was introduced by G. N. Lewis, Nature 118, 874 (1926).
98. A. Einstein, Ann. Phys. (Leipzig) 22, 569 (1907).
99. Although Einstein knew about Plank's blackbody radiation law, he did not deal with it in his first light-quantum paper, but in a following article, submitted to Annalen der Physik on 13 March 1906: A. Einstein, Ann. Phys. (Leipzig) 20, 199 (1906).
100. Although Einstein knew about Plank's blackbody radiation law, he did not deal with it in his first light-quantum paper, but in a following article, submitted to Annalen der Physik on 13 March 1906: A. Einstein, Ann. Phys. (Leipzig) 20, 199 (1906).
101. A. Einstein, Phys. Zs. 10, 185, 817 (1909). His later attempts led him to formulate a completely statistical radiation hypothesis including the concept of "induced" emission: A. Einstein, Verh. Deutsch. Phys. Ges. 18, 318 (1916); Phys. Zv. 18, 121 (1917). The solution of the question concerning the nature of radiation came through the observation that light quanta obey a statistics, different from the Classical one: S. N. Bose, Zeit. f. Phys. 26, 178 (1924). Einstein extended Bose's considerations to material particles: A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 261 (1924); 3, 18 (1925) and found the connection with the work of Louis de Broglie (which he had done in his thesis at Paris, 1924).
102. A. Einstein, Phys. Zs. 10, 185, 817 (1909). His later attempts led him to formulate a completely statistical radiation hypothesis including the concept of "induced" emission: A. Einstein, Verh. Deutsch. Phys. Ges. 18, 318 (1916); Phys. Zv. 18, 121 (1917). The solution of the question concerning the nature of radiation came through the observation that light quanta obey a statistics, different from the Classical one: S. N. Bose, Zeit. f. Phys. 26, 178 (1924). Einstein extended Bose's considerations to material particles: A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 261 (1924); 3, 18 (1925) and found the connection with the work of Louis de Broglie (which he had done in his thesis at Paris, 1924).
103. A. Einstein, Phys. Zs. 10, 185, 817 (1909). His later attempts led him to formulate a completely statistical radiation hypothesis including the concept of "induced" emission: A. Einstein, Verh. Deutsch. Phys. Ges. 18, 318 (1916); Phys. Zv. 18, 121 (1917). The solution of the question concerning the nature of radiation came through the observation that light quanta obey a statistics, different from the Classical one: S. N. Bose, Zeit. f. Phys. 26, 178 (1924). Einstein extended Bose's considerations to material particles: A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 261 (1924); 3, 18 (1925) and found the connection with the work of Louis de Broglie (which he had done in his thesis at Paris, 1924).
104. A. Einstein, Phys. Zs. 10, 185, 817 (1909). His later attempts led him to formulate a completely statistical radiation hypothesis including the concept of "induced" emission: A. Einstein, Verh. Deutsch. Phys. Ges. 18, 318 (1916); Phys. Zv. 18, 121 (1917). The solution of the question concerning the nature of radiation came through the observation that light quanta obey a statistics, different from the Classical one: S. N. Bose, Zeit. f. Phys. 26, 178 (1924). Einstein extended Bose's considerations to material particles: A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 261 (1924); 3, 18 (1925) and found the connection with the work of Louis de Broglie (which he had done in his thesis at Paris, 1924).
105. A. Einstein, Phys. Zs. 10, 185, 817 (1909). His later attempts led him to formulate a completely statistical radiation hypothesis including the concept of "induced" emission: A. Einstein, Verh. Deutsch. Phys. Ges. 18, 318 (1916); Phys. Zv. 18, 121 (1917). The solution of the question concerning the nature of radiation came through the observation that light quanta obey a statistics, different from the Classical one: S. N. Bose, Zeit. f. Phys. 26, 178 (1924). Einstein extended Bose's considerations to material particles: A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 261 (1924); 3, 18 (1925) and found the connection with the work of Louis de Broglie (which he had done in his thesis at Paris, 1924).
106. A. Einstein, Phys. Zs. 10, 185, 817 (1909). His later attempts led him to formulate a completely statistical radiation hypothesis including the concept of "induced" emission: A. Einstein, Verh. Deutsch. Phys. Ges. 18, 318 (1916); Phys. Zv. 18, 121 (1917). The solution of the question concerning the nature of radiation came through the observation that light quanta obey a statistics, different from the Classical one: S. N. Bose, Zeit. f. Phys. 26, 178 (1924). Einstein extended Bose's considerations to material particles: A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 261 (1924); 3, 18 (1925) and found the connection with the work of Louis de Broglie (which he had done in his thesis at Paris, 1924).
107. Johannes Stark, Sommerfeld's successor in the Chair for Technical Mechanics at the Technische Hochschule in Aachen in 1909, was one of the most eager partisans of the quantum concept before 1913. Well known since his discovery of the Doppler effect in canal rays (1905), he is best remembered for the first observation of the splitting of spectral lines in an electric field (Stark effect): J. Stark, Ann. Phys. (Leipzig) 43, 965, 983, 991 (1913); (1914). In contrast to his great experimental skill, which was widely recognized, Stark's highly imaginative theoretical speculations did not find much favor (see, e.g., Ref. 38). The recognition of photochemical processes: J. Stark, Phys. Zs. 9, 767 (1908) brought him to dispute priority claims with Einstein in 1912, though they had been good friends before that: A. Einstein, Ann. Phys. (Leipzig) 37, 932 (1912); 38, 888 (1912).
