The quantum mechanical behavior of a particle in a double well defies our intuition based on classical reasoning. Not surprisingly, an asymmetry in the double well will restore results more consistent with the classical picture. What is surprising, however, is how a very small asymmetry can lead to essentially classical behavior. In this paper, we use the simplest version of a double-well potential to demonstrate these statements. We also show how this system accurately maps onto a two-state system, which we refer to as a “toy model.”

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To be more precise, many textbooks work through in some form or other the eigenvalues of a symmetric double well potential. They will also at least sketch the wave functions corresponding, for example, to the two lowest eigenvalues. As is well known, the eigenvalues are nearly degenerate, and the ground state is as shown in Fig. 1(a) while the first excited state is simply the antisymmetric version of this. The discussion of the asymmetric double well potential is absent from almost all undergraduate textbooks, so of course not even a prescription for determining the eigenvalues is provided. The tacit assumption is that there is very little change in the eigenvalues from the symmetric case, and this is correct, as indicated by the numbers we have given in the text. A further assumption is that the eigenstates also suffer very little change from the symmetric case, and this is incorrect, and the main point of this paper. Also note that almost all textbooks provide a discussion of two-state systems as an illustration of what happens in the symmetric double-well potential, and some even discuss the asymmetric case (see, for example, the next two references), but these tend to fixate on the energies and not the wave functions.

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Note that the ratio of eκb/t remains of order unity, going from 1.3 to 2.1 as V0 changes from 500 to 1000.

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