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Journal of Mathematical Physics

Principles of Limiting Absorption and Limiting Amplitude in Scattering Theory. I. Schrödinger's Equation

Farouk M. Odeh

Published Nov 1, 1961• 8 citations
Publisher

Abstract

The outgoing solution of the time-independent Schrödinger equation, with a suitably restricted real potential, is shown to be the uniform limit of the square-integrable solutions of the same equation with complex energy as the imaginary part of the energy tends to zero. Under further restrictions on the potential, it is also shown that the solution to the initial-value problem for the time-dependent Schrödinger equation tends to the outgoing solution as time increases indefinitely.

DOI
10.1063/1.1724225

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A new interpretation of the wave function: Reply

WAFIK A. WASSEF

I believe that the author of this note is confusing definite and indefinite integrals. Any indefinite integre 'I I can be reduced to the corresponding definite integral by the proper choice of the limits of integration and the constant of integration. The fact that [5] is a solution of [4] can be looked-up directly from any table of elementary integrals; see, for example, ref. 1. For finite values of z i t is easy to show that [3] and [4] are equivalent to the time-dependent Schrodinger's wave equation by taking the time derivative of both sides of [4]. Moreover, the limit z + 0 cannot be allowed, since from [3] this corresponds to an infinite energy E. The meaning of the unit of imaginary numbers in on the right-hand side of [4] is given immediately after this equation.

Canadian Journal of Physics•Apr 1, 1990•0 citations
The outgoing solution of the time-independent Schrödinger equation, with a suitably restricted real potential, is shown to be the uniform limit of the square-integrable solutions of the same equation with complex energy as the imaginary part of the energy tends to zero. Under further restrictions on the potential, it is also shown that the solution to the initial-value problem for the time-dependent Schrödinger equation tends to the outgoing solution as time increases indefinitely.
This chapter deals with the time-independent Schrödinger equation. The introduction of complex ‘Hamiltonians’ is essential for treating resonances that are inherently different from genuine (physical) bound states. True square-integrable bound-state eigenvectors are associated with discrete negative eigenenergies and Hermitean Hamiltonians. Resonant states are also square-integrable, but they are linked to complex energies with positive real parts that belong to a continuum. The concept of the absorbing boundary condition comes to rescue the situation via the introduction of complex ‘Hamiltonians’ with normalizable scattering wavefunctions as proper physical states. A method which can provide an adequate spectral representation of the total Green operator or resolvent G^(ω) would be one of the key inputs into an invaluable practical quantum-mechanical theory for scattering (E = 0) and spectroscopy (E < 0).
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.
Variational principles are designed for the solution of the Schr\"odinger equation when a point source is placed in the presence of an inhomogeneous, absorbing medium represented by an arbitrary complex potential function. When the point source is allowed to recede to infinity, these stationary structures reduce to variational principles for the wave function in the standard scattering problem, namely the outgoing solution to the Schr\"odinger equation for an incident plane wave. Finally in the asymptotic region, the well-known bifunctional variational principles for the transition amplitudes arise automatically from the stationary forms for the wave function describing the standard scattering problem. A few examples leading to variationally improved wave functions are discussed.
Abstract After revisiting the time-dependent Schrödinger equation and introducing the Hamiltonian operator, the time-independent Schrödinger equation (TISE) is obtained. Conditions on acceptable wavefunctions are then stated. The method for solution of the TISE is laid out, and the infinite potential well is solved as an example. A discussion of the physical interpretation of the solutions — as energy eigenstates in position space — follows. This provides an opportunity to introduce overall and relative phases, and to point out that the latter are responsible for time evolution. The TISE is then applied to the important problems of scattering from a step potential and tunneling through a rectangular potential barrier. Use of the TISE for such problems greatly simplifies calculations, but it can also instill misconceptions; both aspects are discussed.
The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.
I believe that the author of this note is confusing definite and indefinite integrals. Any indefinite integre 'I I can be reduced to the corresponding definite integral by the proper choice of the limits of integration and the constant of integration. The fact that [5] is a solution of [4] can be looked-up directly from any table of elementary integrals; see, for example, ref. 1. For finite values of z i t is easy to show that [3] and [4] are equivalent to the time-dependent Schrodinger's wave equation by taking the time derivative of both sides of [4]. Moreover, the limit z + 0 cannot be allowed, since from [3] this corresponds to an infinite energy E. The meaning of the unit of imaginary numbers in on the right-hand side of [4] is given immediately after this equation.