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American Journal of Physics

A novel application of a Fourier integral representation of bound states in quantum mechanics

Guillermo Palma, Ulrich Raff

Published Jan 24, 2011• 12 citations
Publisher

Abstract

A method is developed to find the bound state eigenvalues and eigenfunctions of one-dimensional and rotational symmetric three-dimensional quantum mechanical problems. The method is based on the fact that eigenfunctions of bound states are square integrable. We use this property and a judicious ansatz inspired on its asymptotic behavior to obtain a differential equation that can be solved straightforwardly using a Fourier transform. The main advantage of the method is that it avoids the traditional and tedious convergence analysis using a series representation. The method leads to an integral representation of the wave function and provides insight into concepts such as energy quantization, spectrum degeneracy, and bound states.

DOI
10.1119/1.3531975

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A method is developed to find the bound state eigenvalues and eigenfunctions of one-dimensional and rotational symmetric three-dimensional quantum mechanical problems. The method is based on the fact that eigenfunctions of bound states are square integrable. We use this property and a judicious ansatz inspired on its asymptotic behavior to obtain a differential equation that can be solved straightforwardly using a Fourier transform. The main advantage of the method is that it avoids the traditional and tedious convergence analysis using a series representation. The method leads to an integral representation of the wave function and provides insight into concepts such as energy quantization, spectrum degeneracy, and bound states.
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Bound state solutions of square root power law potential--Wavefunction ansatz method on DDD-dimensional Schrödinger equation
DDD-dimensional Schrödinger equation is addressed for square root power law potential. Bound state unnormalized eigenfunctions and the energy eigenvalues are obtained using wave function ansatz method. Some special cases are studied at the end to ensure the correctness of present work.
The problem of bound states in delta potentials is revisited by means of Fourier transform approach. The problem in a simple delta potential sums up to solve an algebraic equation of degree one for the Fourier transform of the eigenfunction and the problem for more than one delta function also reveals itself to be a simple matter. Quite differently from direct methods, no knowledge about the jump discontinuity of the first derivative of the eigenfunction is required to determine the solution of the problem.