108. Johannes Stark, Sommerfeld's successor in the Chair for Technical Mechanics at the Technische Hochschule in Aachen in 1909, was one of the most eager partisans of the quantum concept before 1913. Well known since his discovery of the Doppler effect in canal rays (1905), he is best remembered for the first observation of the splitting of spectral lines in an electric field (Stark effect): J. Stark, Ann. Phys. (Leipzig) 43, 965, 983, 991 (1913); (1914). In contrast to his great experimental skill, which was widely recognized, Stark's highly imaginative theoretical speculations did not find much favor (see, e.g., Ref. 38). The recognition of photochemical processes: J. Stark, Phys. Zs. 9, 767 (1908) brought him to dispute priority claims with Einstein in 1912, though they had been good friends before that: A. Einstein, Ann. Phys. (Leipzig) 37, 932 (1912); 38, 888 (1912).
109. Johannes Stark, Sommerfeld's successor in the Chair for Technical Mechanics at the Technische Hochschule in Aachen in 1909, was one of the most eager partisans of the quantum concept before 1913. Well known since his discovery of the Doppler effect in canal rays (1905), he is best remembered for the first observation of the splitting of spectral lines in an electric field (Stark effect): J. Stark, Ann. Phys. (Leipzig) 43, 965, 983, 991 (1913); (1914). In contrast to his great experimental skill, which was widely recognized, Stark's highly imaginative theoretical speculations did not find much favor (see, e.g., Ref. 38). The recognition of photochemical processes: J. Stark, Phys. Zs. 9, 767 (1908) brought him to dispute priority claims with Einstein in 1912, though they had been good friends before that: A. Einstein, Ann. Phys. (Leipzig) 37, 932 (1912); 38, 888 (1912).
110. Johannes Stark, Sommerfeld's successor in the Chair for Technical Mechanics at the Technische Hochschule in Aachen in 1909, was one of the most eager partisans of the quantum concept before 1913. Well known since his discovery of the Doppler effect in canal rays (1905), he is best remembered for the first observation of the splitting of spectral lines in an electric field (Stark effect): J. Stark, Ann. Phys. (Leipzig) 43, 965, 983, 991 (1913); (1914). In contrast to his great experimental skill, which was widely recognized, Stark's highly imaginative theoretical speculations did not find much favor (see, e.g., Ref. 38). The recognition of photochemical processes: J. Stark, Phys. Zs. 9, 767 (1908) brought him to dispute priority claims with Einstein in 1912, though they had been good friends before that: A. Einstein, Ann. Phys. (Leipzig) 37, 932 (1912); 38, 888 (1912).
111. J. J. Thomson, Conduction of Electricity Through Gases (Cambridge, 1903), p. 258; Proc. Camb. Phil. Soc. 14, 421 (1908).
112. J. J. Thomson, Conduction of Electricity Through Gases (Cambridge, 1903), p. 258; Proc. Camb. Phil. Soc. 14, 421 (1908).
113. 3-behavior of specific heats at low temperatures, while Born and von Kármán took into account the real lattice structure of the crystal to determine the spectrum of eigenfrequencies. Thanks to the energy and enthusiasm of Nernst, quantum theory was consolidated by two different well-founded observations.
114. 3-behavior of specific heats at low temperatures, while Born and von Kármán took into account the real lattice structure of the crystal to determine the spectrum of eigenfrequencies. Thanks to the energy and enthusiasm of Nernst, quantum theory was consolidated by two different well-founded observations.
115. 3-behavior of specific heats at low temperatures, while Born and von Kármán took into account the real lattice structure of the crystal to determine the spectrum of eigenfrequencies. Thanks to the energy and enthusiasm of Nernst, quantum theory was consolidated by two different well-founded observations.
116. 3-behavior of specific heats at low temperatures, while Born and von Kármán took into account the real lattice structure of the crystal to determine the spectrum of eigenfrequencies. Thanks to the energy and enthusiasm of Nernst, quantum theory was consolidated by two different well-founded observations.
117. 3-behavior of specific heats at low temperatures, while Born and von Kármán took into account the real lattice structure of the crystal to determine the spectrum of eigenfrequencies. Thanks to the energy and enthusiasm of Nernst, quantum theory was consolidated by two different well-founded observations.
118. It is remarkable that in December 1900 Max Planck, then editor of the Annalen der Physik, received a rather speculative paper on "Inferences Drawn from the Phenomena of Capillarity" from an unknown and unemployed author, Albert Einstein. This work was published without delay: Ann. Phys. (Leipzig) 4, 513 (1901), and the same thing happened to his later contribution. Thus, Einstein's name was well known to Planck when he contributed the three papers which made him famous. More than that, Planck immediately promoted the new theory of relativity, starting with a review of it in the Berlin Physical Colloquium in the beginning of the winter semester 1905/1906. Planck himself published several papers on relativistic dynamics, the first one already on 23 March 1906: Verh. Deutsch. Phys. Ges. 8, 136 (1906); reprinted in Phys. Abh. II, 115-120. Planck was less satisfied with Einstein's conception of the light-quantum, and he expressed his negative opinion about it several times in articles and in public, e.g., in his 1911 Solvay lecture. When Planck, together with W. Nernst, H. Rubenst and E. Warburg, proposed to elect Einstein into the Prussian Academy, his letter of nomination, addressed to the Prussian Minister of Education, contained the sentence: "That he [Einstein] may also have surpassed the goal with his speculations, as for example in his hypothesis of light-quanta, should not be held against him, because without taking risk one cannot achieve a real change in the most exact science:" Carl Seelig. Albert Einstein (Zurich, 1954), pp. 174-175). Planck did not revise his point of view before 1927 when the entire quantum mechanics, including the interpretation by Heisenberg, Zeit. f. Phys. 43, 172 (1927), had been completed. In an article submitted to the Journal of the Franklin Institute 204, 13 (1927), he finally wrote: "We shall not be able to escape the fact that light-quanta in vacuum are a physical reality."
