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Progress of Theoretical Physics

Rational Construction and Physical Signification of the Quantal Time Operator

I. Fujiwara

Published Jul 1, 1980• 7 citations
Publisher

Abstract

First a general expression for an eigenstate |t > of the time operator T defined by the commutation relation [ T, H ] = i ℏ is given in terms of the eigenvectors of the Hamiltonian H. Then the time operator itself is constructed as T = ∫dtt | t >≪t|. When a compound state corresponding to transitions is introduced, it leads directly to the time-energy uncertainty relationship formulated previously by the same author.

DOI
10.1143/ptp.64.18

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First a general expression for an eigenstate |t > of the time operator T defined by the commutation relation [ T, H ] = i ℏ is given in terms of the eigenvectors of the Hamiltonian H. Then the time operator itself is constructed as T = ∫dtt | t >≪t|. When a compound state corresponding to transitions is introduced, it leads directly to the time-energy uncertainty relationship formulated previously by the same author.
On the basis of the principles proposed by the first author in 1970 and 1979 the time-energy uncertainty relationship in quantum mechanics is constructed explicitly for transitions between adequately prescribed states of a simple system of a nonrelativistic particle moving in a constant uniform field of force. The implications of the uncertainties in time and energy contemplated for transitions are clarified by contrasting them with ordinary uncertainties of various dynamical variables evaluated in one single state of the system. Decay process and the mean lifetime related to it are discussed too in a particular case when a pair of states, initial and final, for a transition happen to be identical.
Progress of Theoretical Physics Vol. 64 No. 1 (1980) pp. 1-17 The Time as an Observable in Quantum Mechanics Tetsuo Gotō, Shigefumi Naka and Katsuhito Yamaguchi Progress of Theoretical Physics Vol. 64 No. 1 (1980) pp. 18-27 Rational Construction and Physical Signification of the Quantal Time Operator Izuru Fujiwara Progress of Theoretical Physics Vol. 64 No. 2 (1980) pp. 363-379 Explicit Construction of Time-Energy Uncertainty Relationship in Quantum Mechanics Izuru Fujiwara, Kazuki Wakita and Hisao Yoro
Because time does not appear in Schr\"odinger's equation as an operator but only as a parameter, the time-energy uncertainty relation must be formulated in a special way. This problem has in fact been studied by many authors and we give a summary of their treatments. We then criticize the main conclusion of these treatments; viz., that in a measurement of energy carried out in a time interval, $\ensuremath{\Delta}t$, there must be a minimum uncertainty in the transfer of energy to the observed system, given by $\ensuremath{\Delta}({E}^{\ensuremath{'}}\ensuremath{-}E)>~\frac{h}{\ensuremath{\Delta}t}$. We show that this conclusion is erroneous in two respects. First, it is not consistent with the general principles of the quantum theory, which require that all uncertainty relations be expressible in terms of the mathematical formalism, i.e., by means of operators, wave functions, etc. Secondly, the examples of measurement processes that were used to derive the above uncertainty relation are not general enough. We then develop a systematic presentation of our own point of view, with regard to the role of time in the quantum theory, and give a concrete example of a measurement process not satisfying the above uncertainty relation.
The much-studied energy-time uncertainty relation has well-known difficulties that are exacerbated for a system with discrete energy levels. The difficulty in representing time in the abstract sense by an operator raises the related question of whether or not there is some other quantity that is complementary to the Hamiltonian of a quantum system. Such a quantity would have dimensions of time but would be a property of the system itself. We examine this question for a system with discrete energy eigenstates for which the ratios of the energy differences are rational. We find that such a quantity does exist and can be represented both by a probability-operator measure and by an Hermitian operator, but in a state space larger than the minimal space needed to include the states of the system. The uncertainty relation with the energy is slightly more complicated than the momentum-position uncertainty relation, but is readily interpretable. To describe such a quantity the name ``age'' is suggested.
W Pauli pointed out that the existence of a self-adjoint time operator is incompatible with the semi-bounded character of the Hamiltonian spectrum. As a result, there has been much argument about the time–energy uncertainty relation and other related issues. In this paper, we show a way to overcome Pauli's argument. In order to define a time operator, by treating time and space on an equal footing and extending the usual Hamiltonian Ĥ to the generalized Hamiltonian Ĥ μ (with Ĥ 0 = Ĥ), we reconstruct the analytical mechanics and the corresponding quantum (field) theories, which are equivalent to the traditional ones. The generalized Schrödinger equation i∂ μ ψ = Ĥ μ ψ and Heisenberg equation d /d x μ = ∂ μ + i[Ĥ μ , ] are obtained, from which we have: (1) t is to Ĥ 0 as x j is to Ĥ j ( j = 1, 2, 3); likewise, t is to i∂ 0 as x j is to i∂ j ; (2) the proposed time operator is canonically conjugate to i∂ 0 rather than to Ĥ 0 , therefore Pauli's theorem no longer applies; (3) two types of uncertainty relations, the usual Δ x μ Δ p μ ≥ 1/2 and the Mandelstam–Tamm treatment Δ x μ Δ H μ ≥ 1/2, have been formulated.
A general barrier-traversal-time operator is constructed using the time-of-arrival formalism. We study the operator's dynamics and determine the role played by the time-energy commutation relation. It turns out that similar dynamics is observed whether the traversal-time operator is canonically conjugate to the system Hamiltonian or not. We also use the barrier-traversal-time operator to calculate the traversal-time distributions for different cases (free case, and above-the-barrier and under-the-barrier traversals). The peak of the traversal-time distributions coincides with the classical expected traversal times for the free case and the above-the-barrier case. We then present our interpretation of the time-energy uncertainty relation that is consistent with the different traversal-time distributions.