Time in quantum mechanics

American Journal of Physics

Volumn 70, Issue 3, 2002, Pages 301-306

  Hilgevoord, Jan ^a  

^a UTRECHT UNIVERSITY   (Netherlands)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 23044530246     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1430697     Document Type: Article

References (20)

1. J. von Neumann, Mathematical Foundations of Quantum Mechanics (1932). Translated by R. T. Beyer (Princeton University Press, Princeton, NJ, 1955), p. 354.
2. A. Lahiri and P. B. Pal, A First Book of Quantum Field Theory (Alpha Science International Ltd., 2001), p. 2.
3. A fine review is by P. Busch, "On the energy-time uncertainty relation (Parts I, II)," Found. Phys. 21, 1-43 (1990). There is a considerable literature on classical notions of time, such as transit time and arrival time, in a quantum mechanical context. For a review see J. G. Muga and C. R. Leavens, "Arrival time in quantum mechanics," Phys. Rep. 338, 353-438 (2000).
4. The present article elaborates on ideas put forward in an earlier paper, J. Hilgevoord, "The uncertainty principle for energy and time," Am. J. Phys. 64, 1451-1456 (1996).
5. See, for example, H. Goldstein, Classical Mechanics, 2nd ed. (AddisonWesley, New York, 1980), Chap. 9. Reference 5, Sec. 9-5.
6. One may object that the worldline of a particle can be given in a covariant form. But a worldline is not a particle: it has no energy and momentum.
7. This model was suggested by P. Busch (private communication).
8. The schizophrenic character of quantum mechanics in the treatment of space-time coordinates and dynamical variables was explicitly pointed out by Bohr: "The only significant point is that in each case some ultimate measuring instruments, like the scales and clocks which determine the frame of space-time coordination-on which, in the last resort, even the definitions of momentum and energy quantities rest-must always be described entirely on classical lines, and consequently kept outside the system subject to quantum mechanical treatment."
9. See N. Bohr, The causality problem in modern physics. New Theories in Physics (International Institute of Intellectual Cooperation, Paris, 1939), pp. 11-45. This aspect of Bohr's doctrine has often been met with opposition, but it is a remarkable fact that it is firmly embedded in Ihefontialism of standard quantum mechanics.
10. In dealing with operators we shall adopt the level of mathematical precision of elementary texts on quantum mechanics. For our purpose it will suffice to point to analogies with well-known and generally accepted facts.
11. "The notation ψ(x,t) for the wave function may also be responsible for the mysterious expression "second quantization" in quantum field theory where, in fact, the quantization of a field is meant.
12. W. Pauli, General Principles of Quantum Mechanics (1933) (SpringerVerlag, New York, 1980), p. 63, footnote 2.
13. A quantum clock of this type was discussed by E. P. Wigner, "Relativistic Invariance and Quantum Phenomena", Rev. Mod. Phys. 29, 255-268 (1957)
14. See also H. Salecker and E. P. Wigner, "Quantum limitations of the measurement of space-time distances," Phys. Rev. 109, 571-577 (1958); A. Peres, "Measurement of time by quantum clocks," Am. J. Phys. 48, 552-557 (1980);
15. Quantum Theory: Concepts and Methods (Kluwer Academic, New York, 1993).
16. I4 J. Uffink, "Measures of uncertainty and the uncertainty principle," University of Utrecht thesis, 1990.
17. For a discussion of uncertainty relations of a different kind in which the total energy and momentum enter directly, see J. Hilgevoord, "The uncertainty principle for energy and time II," Am. J. Phys. 66, 396-402 (1998).
18. I6 T. D. Newton and E. P. Wigner, "Localized states for elementary systems," Rev. Mod. Phys. 21, 400-406 (1949).
19. G. C. Hegerfeldt, "Causality, particle localization and positivity of the energy," in Irreversibility and Causality in Quantum Theory-Semigroups and Rigged Hilbert Spaces, edited by A. Bohm, H.-D. Doebner, and P. Kielanowski, Springer Lecture Notes in Physics, Vol. 504 (Springer, New York, 1998).
20. L. Foldy and S. Wouthuysen, "On the Dirac theory of spin 1/2 particles and its non-relativistic limit," Phys. Rev. 78, 29-36 (1950).