119. It is remarkable that in December 1900 Max Planck, then editor of the Annalen der Physik, received a rather speculative paper on "Inferences Drawn from the Phenomena of Capillarity" from an unknown and unemployed author, Albert Einstein. This work was published without delay: Ann. Phys. (Leipzig) 4, 513 (1901), and the same thing happened to his later contribution. Thus, Einstein's name was well known to Planck when he contributed the three papers which made him famous. More than that, Planck immediately promoted the new theory of relativity, starting with a review of it in the Berlin Physical Colloquium in the beginning of the winter semester 1905/1906. Planck himself published several papers on relativistic dynamics, the first one already on 23 March 1906: Verh. Deutsch. Phys. Ges. 8, 136 (1906); reprinted in Phys. Abh. II, 115-120. Planck was less satisfied with Einstein's conception of the light-quantum, and he expressed his negative opinion about it several times in articles and in public, e.g., in his 1911 Solvay lecture. When Planck, together with W. Nernst, H. Rubenst and E. Warburg, proposed to elect Einstein into the Prussian Academy, his letter of nomination, addressed to the Prussian Minister of Education, contained the sentence: "That he [Einstein] may also have surpassed the goal with his speculations, as for example in his hypothesis of light-quanta, should not be held against him, because without taking risk one cannot achieve a real change in the most exact science:" Carl Seelig. Albert Einstein (Zurich, 1954), pp. 174-175). Planck did not revise his point of view before 1927 when the entire quantum mechanics, including the interpretation by Heisenberg, Zeit. f. Phys. 43, 172 (1927), had been completed. In an article submitted to the Journal of the Franklin Institute 204, 13 (1927), he finally wrote: "We shall not be able to escape the fact that light-quanta in vacuum are a physical reality."
120. It is remarkable that in December 1900 Max Planck, then editor of the Annalen der Physik, received a rather speculative paper on "Inferences Drawn from the Phenomena of Capillarity" from an unknown and unemployed author, Albert Einstein. This work was published without delay: Ann. Phys. (Leipzig) 4, 513 (1901), and the same thing happened to his later contribution. Thus, Einstein's name was well known to Planck when he contributed the three papers which made him famous. More than that, Planck immediately promoted the new theory of relativity, starting with a review of it in the Berlin Physical Colloquium in the beginning of the winter semester 1905/1906. Planck himself published several papers on relativistic dynamics, the first one already on 23 March 1906: Verh. Deutsch. Phys. Ges. 8, 136 (1906); reprinted in Phys. Abh. II, 115-120. Planck was less satisfied with Einstein's conception of the light-quantum, and he expressed his negative opinion about it several times in articles and in public, e.g., in his 1911 Solvay lecture. When Planck, together with W. Nernst, H. Rubenst and E. Warburg, proposed to elect Einstein into the Prussian Academy, his letter of nomination, addressed to the Prussian Minister of Education, contained the sentence: "That he [Einstein] may also have surpassed the goal with his speculations, as for example in his hypothesis of light-quanta, should not be held against him, because without taking risk one cannot achieve a real change in the most exact science:" Carl Seelig. Albert Einstein (Zurich, 1954), pp. 174-175). Planck did not revise his point of view before 1927 when the entire quantum mechanics, including the interpretation by Heisenberg, Zeit. f. Phys. 43, 172 (1927), had been completed. In an article submitted to the Journal of the Franklin Institute 204, 13 (1927), he finally wrote: "We shall not be able to escape the fact that light-quanta in vacuum are a physical reality."
121. It is remarkable that in December 1900 Max Planck, then editor of the Annalen der Physik, received a rather speculative paper on "Inferences Drawn from the Phenomena of Capillarity" from an unknown and unemployed author, Albert Einstein. This work was published without delay: Ann. Phys. (Leipzig) 4, 513 (1901), and the same thing happened to his later contribution. Thus, Einstein's name was well known to Planck when he contributed the three papers which made him famous. More than that, Planck immediately promoted the new theory of relativity, starting with a review of it in the Berlin Physical Colloquium in the beginning of the winter semester 1905/1906. Planck himself published several papers on relativistic dynamics, the first one already on 23 March 1906: Verh. Deutsch. Phys. Ges. 8, 136 (1906); reprinted in Phys. Abh. II, 115-120. Planck was less satisfied with Einstein's conception of the light-quantum, and he expressed his negative opinion about it several times in articles and in public, e.g., in his 1911 Solvay lecture. When Planck, together with W. Nernst, H. Rubenst and E. Warburg, proposed to elect Einstein into the Prussian Academy, his letter of nomination, addressed to the Prussian Minister of Education, contained the sentence: "That he [Einstein] may also have surpassed the goal with his speculations, as for example in his hypothesis of light-quanta, should not be held against him, because without taking risk one cannot achieve a real change in the most exact science:" Carl Seelig. Albert Einstein (Zurich, 1954), pp. 174-175). Planck did not revise his point of view before 1927 when the entire quantum mechanics, including the interpretation by Heisenberg, Zeit. f. Phys. 43, 172 (1927), had been completed. In an article submitted to the Journal of the Franklin Institute 204, 13 (1927), he finally wrote: "We shall not be able to escape the fact that light-quanta in vacuum are a physical reality."
