Thermodynamics of the harmonic oscillator: Wien's displacement law and the Planck spectrum

American Journal of Physics

Volumn 71, Issue 9, 2003, Pages 866-870

  Boyer, Timothy H ^a  

^a CITY COLLEGE OF NEW YORK   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0042261066     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1566782     Document Type: Article

References (30)

1. See, for example, M. Planck, The Theory of Heat Radiation (Dover, New York, 1959), pp. 61-63; R. Becker and G. Leibfried, Theory of Heat (Springer, New York, 1967), 2nd ed., pp. 16 and 17; P. M. Morse, Thermal Physics (Benjamin/Cummings, Reading, MA, 1969), 2nd ed., pp. 78 and 79.
2. See, for example, M. Planck, The Theory of Heat Radiation (Dover, New York, 1959), pp. 61-63; R. Becker and G. Leibfried, Theory of Heat (Springer, New York, 1967), 2nd ed., pp. 16 and 17; P. M. Morse, Thermal Physics (Benjamin/Cummings, Reading, MA, 1969), 2nd ed., pp. 78 and 79.
3. See, for example, M. Planck, The Theory of Heat Radiation (Dover, New York, 1959), pp. 61-63; R. Becker and G. Leibfried, Theory of Heat (Springer, New York, 1967), 2nd ed., pp. 16 and 17; P. M. Morse, Thermal Physics (Benjamin/Cummings, Reading, MA, 1969), 2nd ed., pp. 78 and 79.
4. See, for example, E. A. Power, Introductory Quantum Electrodynamics (American Elsevier, New York, 1964), pp. 18-22.
5. See, for example, M. Planck in Ref. 1, pp. 72-83, or F. K. Richtmyer, E. H. Kennard, and T. Lauritsen, Introduction to Modern Physics (McGraw-Hill, New York, 1955), pp. 113-118; B. H. Lavenda, Statistical Physics: A Probabilistic Approach (Wiley, New York, 1991), pp. 67-69.
6. See, for example, M. Planck in Ref. 1, pp. 72-83, or F. K. Richtmyer, E. H. Kennard, and T. Lauritsen, Introduction to Modern Physics (McGraw-Hill, New York, 1955), pp. 113-118; B. H. Lavenda, Statistical Physics: A Probabilistic Approach (Wiley, New York, 1991), pp. 67-69.
7. See, for example, M. Planck in Ref. 1, pp. 72-83, or F. K. Richtmyer, E. H. Kennard, and T. Lauritsen, Introduction to Modern Physics (McGraw-Hill, New York, 1955), pp. 113-118; B. H. Lavenda, Statistical Physics: A Probabilistic Approach (Wiley, New York, 1991), pp. 67-69.
8. See, for example, R. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 55-56, 251-253; R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (Wiley, New York, 1985), 2nd ed., pp. 6-19.
9. See, for example, R. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York, 1965), pp. 55-56, 251-253; R. Eisberg and R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (Wiley, New York, 1985), 2nd ed., pp. 6-19.
10. I do not know of any text where the harmonic oscillator is treated as a thermodynamics system apart from statistical mechanics.
11. I am not aware of any thermodynamic treatment of the quasistatic change of the harmonic oscillator frequency. The familiar derivations of the Wien law take the Doppler-shift form given in Ref. 3.
12. P. C. W. Davies, "Scalar particle production in Schwarzschild and Rindler metrics," J. Phys. A 8, 609-616 (1975); W. G. Unruh, "Notes on black-hole evaporation," Phys. Rev. D 14, 870-892 (1976); T. H. Boyer, "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," ibid. 29, 1089-1095 (1984); D. C. Cole, "Properties of a classical charged harmonic oscillator accelerated through classical electromagnetic zero-point radiation," ibid. 31, 1972-1981 (1985).
13. P. C. W. Davies, "Scalar particle production in Schwarzschild and Rindler metrics," J. Phys. A 8, 609-616 (1975); W. G. Unruh, "Notes on black-hole evaporation," Phys. Rev. D 14, 870-892 (1976); T. H. Boyer, "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," ibid. 29, 1089-1095 (1984); D. C. Cole, "Properties of a classical charged harmonic oscillator accelerated through classical electromagnetic zero-point radiation," ibid. 31, 1972-1981 (1985).
