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Physical Review A

Quantum-mechanical results for a free particle inside a box with general boundary conditions

M. G. E. da Luz, Bin Kang Cheng

Published Mar 1, 1995• 19 citations
Publisher

Abstract

The wave functions with the most general boundary conditions consistent with the conservation of probability for a free particle inside a box [Phys. Rev. D 42, 1194 (1990)] are calculated. The exact Green's functions and propagators for some special cases are obtained and a semiclassical approach for the propagators is considered. Finally, the influence of the boundary conditions over the path integral's formalism is briefly discussed.

DOI
10.1103/physreva.51.1811

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The wave functions with the most general boundary conditions consistent with the conservation of probability for a free particle inside a box [Phys. Rev. D 42, 1194 (1990)] are calculated. The exact Green's functions and propagators for some special cases are obtained and a semiclassical approach for the propagators is considered. Finally, the influence of the boundary conditions over the path integral's formalism is briefly discussed.
This chapter discusses the problem of one particle in a one-dimensional box free of the potential energy inside the box, that is, no external force is exerted on the particle inside the box. The corresponding Schrödinger equation is given along with the boundary conditions which yields an elliptic second-order differential equation and the solutions are given next: the wave function, quantum numbers and corresponding energy. The normalized wave function is also given which is similar to that of the one-particle harmonic oscillator. Afterwards, the Hamiltonian function of the particle in a bidimensional box is given along with its wave function that is the product of two wave function of the particle in a one-dimensional box and the energy is the sum of two energy components, x and y, of the particle in a one-dimensional box. Finally, there is a couple of exercises to plot the wave functions of a rectangular box to find the probability density of the wave function for a symmetrical box. (164 words).
We study the Feynman path-integral formalism for the constrained problem of a free particle moving on the half-line. It is shown that the effect of the boundary condition at the origin can be incorporated into the path integral by a simple modification of the action. The small-time behavior of the Green's function can be obtained from the stationary-phase evaluation of our expression for the path integral, which in this case includes contributions from both the direct and reflected classical paths.
A free nonrelativistic particle moving in a one-dimensional box can be described by any of a four-parameter family of self-adjoint Hamiltonians each of which guarantees that probability does not leave the box. Included are cases where the particle striking one wall may reappear at the other. We construct a four-parameter family of Brownian functional integrals which include paths that jump from wall to wall. Their analytic continuation in time results in the same quantum Green's functions as those which arise directly from the Hamiltonians.
For a particle inside a one-dimensional finite-deep square well potential, we try to substitute the continuity condition of the first-order derivative of logarithm of wave functions by the general boundary conditions. It seems more reasonable than that replaceing the usual boundary condition forcing wave functions to vanish at two walls for a free particle confined in a one-dimensional box. It is found that the continuity condition of probability current density might be damaged, if the values of parameters ρ and θ are arbitrary. In addition, the constrained conditions for the parameters ρ and θ are given and a new class of analytical solutions is obtained.\;
A free nonrelativistic particle moving in two dimensions on a half-plane can be described by self-adjoint Hamiltonians characterized by boundary conditions imposed on the systems. The most general boundary condition is parameterized in terms of the elements of an infinite-dimensional matrix. We construct the Brownian functional integral for each of these self-adjoint Hamiltonians. Nonlocal boundary conditions are implemented by allowing the paths striking the boundary to jump to other locations on the boundary. Analytic continuation in time results in the Green’s functions of the Schrödinger equation satisfying the boundary condition characterizing the self-adjoint Hamiltonian.
Rather general expressions are derived which represent the semiclassical time-dependent propagator as an integral over initial conditions for classical trajectories. These allow one to propagate time-dependent wave functions without searching for special trajectories that satisfy two-time boundary conditions. In many circumstances, the integral expressions are free of singularities and provide globally valid uniform asymptotic approximations. In special cases, the expressions for the propagators are related to existing semiclassical wave function propagation techniques. More generally, the present expressions suggest a large class of other, potentially useful methods. The behavior of the integral expressions in certain limiting cases is analyzed to obtain simple formulas for the Maslov index that may be used to compute the Van Vleck propagator in a variety of representations.