Prog. Theor. Phys. Vol. 64 No. 1 (1980) pp. 1-17
The Time as an Observable in Quantum Mechanics
Atomic Energy Research Institute, College of Science and Technology, Nihon University, Tokyo 101
(Received February 16, 1980)
Following homogeneous canonical formalism where the time is treated on an equal footing with the space coordinates, we propose an alternative formulation of quantum mechanics in which the time is considered as a dynamical variable. The basic equation corresponding to and being equivalent to the ordinary Schrödinger equation is obtained when we consider the space coordinates x as a parameter corresponding to the time t in the ordinary formulation. The formal solution of this equation is given and examined. By making use of this solution the scattering theory is formulated in our formalism.
It is shown that observables corresponding to the time and space coordinates can be defined. A short comment on the energy-time uncertainty relation is also given.
DOI : 10.1143/PTP.64.1
Y. Aharonov and D. Bohm, Phys. Rev. 122 (1961), 1649[APS].
V. A. Fock, Sov. Phys. -JETP 15 (1962), 784.
L. Susskind and J. Glogower, Physics 1 (1964), 49.
M. Razavy, Nuovo Cim. B 63 (1969), 271.
M. Razavy, Can. J. Phys. 49 (1971), 3075.
D. R. Rosenbaum, J. Math. Phys. 10 (1969), 1127[CrossRef].
G. R. Allcock, Ann. of Phys. 53 (1969), 253[CrossRef];
ibid. 53 (1969), 286[CrossRef]; ibid. 53 (1969), 331.
R. Carruther and M. M. Nieto, Rev. Mod. Phys. 40 (1968), 411[APS].
I. Fujiwara, Prog. Theor. Phys. 44 (1970), 1701[PTP].
I. Fujiwara, Prog. Theor. Phys. 62 (1979), 1179[PTP].
- P. A. M. Dirac, Can. J. Math. 2 (1950), 129.
P. A. M. Dirac, Lectures on Quantum Mechanics, (Belfer Graduate School of Science, Yeshiva Univ., New York, 1964).
A. Mercier, Analytical Canonical Formalism of Physics (North-Holland, Amsterdam, 1959).
- J. Schwinger, Nuovo Cim. 37 (1965), 278.
A. Katz, Nuovo Cim. 37 (1965), 745,
- L. D. Faddeev, Theor. Math. Phys. 1 (1970), 1.
S. Naka and T. Gotō, Phys. Lett. B 92 (1980), 139[CrossRef].
- J. Rayski and J. M. Rayski Jr., The Uncertainty Principle and Foundations of Quantum Mechanics, A Fifty Years Survey, ed. by W. C. Price and S. S. Chissick, (John Wiley & Sons, 1977).
E. P. Wigner, Aspect of Quantum Theory, ed. by A. Salam and E. P. Wigner, (Cambridge Univ. Press, 1972).
Citing Article(s) :
Progress of Theoretical Physics Vol. 66 No. 5 (1981) pp. 1525-1538
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Progress of Theoretical Physics Vol. 66 No. 6 (1981) pp. 1915-1925
On the Time Operator in Quantum Mechanics. II
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Progress of Theoretical Physics Vol. 69 No. 1 (1983) pp. 158-170
Time Compound Nucleus for High Energy Nuclear Reactions
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