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American Journal of Physics

The Quantum-Mechanical Problem of a Particle in Two Adjacent Potential Minima I. Direct Solution

D. S. Carter, G. M. Volkoff

Published Apr 1, 1949• 1 citation
Publisher
DOI
10.1119/1.1989547

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The problem of a particle in two adjacent one-dimensional rectangular, potential "boxes" is an exactly soluble representative of a class of two-minima problems of considerable physical interest which have not been solved exactly. It therefore affords a valuable opportunity for a critical examination of the extent of applicability of perturbation theory methods to such problems. An exact implicit solution of the problem is obtained, and is reduced to explicit approximate form in two important special cases. These approximations are reproduced by perturbation theory methods, and their ranges of validity are demonstrated by comparison with the exact solution. The application of the model to a physical system is demonstrated by using the identical two-box problem as a basis for calculation of some constants of the ammonia molecule.
We find the exact quasiparticle spectrum, elastic S matrix, and free energy for the continuum Kondo problem of k species of electrons coupled to an impurity of spin S. Here, the impurity becomes an immobile quasiparticle sitting on the boundary. The particles are ``kinks,'' which can be thought of as field configurations interpolating between adjacent wells of a potential with k+1 degenerate minima. For the overscreened case k>2S, the impurity in the continuum is a kink as well, which explains the noninteger number of boundary states.