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Physical Review A

Time-dependent invariants and the Feynman propagator

Asish Kumar Dhara, S. V. Lawande

Published Jul 1, 1984• 45 citations
Publisher

Abstract

A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.

DOI
10.1103/physreva.30.560

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A class of recently discussed time-dependent classical Lagrangians possessing invariants is considered from a quantum-mechanical point of view. Quantum mechanics is introduced directly through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out explicit path integration, that the propagator for these time-dependent problems is related to the propagator of associated time-independent problems. The expansion of the propagator in terms of the eigenfunctions of the invariant operator and the quantum superposition principle follow naturally in our scheme. The theory is applied to obtain explicitly exact propagators for some illustrative examples.
A class of time-dependent classical Lagrangians possessing an invariant quadratic in momentum is considered from a quantal point of view. Quantum mechanics is introduced through the Feynman propagator defined as a path integral involving the classical action. It is shown, without carrying out an explicit path integration, that the propagator for such a time-dependent system is related to the propagator of an associated time-independent problem. The expansion of the propagator in terms of the eigenfunctions of the invariant operator is derived and the equivalence of the present theory to that of Lewis and Reisenfeld (1969) is discussed. Explicit analytic forms of propagators are obtained for some cases to illustrate the application of the present approach.
A class of time-dependent problems in two space dimensions possessing time-dependent invariants, bilinear in momenta, is considered. Explicit expression of the potentials and the corresponding invariants are derived. Quantum mechanics is introduced in these time-dependent problems directly through a Feynman propagator defined as a path integral involving the classical action. The propagators are shown to admit expansions in terms of the eigenfunctions of the corresponding invariant operators. Equivalence of the present theory to that of Lewis and Riesenfeld [H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)] is discussed.
The conventional approach to quantum mechanics begins with the classical Hamiltonian and changes observables to noncommuting operators. Canonically conjugate observables have to be noncommuting operators in order to satisfy the Heisenberg uncertainty principle. The dynamics of a quantum mechanical system is given by the time-dependent Schrödinger equation. In the early 1940s Feynman realized that it would be possible to construct a quantum model using the classical Lagrangian approach (a method closely related to the action integral). Dirac had earlier pointed out that eit times the Lagrangian was analogous to a transformation function for the wave function in which the wave function at one instant could be related to the wave function at a next instant in a time interval of t by multiplying with such an exponential function. Feynman found the exponential phase factor to be the action S which is the time integral of the Lagrangian. Further, he noticed that it was necessary to perform integrals over all space variables at every instant of time to find the transition amplitudes. He developed a way for describing and evaluating the integration using the idea of integral over all paths. In Feynman’s approach, the particles are described using amplitudes calculated along the paths they may or may not follow. These amplitudes behave like waves, their phase changes as the quantum system moves along the path and the amplitudes for all paths superpose and interfere. This interference between all paths is called as sum-over-histories and the resulting amplitude is connected in the same way that the amplitude of a wave function is linked to probability as magnitude-squared.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.
In this thesis we are investigating a different formalism of non-relativistic quantum mechanics called the path integral formalism.It is a generalization of the classical least action principle.The introduction to this subject begins with the construction of the path integral in terms of the idea of probability amplitudes whose absolute square gives the probability of finding a system in a particular state.Then we show that if the Lagrangian is a quadratic form one needs only to calculate the classical action besides from a time-dependent normalization constant to find the explicit expression of the path integral.We look in to the subject of two kinds of slit-experiments: The square slit, the single-and the double-Gaussian slit.Also, the propagator for constrained paths is calculated and applied to the Aharonov-Bohm effect, which shows that the vector potential defined in classical electrodynamics have a physical meaning in quantum mechanics.It is also shown that the path integral formulation is equivalent to the Schrödinger description of quantum mechanics, by deriving the Schrödinger equation from the path integral.Further applications of the path integral are discussed.