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The Schrödinger Equation in Three Dimensions

Published Apr 13, 2006• 0 citations
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Abstract

Abstract Most “real-world “ applications of quantum mechanics arise in three- dimensional systems where the Schrödinger equation for one particle takes the formwhere the Hamiltonianare intrinsically no more difficult as the use of relative and center-of-mass coordinates,just as in one dimension (Section 14.3), leads to an effective one-particle Schrödinger equationfor the relative coordinaterrel with the reduced mass µ as the mass parameter.

DOI
10.1093/oso/9780198530978.003.0016

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Abstract Most “real-world “ applications of quantum mechanics arise in three- dimensional systems where the Schrödinger equation for one particle takes the formwhere the Hamiltonianare intrinsically no more difficult as the use of relative and center-of-mass coordinates,just as in one dimension (Section 14.3), leads to an effective one-particle Schrödinger equationfor the relative coordinaterrel with the reduced mass µ as the mass parameter.
In this work, the quantum-mechanical problem of the motion of two material points of different masses on a three-dimensional sphere with a non-fixed position of the center of mass of the system is formulated on the basis of the previously solved classical problem. It is shown that the established Schrödinger equation includes two different reduced masses, depending on the distance between the points. For the case of the interaction potential of points, depending only on the distance between them, this equation allows the separation of variables into a radial, depending on the relative distance and both the reduced masses and the spherical part. The equation for the spherical part depends only on one of the above reduced mass and allows one to formulate and solve the problem of a rigid rotator - the distance between the points is fixed. The solution and spectrum of the problem of a rigid rotator are found. It is shown that the spectrum of the system has an upper limit that does not depend on the distance between points, in contrast to the spectrum in a flat space.