122. It is remarkable that in December 1900 Max Planck, then editor of the Annalen der Physik, received a rather speculative paper on "Inferences Drawn from the Phenomena of Capillarity" from an unknown and unemployed author, Albert Einstein. This work was published without delay: Ann. Phys. (Leipzig) 4, 513 (1901), and the same thing happened to his later contribution. Thus, Einstein's name was well known to Planck when he contributed the three papers which made him famous. More than that, Planck immediately promoted the new theory of relativity, starting with a review of it in the Berlin Physical Colloquium in the beginning of the winter semester 1905/1906. Planck himself published several papers on relativistic dynamics, the first one already on 23 March 1906: Verh. Deutsch. Phys. Ges. 8, 136 (1906); reprinted in Phys. Abh. II, 115-120. Planck was less satisfied with Einstein's conception of the light-quantum, and he expressed his negative opinion about it several times in articles and in public, e.g., in his 1911 Solvay lecture. When Planck, together with W. Nernst, H. Rubenst and E. Warburg, proposed to elect Einstein into the Prussian Academy, his letter of nomination, addressed to the Prussian Minister of Education, contained the sentence: "That he [Einstein] may also have surpassed the goal with his speculations, as for example in his hypothesis of light-quanta, should not be held against him, because without taking risk one cannot achieve a real change in the most exact science:" Carl Seelig. Albert Einstein (Zurich, 1954), pp. 174-175). Planck did not revise his point of view before 1927 when the entire quantum mechanics, including the interpretation by Heisenberg, Zeit. f. Phys. 43, 172 (1927), had been completed. In an article submitted to the Journal of the Franklin Institute 204, 13 (1927), he finally wrote: "We shall not be able to escape the fact that light-quanta in vacuum are a physical reality."
123. It is remarkable that in December 1900 Max Planck, then editor of the Annalen der Physik, received a rather speculative paper on "Inferences Drawn from the Phenomena of Capillarity" from an unknown and unemployed author, Albert Einstein. This work was published without delay: Ann. Phys. (Leipzig) 4, 513 (1901), and the same thing happened to his later contribution. Thus, Einstein's name was well known to Planck when he contributed the three papers which made him famous. More than that, Planck immediately promoted the new theory of relativity, starting with a review of it in the Berlin Physical Colloquium in the beginning of the winter semester 1905/1906. Planck himself published several papers on relativistic dynamics, the first one already on 23 March 1906: Verh. Deutsch. Phys. Ges. 8, 136 (1906); reprinted in Phys. Abh. II, 115-120. Planck was less satisfied with Einstein's conception of the light-quantum, and he expressed his negative opinion about it several times in articles and in public, e.g., in his 1911 Solvay lecture. When Planck, together with W. Nernst, H. Rubenst and E. Warburg, proposed to elect Einstein into the Prussian Academy, his letter of nomination, addressed to the Prussian Minister of Education, contained the sentence: "That he [Einstein] may also have surpassed the goal with his speculations, as for example in his hypothesis of light-quanta, should not be held against him, because without taking risk one cannot achieve a real change in the most exact science:" Carl Seelig. Albert Einstein (Zurich, 1954), pp. 174-175). Planck did not revise his point of view before 1927 when the entire quantum mechanics, including the interpretation by Heisenberg, Zeit. f. Phys. 43, 172 (1927), had been completed. In an article submitted to the Journal of the Franklin Institute 204, 13 (1927), he finally wrote: "We shall not be able to escape the fact that light-quanta in vacuum are a physical reality."
124. See Planck's first relativity paper, cited in Ref. 50.
125. J. W. Gibbs had sent a copy of his book Elementary Principles in Statistical Mechanics (Yale University Press, New Haven, 1902 ) to many renowned scientists, and there exists a letter of Planck in which he thanked Gibbs (Beinecke Rare Books Library, Yale University). Planck's assistants Ernst Zermelo, translated Gibbs' book into German. The other book on statistical mechanics on which Planck could rely was L. Boltzmann's Lectures on Gas Theory. Vol. I (Leipzig, 1896); Vol. II (Leipzig, 1898).
126. J. W. Gibbs had sent a copy of his book Elementary Principles in Statistical Mechanics (Yale University Press, New Haven, 1902 ) to many renowned scientists, and there exists a letter of Planck in which he thanked Gibbs (Beinecke Rare Books Library, Yale University). Planck's assistants Ernst Zermelo, translated Gibbs' book into German. The other book on statistical mechanics on which Planck could rely was L. Boltzmann's Lectures on Gas Theory. Vol. I (Leipzig, 1896); Vol. II (Leipzig, 1898).
