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American Journal of Physics

Integral Equations for the Schrödinger Wave Function

William Tobocman, Leslie L. Foldy

Published Oct 1, 1959• 6 citations
Publisher

Abstract

For many problems in quantum mechanics it proves to be useful to write the Schrödinger equation as an integral equation rather than as a differential equation. In this paper we describe how the transformation of the Schrödinger equation from differential form to integral form may be accomplished. Two methods are described, one appropriate to bound-state theory and one appropriate to scattering theory. It is shown that the bound-state wave function is also a solution of the integral equation (Lippman-Schwinger equation) appropriate to scattering theory.

DOI
10.1119/1.1996206

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For many problems in quantum mechanics it proves to be useful to write the Schrödinger equation as an integral equation rather than as a differential equation. In this paper we describe how the transformation of the Schrödinger equation from differential form to integral form may be accomplished. Two methods are described, one appropriate to bound-state theory and one appropriate to scattering theory. It is shown that the bound-state wave function is also a solution of the integral equation (Lippman-Schwinger equation) appropriate to scattering theory.
Laplace transform techniques for solving differential equations do not seem to have been directly applied to the Schrödinger equation in quantum mechanics. This may be because the Laplace transform of a wave function, in contrast to the Fourier transform, has no direct physical significance. However, this paper will show that scattering phase shifts and bound state energies can be determined from the singularities of the Laplace transform of the wave function. The Laplace transform method can thereby simplify calculations if the potential allows a straightforward solution of the transformed Schrödinger equation. Suitable cases are the Coulomb, oscillator and exponential potentials and the Yamaguchi separable non-local potential.
The author discusses a method for solving the Schrodinger equation with spherically symmetric potentials. The method is based on the use of a certain integral transform for the radial wavefunction. In the representation defined by this transform the Schrodinger equation becomes an integral equation of Volterra type. In a number of cases this equation can be solved exactly without any use of perturbation theory. The resulting solution is represented by a locally finite sum of functions, which can be easily calculated since they are defined by integrals involving simple algebraic functions. For this reason, the method is well suited for practical calculations. The author derives exact solutions for the Yukawa and exponential potentials. As an application the author calculates the bound state spectrum and the scattering cross-sections.
The Schrödinger equation for the scattered particle is converted into an integral equation, in which the kernel is obtained by the use of the condition that the events described by the solutions of the equation must be causally related, i.e., that causes must always precede the effects which they produce. The integral equation is iterated once, and it is found that its solution corresponds to the usual description of a scattering process in terms of an undeflected wave traversing the system plus a radially scattered wave moving outwards. In the second part, it is shown that a simple relation exists between solutions of Schrödinger's equation of the sort just described and the solution of the initial-value problem associated with the equation of motion for a free particle. By this means a solution of the scattering problem is set up which describes how the wave function develops as the wave which it describes propagates through the system. The series which results was first given by Feynman. As an example, the first Born approximation for the scattering problem is developed by each approach.
Quantum mechanics heavily depends on integral equations because they supply advanced mathematical solutions for solving complex theoretical physics problems. This paper examines the development of integral equations starting from their inception through their mathematical structure until their utility in solving Schrödinger's equation and its adjoint variants. The research analyzes integral equations by combining sophisticated derivations with numerical approximation methods and case-based examinations, and multidisciplinary connections to explain their use in scattering theory and bound states, as well as quantum field theory and quantum electrodynamics and condensed matter physics and quantum information theory, and new fields of quantum computing and quantum gravity. The assessment demonstrates that integral equations continue to adapt for quantum mechanical research as they advance contemporary knowledge in this field.
The Schrödinger equation is a linear partial differential equation for the wave function. Therefore, any linear combination of two solutions is also a solution. The possibility of forming linear combinations is a well-known property of vectors. Moreover, we normalize solutions for the statistical interpretation, and we have seen several examples of orthogonality relations between bound and other states. These concepts are reminiscent of the length of a vector and the angle between two vectors or the dot product. Scattering is described by a matrix, another concept from linear algebra.