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Canadian Journal of Physics

On the operators for time of motion and delay time induced by scattering

Tran Trong Gien

Published Feb 1, 1969• 7 citations

Abstract

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.

DOI: 10.1139/p69-037

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If the lifetime of a collision is defined as the difference between the times in which a particle moves in free space and under an interaction from a scattering center to the outside (or in the reverse direction), this paper shows that exactly the same relation between the lifetime operator Q and the scattering matrix S as was found by Smith (1960) can be derived. This can be done in a natural way provided that the time operator is defined as [Formula: see text] and the manifold of the asymptotic outgoing (or incoming) states is chosen to be the set of corresponding eigenstates. The part of the time needed for the particle moving freely from the scattering center to R is equal to R/v.

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We consider time delay and symmetrized time delay (defined in terms of sojourn times) for quantum scattering pairs {H0 = h(P), H}, where h(P) is a dispersive operator of hypoelliptic-type. For instance, h(P) can be one of the usual elliptic operators such as the Schrödinger operator h(P) = P2 or the square-root Klein–Gordon operator [Formula: see text]. We show under general conditions that the symmetrized time delay exists for all smooth even localization functions. It is equal to the Eisenbud–Wigner time delay plus a contribution due to the non-radial component of the localization function. If the scattering operator S commutes with some function of the velocity operator ∇h(P), then the time delay also exists and is equal to the symmetrized time delay. As an illustration of our results, we consider the case of a one-dimensional Friedrichs Hamiltonian perturbed by a finite rank potential. Our study puts into evidence an integral formula relating the operator of differentiation with respect to the kinetic energy h(P) to the time evolution of localization operators.

We study the problem of computing the probability for the time-of-arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle, and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schr\\"odinger current. We derive a well defined uncertainty relation between time-of-arrival and energy; this result shows that the well known arguments against the existence of such a relation can be circumvented. Finally, we define a ``time-representation'' of the quantum mechanics of a free particle, in which the time-of-arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics.

We study the problem of computing the probability for the time of arrival of a quantum particle at a given spatial position. We consider a solution to this problem based on the spectral decomposition of the particle's (Heisenberg) state into the eigenstates of a suitable operator, which we denote as the ``time-of-arrival'' operator. We discuss the general properties of this operator. We construct the operator explicitly in the simple case of a free nonrelativistic particle and compare the probabilities it yields with the ones estimated indirectly in terms of the flux of the Schr\"odinger current. We derive a well-defined uncertainty relation between time of arrival and energy; this result shows that the well-known arguments against the existence of such a relation can be circumvented. Finally, we define a ``time representation'' of the quantum mechanics of a free particle, in which the time of arrival is diagonal. Our results suggest that, contrary to what is commonly assumed, quantum mechanics exhibits a hidden equivalence between independent (time) and dependent (position) variables, analogous to the one revealed by the parametrized formalism in classical mechanics. \textcopyright{} 1996 The American Physical Society.