127. J. W. Gibbs had sent a copy of his book Elementary Principles in Statistical Mechanics (Yale University Press, New Haven, 1902 ) to many renowned scientists, and there exists a letter of Planck in which he thanked Gibbs (Beinecke Rare Books Library, Yale University). Planck's assistants Ernst Zermelo, translated Gibbs' book into German. The other book on statistical mechanics on which Planck could rely was L. Boltzmann's Lectures on Gas Theory. Vol. I (Leipzig, 1896); Vol. II (Leipzig, 1898).
128. See the German edition of the Solvay Conference Proceedings, cited in Reference 1, p. 94.
129. See German version of the Solvay Conference Proceedings, p. 365.
130. See Ref. 38. His first contribution, including the quantum hypothesis, was presented to the Bavarian Academy on 7 January 1911. Already in September of the same year Sommerfeld gave a general report on "Planck's Quantum of Action and Its General Significance for Molecular Physics" to the 83. Naturforscherversammlung in Karlsruhe. In the meantime he exchanged letters with Max Planck in which his final ideas were formed. Sommerfeld might have been influenced in favor of quantum theory by his student Peter Debye who gave an elegant and pointed derivation of Planck's law in fall 1910: Debye, Ann. Phys. (Leipzig) 33, 1427 (1910). Sommerfeld was perhaps too much of an applied mathematician to have developed entirely new physical theories, but besides being a most effective teacher (Debye, Pauli, Heisenberg, Wentzem, and Bethe were among his students) he was a master of applying powerful mathematical techniques to new fields.
131. In this connection, it is remarkable that both Sommerfeld and Lorentz defended the ideas of Arthur Haas to consider an atomic model on the basis of the quantum theory against the criticism of Planck. See the German version of the proceedings of the first Solvay Conference, pp. 102, 103, 108.
132. See the proceedings of the first Solvay Conference, German version, pp. 252-253.
133. P. Debye and A. Sommerfeld, Ann. Phys. (Leipzig) 41, 873 (1913).
134. A. Joffe, Sitz. Ber. Bayer. Akad. Wiss. (München), Math. Phys. Kl. 43, 19 (1913); E. Meyer and W. Gerlach, Ann. Phys. (Leipzig) 45, 177 (1914); 48, 718 (1915).
135. A. Joffe, Sitz. Ber. Bayer. Akad. Wiss. (München), Math. Phys. Kl. 43, 19 (1913); E. Meyer and W. Gerlach, Ann. Phys. (Leipzig) 45, 177 (1914); 48, 718 (1915).
136. A. Joffe, Sitz. Ber. Bayer. Akad. Wiss. (München), Math. Phys. Kl. 43, 19 (1913); E. Meyer and W. Gerlach, Ann. Phys. (Leipzig) 45, 177 (1914); 48, 718 (1915).
137. N. Bohr, Phil. Mag. 26, 476, 857 (1913).
138. A. Sommerfeld, Sitz. Ber. Bayer. Akad. Wiss. (München), Math. Phys. Kl. 425 (1915).
139. K. Wilson, Phil. Mag. 29, 795 (1915); 31, 156 (1916); M. Planck, Verh. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Ph. (Leipzig) 50, 385 (1916).
140. K. Wilson, Phil. Mag. 29, 795 (1915); 31, 156 (1916); M. Planck, Verh. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Ph. (Leipzig) 50, 385 (1916).
141. K. Wilson, Phil. Mag. 29, 795 (1915); 31, 156 (1916); M. Planck, Verh. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Ph. (Leipzig) 50, 385 (1916).
142. K. Wilson, Phil. Mag. 29, 795 (1915); 31, 156 (1916); M. Planck, Verh. Deutsch. Phys. Ges. 17, 407, 438 (1915); Ann. Ph. (Leipzig) 50, 385 (1916).
143. A. Einstein made this remark at the Solvay Conference (German version, p. 337, footnote 1).
144. Nernst's Lehrbuch der Theoretischen Chemie (Encke, Stuttgart) was first published in 1893 and went through numerous editions.
145. Sommerfeld called it the "most ingenious extension of classical thermodynamics in this century": A. Sommerfeld, Thermodynamik und Statistik (Leipzig, 1952).
146. C. Seelig, quoted in Ref. 50, p. 137.
147. W. Nernst and F. A. Lindemann, Zs. Elektrochemie 17, 817 (1911)
148. N. Bjerrum, Nernst Festschriftt (Halle, 1912), p. 90.
149. The factor 1/2 appears because the rotator, in contrast to the oscillator, only has kinetic energy.
150. See the experiments of S. P. Langley, Ann. Astrophys. Obs. Smithsonian Inst. I, p. 127. Table XX(1900); F. Paschen, Wiedemann Ann. Phys. 51, 1; 52, 209; 53, 335 (1894); H. Rubens, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 513 (1913); E. von Bahr, Verh. Deutsch. Phys. Ges. 15, 1159 (1913).
151. See the experiments of S. P. Langley, Ann. Astrophys. Obs. Smithsonian
152. See the experiments of S. P. Langley, Ann. Astrophys. Obs. Smithsonian Inst. I, p. 127. Table XX(1900); F. Paschen, Wiedemann Ann. Phys. 51, 1; 52, 209; 53, 335 (1894); H. Rubens, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 513 (1913); E. von Bahr, Verh. Deutsch. Phys. Ges. 15, 1159 (1913).
