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Quantum Mechanics of One‐Dimensional Two‐Particle Models. Electrons Interacting in an Infinite Square Well

Diestler, Dennis J. and McKoy, Vincent (1967) Quantum Mechanics of One‐Dimensional Two‐Particle Models. Electrons Interacting in an Infinite Square Well. Journal of Chemical Physics, 47 (2). pp. 454-467. ISSN 0021-9606. http://resolver.caltech.edu/CaltechAUTHORS:20151028-114741809

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Abstract

Solutions of Schrödinger's equation for the system of two particles bound in a one‐dimensional infinite square well and repelling each other with a Coulomb force are obtained by the method of finite differences. For the case of a 4.0‐a.u. well, the energy levels are shifted above those of the noninteracting‐particle model by as much as a factor of 4 although the excitation energies are only about 50% greater. The analytical form of the solutions is also obtained and it is shown that every eigenstate is doubly degenerate due to the "pathological'' nature of the one‐dimensional Coulomb potential. This degeneracy is verified numerically by the finite‐difference method. The properties of the model system are compared with those of the free‐electron and hard‐sphere models; perturbation and variational treatments are also carried out using the hard‐sphere Hamiltonian as a zeroth‐order approximation. The lowest several finite‐difference eigenvalues converge from below with decreasing mesh size to energies below those of the "best'' linear variational function consisting of hard‐sphere eigenfunctions. The finite‐difference solutions in general give expectation values and matrix elements more accurately than do the other approximations.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1063/1.1711916DOIArticle
http://scitation.aip.org/content/aip/journal/jcp/47/2/10.1063/1.1711916PublisherArticle
Additional Information:© 1967 American Institute of Physics. (Received 14 November 1966) We are grateful to Dr. Russell M. Pitzer and Dr. William A. Goddard for helpful discussions and also to Mr. Nicholas W. Winter for valuable discussions and assistance in some of the computations. We thank Mr. David Cartwright for providing the computer programs to generate the three-dimensional plots on Fig. 4. National Science Foundation Predoctoral Fellow 1964-1967.
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NSF Predoctoral FellowshipUNSPECIFIED
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Caltech Gates and Crellin Laboratories of Chemistry3434
Record Number:CaltechAUTHORS:20151028-114741809
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20151028-114741809
Official Citation:Quantum Mechanics of One‐Dimensional Two‐Particle Models. Electrons Interacting in an Infinite Square Well Diestler, Dennis J. and McKoy, Vincent, The Journal of Chemical Physics, 47, 454-467 (1967), DOI:http://dx.doi.org/10.1063/1.1711916
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ID Code:61682
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:28 Oct 2015 22:06
Last Modified:13 Feb 2019 18:04

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