The uncertainty principle for energy and time

American Journal of Physics

Volumn 64, Issue 12, 1996, Pages 1451-1456

  Hilgevoord, Jan ^a  

^a UTRECHT UNIVERSITY   (Netherlands)

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[No Author keywords available]

Indexed keywords

EID: 0030523945     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18410     Document Type: Article

References (22)

1. W. Heisenberg, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik," Z. Phys. 43, 172-198 (1927). For an English translation of this article see Ref. 2.
2. J. A. Wheeler and W. H. Zurek, Quantum Theory and Measurement (Princeton U.P., Princeton, NJ, 1983), p. 62.
3. John von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, 1955), pp. 353, 354. Translated from the original German edition: Johann von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932), pp. 187 and 188. These two pages provide an easy reference for the views we want to criticize in this article, but similar views are widespread in the literature.
4. John von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University, Princeton, 1955), pp. 353, 354. Translated from the original German edition: Johann von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932), pp. 187 and 188. These two pages provide an easy reference for the views we want to criticize in this article, but similar views are widespread in the literature.
5. In view of the difficulties that beset the definition of action and angle variables as self-adjoint operators in quantum mechanics we prefer the more general term "observable" to distinguish them from c numbers.
6. P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford U.P., Oxford, 1947), 3rd ed., Chap. IV.
7. Niels Bohr, "Discussion with Einstein on epistemological problems in atomic physics," in Albert Einstein: Philosopher-Scientist, edited by P. A. Schilpp (Open Court, La Salle, IL, 1949), p. 214. Reprinted in Ref. 2, p. 22.
8. O. D. Johns, "Canonical transformations with time as a coordinate," Am. J. Phys. 57, 204-215 (1989).
9. J. L. Synge, "Classical Dynamics," in Vol. 3, Part 1 of Encyclopedia of Physics, edited by S. Flügge (Springer-Verlag, Berlin, 1960), p. 105. Somewhat disappointingly, Synge confines his treatment of relativistic particle dynamics to one-particle systems, a most atypical case.
10. i and t are functions. It is then possible to find an appropriate Lagrangian for the system. From this Lagrangian the momentum conjugate to t may be calculated and it turns out to be - H. This would then be a ground for considering t and -H as forming a fourth canonical pair. However, the next step, the inversion of the Lagrangian to obtain the generalized velocities as functions of the generalized coordinates and momenta, turns out to be impossible, that is, the equations of motion cannot be written in the usual Hamiltonian form.
11. E. C. G. Sudarshan and N. Mukunda, Classical Dynamics: A Modern Perspective (Wiley, New York, 1974), Chap. 21.
12. F. London, "Ueber die Jacobischen Transformationen der Quantenmechanik," Z. Phys. 37, 915 (1926).
13. Pauli has already remarked that if the spectrum of t is the whole real axis and t and H satisfy the commutation relation in (1), then the spectrum of H cannot contain discrete eigenvalues. In fact, the restrictions following from H≥0 are even more severe.
14. See Ref. 5, Sec. 25. For a modern treatment: L. E. Ballentine, Quantum Mechanics (Prentice-Hall, Englewood Cliffs, NJ, 1990), Chap. 3.
15. J. Uffink, "The rate of evolution of a quantum state," Am. J. Phys. 61, 935-936 (1993).
16. J. Hilgevoord and J. Uffink, "A new view on the uncertainty principle," in Sixty-Two Years of Uncertainty, Historical and Physical Inquiries into the Foundations of Quantum Mechanics, edited by A. I. Miller (Plenum, New York, 1990), pp. 121-139.
17. J. Hilgevoord and J. Uffink, "Uncertainty in prediction and in inference," Found. Phys. 21, 323-341 (1991).
18. J. Hilgevoord and J. Uffink, "The mathematical expression of the uncertainty principle," in Microphysical Reality and Quantum Formalism, edited by A. van der Merwe, F. Selleri, and G. Tarozzi (Kluwer, Dordrecht, 1988), pp. 91-114.
19. L. D. Landau and E. M. Lifshitz, Quantum Mechanics, Volume 3 of Course of Theoretical Physics (Pergamon, Oxford, 1975), 2nd ed., Sec. 44.
20. L. Mandelstam and I. Tamm, "The uncertainty relation between energy and time in nonrelativistic quantum mechanics," J. Phys. (USSR) 9, 249-254 (1945).
21. P. Busch, "On the energy-time uncertainty relation," Part I and II, Found. Phys. 20, 1-43 (1990).
22. A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht, 1993).