153. See the experiments of S. P. Langley, Ann. Astrophys. Obs. Smithsonian Inst. I, p. 127. Table XX(1900); F. Paschen, Wiedemann Ann. Phys. 51, 1; 52, 209; 53, 335 (1894); H. Rubens, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 513 (1913); E. von Bahr, Verh. Deutsch. Phys. Ges. 15, 1159 (1913).
154. See the experiments of S. P. Langley, Ann. Astrophys. Obs. Smithsonian Inst. I, p. 127. Table XX(1900); F. Paschen, Wiedemann Ann. Phys. 51, 1; 52, 209; 53, 335 (1894); H. Rubens, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 513 (1913); E. von Bahr, Verh. Deutsch. Phys. Ges. 15, 1159 (1913).
155. See the experiments of S. P. Langley, Ann. Astrophys. Obs. Smithsonian Inst. I, p. 127. Table XX(1900); F. Paschen, Wiedemann Ann. Phys. 51, 1; 52, 209; 53, 335 (1894); H. Rubens, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 513 (1913); E. von Bahr, Verh. Deutsch. Phys. Ges. 15, 1159 (1913).
156. A. Eucken, Verh. Deutsch. Phys. Ges. 15, 1159 (1913). See also his appendix to the German version of the Solvay Proceedings on "The Development of the Quantum Theory from Fall 1911 to Summer 1913" (Halle, 1914, p. 371).
157. A. Eucken, Verh. Deutsch. Phys. Ges. 15, 1159 (1913). See also his appendix to the German version of the Solvay Proceedings on "The Development of the Quantum Theory from Fall 1911 to Summer 1913" (Halle, 1914, p. 371).
158. E. S. Imes, Ap. J. 50, 25 (1919).
159. A. Eucken, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 141 (1912).
160. A. Eucken, Appendix to the Solvay Report, p. 392.
161. A. Einstein, Ann. Phys. (Leipzig) 40, 551 (1913).
162. A. Einstein, loc.cit., p. 560.
163. P. Ehrenfest, Verh. Deutsch. Phys. Ges. 15, 451 (1913).
164. Reiche made use of Planck's first hypothesis and Bohr's dumb-bell model with two degrees of freedom, and could exclude the decrease of the theoretical specific heat curve under certain subsidiary assumptions: Reiche, Ann. Phys. (Leipzig) 58, 657 (1919). J. V. Weissenhoff found a montonically increasing specific heat with Planck's second hypothesis: Weissenhoff, Ann. Phys. (Leipzig) 51, 285 (1916) . Also, the attempts of M. Planck, Verh. Deutsch. Phys. Ges. 17, 407 (1915); S. Rotszayn, Ann. Phys. (Leipzig) 57, 81 (1918): P. Epstein, Verh. Deutsch. Phys. Ges. 18, 398 (1916), did not solve the problem satisfactorily.
165. Reiche made use of Planck's first hypothesis and Bohr's dumb-bell model with two degrees of freedom, and could exclude the decrease of the theoretical specific heat curve under certain subsidiary assumptions: Reiche, Ann. Phys. (Leipzig) 58, 657 (1919). J. V. Weissenhoff found a montonically increasing specific heat with Planck's second hypothesis: Weissenhoff, Ann. Phys. (Leipzig) 51, 285 (1916) . Also, the attempts of M. Planck, Verh. Deutsch. Phys. Ges. 17, 407 (1915); S. Rotszayn, Ann. Phys. (Leipzig) 57, 81 (1918): P. Epstein, Verh. Deutsch. Phys. Ges. 18, 398 (1916), did not solve the problem satisfactorily.
166. Reiche made use of Planck's first hypothesis and Bohr's dumb-bell model with two degrees of freedom, and could exclude the decrease of the theoretical specific heat curve under certain subsidiary assumptions: Reiche, Ann. Phys. (Leipzig) 58, 657 (1919). J. V. Weissenhoff found a montonically increasing specific heat with Planck's second hypothesis: Weissenhoff, Ann. Phys. (Leipzig) 51, 285 (1916) . Also, the attempts of M. Planck, Verh. Deutsch. Phys. Ges. 17, 407 (1915); S. Rotszayn, Ann. Phys. (Leipzig) 57, 81 (1918): P. Epstein, Verh. Deutsch. Phys. Ges. 18, 398 (1916), did not solve the problem satisfactorily.
167. Reiche made use of Planck's first hypothesis and Bohr's dumb-bell model with two degrees of freedom, and could exclude the decrease of the theoretical specific heat curve under certain subsidiary assumptions: Reiche, Ann. Phys. (Leipzig) 58, 657 (1919). J. V. Weissenhoff found a montonically increasing specific heat with Planck's second hypothesis: Weissenhoff, Ann. Phys. (Leipzig) 51, 285 (1916) . Also, the attempts of M. Planck, Verh. Deutsch. Phys. Ges. 17, 407 (1915); S. Rotszayn, Ann. Phys. (Leipzig) 57, 81 (1918): P. Epstein, Verh. Deutsch. Phys. Ges. 18, 398 (1916), did not solve the problem satisfactorily.
168. Reiche made use of Planck's first hypothesis and Bohr's dumb-bell model with two degrees of freedom, and could exclude the decrease of the theoretical specific heat curve under certain subsidiary assumptions: Reiche, Ann. Phys. (Leipzig) 58, 657 (1919). J. V. Weissenhoff found a montonically increasing specific heat with Planck's second hypothesis: Weissenhoff, Ann. Phys. (Leipzig) 51, 285 (1916) . Also, the attempts of M. Planck, Verh. Deutsch. Phys. Ges. 17, 407 (1915); S. Rotszayn, Ann. Phys. (Leipzig) 57, 81 (1918): P. Epstein, Verh. Deutsch. Phys. Ges. 18, 398 (1916), did not solve the problem satisfactorily.
