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Annalen der Physik

On Analytic Continuability of Wave Functions

Julian Ławrynowicz, L. Wojtczak

Published Jan 1, 1967• 1 citation
Publisher

Abstract

Abstract In this paper we discuss the general form of a wave equation which in particular may be a SCHRÖDINGER, DIRAC, KLEIN‐GORDON equation. Under certain assumptions we obtain the necessary and sufficient condition for analytic continuability of its solutions considered as meromorphic functions. Next, we obtain some general formulae in the scattering theory for a potential of arbitrary symmetry. The values of scattering matrix elements are determined by the wave functions for the discrete energy spectrum and by their analytic continuations. Our theoretical results are applied to the deuteron disintegration problem and the phase shift calculation for a modified YUKAWA potential.

DOI
10.1002/andp.19674740110

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Abstract In this paper we discuss the general form of a wave equation which in particular may be a SCHRÖDINGER, DIRAC, KLEIN‐GORDON equation. Under certain assumptions we obtain the necessary and sufficient condition for analytic continuability of its solutions considered as meromorphic functions. Next, we obtain some general formulae in the scattering theory for a potential of arbitrary symmetry. The values of scattering matrix elements are determined by the wave functions for the discrete energy spectrum and by their analytic continuations. Our theoretical results are applied to the deuteron disintegration problem and the phase shift calculation for a modified YUKAWA potential.
The Schrödinger equation with the complex momentum k leads to an S matrix with very simple analytical properties. It differs from the conventional S matrix as little as one wishes on the real k axis, but it has, in general, completely different analytical behavior outside the real axis. The present formulation removes some of the unsatisfactory features of the conventional formalism in the sense that no redundant poles can occur and a phase shift determines the scattering potential uniquely. The complete analytical behavior of the S matrix, in particular at infinity, is discussed and the theory is extended to Klein-Gordon and Dirac equations with central potential.
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One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrödinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from the locality of the potential in the wave equation and represent the effect of causality in the time dependence of the scattering process.
In this paper we extend our previous analysis of the scattering of wave packets in one dimension to the case of the square-well potential. The analytic properties of the general scattering solution are emphasized thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. The square-well model is particularly interesting because of its application to the deuteron problem. Resonance scattering, barrier penetration, time delay, and line shape are discussed at the level of the first-year graduate student.
The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.