14. P. C. W. Davies, "Scalar particle production in Schwarzschild and Rindler metrics," J. Phys. A 8, 609-616 (1975); W. G. Unruh, "Notes on black-hole evaporation," Phys. Rev. D 14, 870-892 (1976); T. H. Boyer, "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," ibid. 29, 1089-1095 (1984); D. C. Cole, "Properties of a classical charged harmonic oscillator accelerated through classical electromagnetic zero-point radiation," ibid. 31, 1972-1981 (1985).
15. P. C. W. Davies, "Scalar particle production in Schwarzschild and Rindler metrics," J. Phys. A 8, 609-616 (1975); W. G. Unruh, "Notes on black-hole evaporation," Phys. Rev. D 14, 870-892 (1976); T. H. Boyer, "Thermal effects of acceleration for a classical dipole oscillator in classical electromagnetic zero-point radiation," ibid. 29, 1089-1095 (1984); D. C. Cole, "Properties of a classical charged harmonic oscillator accelerated through classical electromagnetic zero-point radiation," ibid. 31, 1972-1981 (1985).
16. The quasistatic invariance of U/ω can be obtained by action-angle variables, See, for example, H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980), 2nd ed., p. 462. The situation involving quasistatic change in frequency is most famous in the case of a pendulum length which is changed slowly. See, for example, Problem 10, p. 543. An elementary derivation is given, for example, by T. H. Boyer, "Connection between the quasistatic hypothesis, of old quantum theory and classical electrodynamics with classical electromagnetic zero-point radiation," Phys. Rev. A 18, 1238-1245 (1978), Eqs. (1)-(9).
17. The quasistatic invariance of U/ω can be obtained by action-angle variables, See, for example, H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980), 2nd ed., p. 462. The situation involving quasistatic change in frequency is most famous in the case of a pendulum length which is changed slowly. See, for example, Problem 10, p. 543. An elementary derivation is given, for example, by T. H. Boyer, "Connection between the quasistatic hypothesis, of old quantum theory and classical electrodynamics with classical electromagnetic zero-point radiation," Phys. Rev. A 18, 1238-1245 (1978), Eqs. (1)-(9).
18. The quasistatic invariance of U/ω can be obtained by action-angle variables, See, for example, H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980), 2nd ed., p. 462. The situation involving quasistatic change in frequency is most famous in the case of a pendulum length which is changed slowly. See, for example, Problem 10, p. 543. An elementary derivation is given, for example, by T. H. Boyer, "Connection between the quasistatic hypothesis, of old quantum theory and classical electrodynamics with classical electromagnetic zero-point radiation," Phys. Rev. A 18, 1238-1245 (1978), Eqs. (1)-(9).
19. C. Garrod. Statistical Mechanics and Thermodynamics (Oxford U.P., New York, 1995), p. 128.
20. M. J. Sparnaay, "Measurement of the attractive forces between flat plates," Physica (Amsterdam) 24, 751-764 (1958); S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6/μm range," Phys. Rev. Lett. 78, 5-8 (1997), 81, 5475-5476 (1998); U. Mohideen, "Precision measurement of the Casimir force from 0.1 to 0.9 μm," Phys. Rev. Lett. 81, 4549-4552 (1998); H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectro-mechanical systems by the Casimir force," Science 291, 1941-1944 (2001); G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, "Measurement of the Casimir force between parallel metallic surfaces, " Phys. Rev. Lett. 88, 041804 (4 pages) (2002).
21. M. J. Sparnaay, "Measurement of the attractive forces between flat plates," Physica (Amsterdam) 24, 751-764 (1958); S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6/μm range," Phys. Rev. Lett. 78, 5-8 (1997), 81, 5475-5476 (1998); U. Mohideen, "Precision measurement of the Casimir force from 0.1 to 0.9 μm," Phys. Rev. Lett. 81, 4549-4552 (1998); H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectro-mechanical systems by the Casimir force," Science 291, 1941-1944 (2001); G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, "Measurement of the Casimir force between parallel metallic surfaces, " Phys. Rev. Lett. 88, 041804 (4 pages) (2002).