169. D. M. Dennison, Rev. Mod. Phys. 3, 280 (1931). In order to solve the problem the following steps had to be taken: (1) introduction of the correct quantization of the rotator; (2) introduction of nuclear spin leading to ortho- and para-hydrogen; (3) determination of the relevant mixture of ortho- and para-hydrogen.
170. O. Stern, Phys. Zs. 14, 629 (1913). Stern returned to this question in 1919: Zs. Elektrochem. 25, 66 (1919).
171. O. Stern, Phys. Zs. 14, 629 (1913). Stern returned to this question in 1919: Zs. Elektrochem. 25, 66 (1919).
172. See Stern's second paper, quoted in Ref. 80.
173. F. A. Lindemann. Phil. Mag. 38, 173 (1913).
174. F. W. Aston and F. A. Lindemann, Phil. Mag. 37, 523 (1919).
175. According to C. Enz and A. Thellung. Helv. Phys. Acta. 33, 839 (1969), this fact was already known to Stern before Lindemann published his paper.
176. I. Prigogine, R. Bingen, and J. Jeener, Physica 20, 33 (1954); I. Prigogine and J. Jeener, Physica 20, 514 (1954).
177. I. Prigogine, R. Bingen, and J. Jeener, Physica 20, 33 (1954); I. Prigogine and J. Jeener, Physica 20, 514 (1954).
178. 4, for example, exhibits a contraction at low temperatures, which can be used for a separation below a certain critical temperature, calculated to be of the order of 1 K. G. K. Walters and F. M. Fairbank have verified these predictions and obtained an isotope separation below a critical point of approximately 0.8 K: Phys. Rev. 103, 262 (1956). Further experiments have been performed by D. P. Edwards, A. S. McWilliams, and J. G. Daunt, Phys. Rev. Lett. 9, 155 (1962), and theoretical studies have been undertaken by R. A. Coldwell-Horsfall, Low Temp. Phys. 9B, 1110 (1965), as well as in collaboration with A. A. Maradudin, Technical Report of the Westinghouse Research Laboratory, Pittsburg, on "The Configuration Average of the Ground State Energy in an Isotropically Disordered Lattice, etc." (July 1961).
179. 4, for example, exhibits a contraction at low temperatures, which can be used for a separation below a certain critical temperature, calculated to be of the order of 1 K. G. K. Walters and F. M. Fairbank have verified these predictions and obtained an isotope separation below a critical point of approximately 0.8 K: Phys. Rev. 103, 262 (1956). Further experiments have been performed by D. P. Edwards, A. S. McWilliams, and J. G. Daunt, Phys. Rev. Lett. 9, 155 (1962), and theoretical studies have been undertaken by R. A. Coldwell-Horsfall, Low Temp. Phys. 9B, 1110 (1965), as well as in collaboration with A. A. Maradudin, Technical Report of the Westinghouse Research Laboratory, Pittsburg, on "The Configuration Average of the Ground State Energy in an Isotropically Disordered Lattice, etc." (July 1961).
180. 4, for example, exhibits a contraction at low temperatures, which can be used for a separation below a certain critical temperature, calculated to be of the order of 1 K. G. K. Walters and F. M. Fairbank have verified these predictions and obtained an isotope separation below a critical point of approximately 0.8 K: Phys. Rev. 103, 262 (1956). Further experiments have been performed by D. P. Edwards, A. S. McWilliams, and J. G. Daunt, Phys. Rev. Lett. 9, 155 (1962), and theoretical studies have been undertaken by R. A. Coldwell-Horsfall, Low Temp. Phys. 9B, 1110 (1965), as well as in collaboration with A. A. Maradudin, Technical Report of the Westinghouse Research Laboratory, Pittsburg, on "The Configuration Average of the Ground State Energy in an Isotropically Disordered Lattice, etc." (July 1961).
181. 4, for example, exhibits a contraction at low temperatures, which can be used for a separation below a certain critical temperature, calculated to be of the order of 1 K. G. K. Walters and F. M. Fairbank have verified these predictions and obtained an isotope separation below a critical point of approximately 0.8 K: Phys. Rev. 103, 262 (1956). Further experiments have been performed by D. P. Edwards, A. S. McWilliams, and J. G. Daunt, Phys. Rev. Lett. 9, 155 (1962), and theoretical studies have been undertaken by R. A. Coldwell-Horsfall, Low Temp. Phys. 9B, 1110 (1965), as well as in collaboration with A. A. Maradudin, Technical Report of the Westinghouse Research Laboratory, Pittsburg, on "The Configuration Average of the Ground State Energy in an Isotropically Disordered Lattice, etc." (July 1961).
182. 4, for example, exhibits a contraction at low temperatures, which can be used for a separation below a certain critical temperature, calculated to be of the order of 1 K. G. K. Walters and F. M. Fairbank have verified these predictions and obtained an isotope separation below a critical point of approximately 0.8 K: Phys. Rev. 103, 262 (1956). Further experiments have been performed by D. P. Edwards, A. S. McWilliams, and J. G. Daunt, Phys. Rev. Lett. 9, 155 (1962), and theoretical studies have been undertaken by R. A. Coldwell-Horsfall, Low Temp. Phys. 9B, 1110 (1965), as well as in collaboration with A. A. Maradudin, Technical Report of the Westinghouse Research Laboratory, Pittsburg, on "The Configuration Average of the Ground State Energy in an Isotropically Disordered Lattice, etc." (July 1961).
