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Physical Review A

Inverse-square potential and the quantum vortex

Hua Wu, Donald W. L. Sprung

Published Jun 1, 1994• 38 citations
Publisher

Abstract

The vortex structure of the steady-state probability flow for a general one-particle system in quantum mechanics is introduced and its relationship to the inverse-square potential is discussed. The relationship is made clear by classifying the corresponding solutions of the related classical inverse-square-potential problem.
DOI
10.1103/physreva.49.4305

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The quantum mechanics of inverse square potentials in one dimension is usually studied through renormalization, self-adjoint extension and WKB approximation. This paper shows that such potentials may be investigated within the framework of the position-dependent mass quantum mechanics formalism under the usual boundary conditions. As a result, exact discrete bound state solutions are expressed in terms of associated Laguerre polynomials with negative energy spectrum using the Nikiforov-Uvarov method for the repulsive inverse square potential.
Abstract We discuss a general description of the solutions to the 1D time-independent Schrödinger equation that does not a priori distinguish between scattering states and bound states and emphasizes and reinforces their relationship and connection to each other. This manuscript also introduces the concept of transfer matrices, which it presents as a logical extension of the traditional approach to evaluating 1D potentials. Using the transfer matrix method and a finite step approximation allows for a simple and straight-forward numerical solution of arbitrary 1D potentials. It also separates the process of solving the Schrödinger equation from selecting physically relevant solutions, which is a critical skill in quantum mechanics and is at the core of physics problems in general.
Steady state scenarios exist in classical physics and even though there is motion (flows) (i.e. changes in time) overall the system does not change in time. Such systems are usually not described in terms of probability. Furthermore, a single particle moving in one direction is described by x(t), i.e. x=vt, so there is no probability and no steady stream scenario.Such a particle, however, may give rise to a steady stream scenario if it encounters a V1-V2 discontinuity (V1, V2 are two constant potentials) and E (energy) remains constant. In such a case, there is a probability to reflect or to pass into the V2 region with a different p, but the same E. This situation automatically introduces probability and may be associated with a steady stream picture, even in classical physics. As a result, classical physics is not as deterministic as it may seem. If probabilities arise at the discontinuity, one may ask: Is there a steady-stream (flux) type of probability associated with a moving particle? In a steady stream situation there is no change in time, but a moving particle clearly changes in time x(t). How does one associate probabilities which do not contain time to flow problems which clearly contain motion? We investigate these questions and try to argue that the quantum exp(ipx) form and the Born W*(x)W(x) wavefunction picture are associated with these questions.
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