22. M. J. Sparnaay, "Measurement of the attractive forces between flat plates," Physica (Amsterdam) 24, 751-764 (1958); S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6/μm range," Phys. Rev. Lett. 78, 5-8 (1997), 81, 5475-5476 (1998); U. Mohideen, "Precision measurement of the Casimir force from 0.1 to 0.9 μm," Phys. Rev. Lett. 81, 4549-4552 (1998); H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectro-mechanical systems by the Casimir force," Science 291, 1941-1944 (2001); G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, "Measurement of the Casimir force between parallel metallic surfaces, " Phys. Rev. Lett. 88, 041804 (4 pages) (2002).
23. M. J. Sparnaay, "Measurement of the attractive forces between flat plates," Physica (Amsterdam) 24, 751-764 (1958); S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6/μm range," Phys. Rev. Lett. 78, 5-8 (1997), 81, 5475-5476 (1998); U. Mohideen, "Precision measurement of the Casimir force from 0.1 to 0.9 μm," Phys. Rev. Lett. 81, 4549-4552 (1998); H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectro-mechanical systems by the Casimir force," Science 291, 1941-1944 (2001); G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, "Measurement of the Casimir force between parallel metallic surfaces, " Phys. Rev. Lett. 88, 041804 (4 pages) (2002).
24. M. J. Sparnaay, "Measurement of the attractive forces between flat plates," Physica (Amsterdam) 24, 751-764 (1958); S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6/μm range," Phys. Rev. Lett. 78, 5-8 (1997), 81, 5475-5476 (1998); U. Mohideen, "Precision measurement of the Casimir force from 0.1 to 0.9 μm," Phys. Rev. Lett. 81, 4549-4552 (1998); H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectro-mechanical systems by the Casimir force," Science 291, 1941-1944 (2001); G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, "Measurement of the Casimir force between parallel metallic surfaces, " Phys. Rev. Lett. 88, 041804 (4 pages) (2002).
25. M. J. Sparnaay, "Measurement of the attractive forces between flat plates," Physica (Amsterdam) 24, 751-764 (1958); S. K. Lamoreaux, "Demonstration of the Casimir force in the 0.6 to 6/μm range," Phys. Rev. Lett. 78, 5-8 (1997), 81, 5475-5476 (1998); U. Mohideen, "Precision measurement of the Casimir force from 0.1 to 0.9 μm," Phys. Rev. Lett. 81, 4549-4552 (1998); H. B. Chan, V. A. Aksyuk, R. N. Kleiman, D. J. Bishop, and F. Capasso, "Quantum mechanical actuation of microelectro-mechanical systems by the Casimir force," Science 291, 1941-1944 (2001); G. Bressi, G. Carugno, R. Onofrio, and G. Ruoso, "Measurement of the Casimir force between parallel metallic surfaces, " Phys. Rev. Lett. 88, 041804 (4 pages) (2002).
26. See the discussion of natural units by C. Garrod, Ref. 9, p. 120. The choice ℏ = 1 is familiar to particle physicists. The measurement of temperature in energy units is familiar in thermodynamics where our choice corresponds to the use of what is usually termed τ instead of T. See also, for example, C. Kittel, Elementary Statistical Physics (Wiley, New York. 1958), p. 27.
27. See the discussion of natural units by C. Garrod, Ref. 9, p. 120. The choice ℏ = 1 is familiar to particle physicists. The measurement of temperature in energy units is familiar in thermodynamics where our choice corresponds to the use of what is usually termed τ instead of T. See also, for example, C. Kittel, Elementary Statistical Physics (Wiley, New York. 1958), p. 27.
28. See, for example, the discussion by B. H. Lavenda in Ref. 3, pp. 74-76. See also, T. H. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912 (Oxford U.P., New York, 1978).
29. See, for example, the discussion by B. H. Lavenda in Ref. 3, pp. 74-76. See also, T. H. Kuhn, Black-Body Theory and the Quantum Discontinuity, 1894-1912 (Oxford U.P., New York, 1978).
30. I am not aware of any textbook treatment of blackbody radiation that mentions zero-point energy as part of the spectrum involved.