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191. In a lecture at the University of Texas, Austin, R. S. Mulliken reminisced: "Once Hulthén told me that they wanted to award the Nobel Prize to me. I guess it would have been for the half-integral quantum numbers in the BO-spectra" (Mulliken, 29 April 1970).
192. R. W. James and Miss Firth, Proc. Roy. Soc. (London) A117, 62 (1927).
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196. F signifies the ratio of the amplitude of the coherent X-radiation scattered from the atom, in the state of rest, to the amplitude of the coherent X-radiation from a free classical electron. In F-curves, the ratio is plotted vs. sin Θ/λ, where Θ is the glancing angle of the lattice and λ the wavelength of X-rays.
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210. W. Heisenberg, quoted in Ref. 104, p. 172.
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216. O. Sackur, Ber. Deutsch. Chem. Ges. 47, 1318 (1914); O. Sackur, H. Tetrode, quoted in Ref. 102; W. H. Keesom, Phys. Zs 15, 695 (1914); A. Sommerfeld, Wolfskehl lectures 1913, Leipzig 1914; P. Scherrer, Nachr. Akad. Wiss. Göttingen (1916); M. Planck, Sitz. Ber. Preuss. Akad. Wiss. 652 (1916).
217. O. Sackur, Ber. Deutsch. Chem. Ges. 47, 1318 (1914); O. Sackur, H. Tetrode, quoted in Ref. 102; W. H. Keesom, Phys. Zs 15, 695 (1914); A. Sommerfeld, Wolfskehl lectures 1913, Leipzig 1914; P. Scherrer, Nachr. Akad. Wiss. Göttingen (1916); M. Planck, Sitz. Ber. Preuss. Akad. Wiss. 652 (1916).
218. O. Sackur, Ber. Deutsch. Chem. Ges. 47, 1318 (1914); O. Sackur, H. Tetrode, quoted in Ref. 102; W. H. Keesom, Phys. Zs 15, 695 (1914); A. Sommerfeld, Wolfskehl lectures 1913, Leipzig 1914; P. Scherrer, Nachr. Akad. Wiss. Göttingen (1916); M. Planck, Sitz. Ber. Preuss. Akad. Wiss. 652 (1916).
219. O. Sackur, Ber. Deutsch. Chem. Ges. 47, 1318 (1914); O. Sackur, H. Tetrode, quoted in Ref. 102; W. H. Keesom, Phys. Zs 15, 695 (1914); A. Sommerfeld, Wolfskehl lectures 1913, Leipzig 1914; P. Scherrer, Nachr. Akad. Wiss. Göttingen (1916); M. Planck, Sitz. Ber. Preuss. Akad. Wiss. 652 (1916).
220. A. Sommerfeld, Z. Phys. 47, 1 (1928).
221. A. Sommerfeld and H. A. Bethe, "Elektronentheorie der Metalle," in H. Geiger and K. Scheel, eds. Handbuch der Physik, Vol. 24/2 (Berlin, 1933).
222. F. Simon, Nature 133, 529 (1934).
223. 4 on this basis to explain superfluidity.
224. 4 on this basis to explain superfluidity.
225. R. Eisenschitz and F. London, Zeit. f. Phys. 60, 491 (1930); F. London, Zeit f. Phys. 63, 245 (1930).
226. R. Eisenschitz and F. London, Zeit. f. Phys. 60, 491 (1930); F. London, Zeit f. Phys. 63, 245 (1930).
227. H. B. G. Casimir and D. Polder, Phys. Rev. 73, 360 (1948).
228. H. B. G. Casimir, J. Chim. Phys. 46, 407 (1949).
229. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
230. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
231. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
232. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
233. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
234. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
235. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
236. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
237. -4 cm: Acta Phys. Austr. 27, 341 (1968); "The General Theory of London-van der Waals Forces and Certain Features of the Equation of State for Fases," University of Neuchâtel, Switzerland (1963).
238. C. P. Enz and A. Thellung, Helv. Phys. Acta 33, 839 (1960). This question arises naturally in the context of relativistic quantum field theory. Here the concept of the zero-point energy leads to complications and is, therefore, avoided by using a normal or ordered Hamiltonian.
239. In classical theory, the position of the variables p and q does not play any role. If we "translate" a Classical Hamiltonian into a quantum one, in accordance with the correspondence principle, we are left with an ambiguity which we have to resolve.
240. W. Pauli, "Die allgemeinen prinzipien der Wellenmechanik," in H. Geiger and K. Scheel eds., Handhuch der Physik, Vol. 24/1 (Berlin 1933), p. 250.
241. The story of the application of zero-point energy in quantum field theory, though it touches only the periphery of the development, offers many interesting perspectives. In the last two decades of his life, Julian Schwinger became seriously interested in investigating various aspects of the Casimir effect. We have treated Schwinger's work in this field in Chap. 15, entitled "Taking the Road Less Traveled By" in a forthcoming work: Jagdish Mehra and Kimball A. Milton, Climbing the Mountain: The Scientific Biography of Julian Schwinger.