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American Journal of Physics

Quantum-Mechanical Time Operator

Mohsen Razavy

Published Oct 1, 1967• 26 citations
Publisher

Abstract

A first integral of motion in classical dynamics which contains the time explicitly and is conjugate to the Hamiltonian is quantized. The resulting quantal operator is applied to study the time delay caused by scattering, and in a simple case the expectation value of this operator is obtained.
DOI
10.1119/1.1973651

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Recently, Lippmann showed that a time operator might exist in quantum theory and that its explicit form is i ∂/∂E in the energy representation. In this paper, we wish to stress that i ∂/∂E actually represents only the time which the particle used to move from the scattering center to a certain point at the outside. (Later on, we shall call this time the "time of motion.") We also show that, although it is possible to define an operator for the "time of motion" of the particle in the energy representation, it is necessary to introduce a new conjugate representation of the energy representation, the "time representation," to overcome difficulties which may arise in the evaluation of the mean time of motion [Formula: see text]. If the position representation is used for the calculation instead, we shall meet these difficulties. The reason is that time of motion T, position x, and energy E are not two-by-two compatible variables. Bearing in mind that i ∂/∂E only represents the time of motion of the particle, we can then deduce the explicit form of the delay time operator Q in terms of the scattering matrix S directly from the time of motion operator by simple reasoning. This form is Q = −i(∂S/∂E)·S† as was found previously. Some interesting features of the delay time operator will also be sketched at the end of the paper.
From the equation of motion and the canonical commutation relation for the position of a particle and its conjugate momentum, different first integrals of motion can be constructed. In addition to the proper Hamiltonian, there are other operators that can be considered as the generators of motion for the position operator (q-equivalent Hamiltonians). All of these operators have the same classical limit for the probability density of the coordinate of the particle, and many of them are symmetric and self-adjoint operators or have self-adjoint extensions. However, they do not satisfy the Heisenberg rule of quantization, and lead to incorrect commutation relations for velocity and position operators. Therefore, it is concluded that the energy first integral and the potential, rather than the equation of motion and the force law, are the physically significant operators in quantum mechanics.
The information-theoretic maximal-entropy procedure for the analysis of collision processes is derived as a consequence of the dynamics, be they quantal or classical. The method centers attention on the minimal number of operators (the "dynamic constraints") whose expectation values are both necessary and sufficient to completely characterize the collision dynamics. For a given Hamiltonian and initial state, the constraints required to obtain an exact solution of the equations of motion are determined by a purely algebraic procedure. It is furthermore found possible to derive equations of motion for the conjugate Lagrange parameters. Immediate applications are noted, e.g., a family of similar reactions is shown to have a common set of dynamic constraints and simple illustrative applications are provided. The determination of the scattering matrix is discussed, with examples. The general formalism consists in solving the scattering problem in two stages. The first is purely algebraic. At the end of this stage one obtains the functional form of, say, the scattering matrix or of the density matrix after the collision expressed in terms of parameters whose number equals the number of dynamic constraints. The end result of this algebraic stage suffices to analyze the scattering pattern for any initial state. The second stage is the predictive procedure. Explicit coupled first-order nonlinear differential equations are obtained for the parameters.
The vacuum field’s role in the radiative properties of two-level atoms located externally to a phase-conjugate mirror (PCM) is investigated. The basic result of Gaeta and Boyd [ Phys. Rev. Lett.60, 2618 ( 1988)], that the PCM emits an average of R (where R is reflectivity) noise photons per mode of the radiation field, is modified. Stimulated emission of the combined PCM–atom system takes place when the atom is in its ground state; this emission is caused by a scattering–conjugation process of virtual photons. The expression for the power emitted by a system of N (N > 1) two-level atoms also contains terms that are the expectation values of products of two atomic raising or lowering operators in addition to the usual superradiant terms. The case N = 2 is solved exactly in the limit of a vanishing separation-to-wavelength ratio. The expectation values of individual and pair-product operators are derived from the Heisenberg equations of motion, and the solutions are classified according to the value of an integral of motion. When R < 1, the time evolution is characterized by four real-time constants, of which two are a complex-conjugate pair for R > 1, indicating relaxation by damped oscillations.
We explore the properties of quantum states and operators that are conjugate to the Hamiltonian eigenstates and operator when the Hamiltonian spectrum is continuous, i.e., we find time-like operators T^ such that [T^,H^]=iℏ. This is a property expected for a time operator. We explicitly unfold the momentum sign degeneracy of energy states. We consider the free-particle case, and we find, among other things, that the time states are also the solution of the quantized version of the classical motion of the particle.