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Unformatted text preview: well is very deep), the solutions to Equation (1190) asymptote to the roots of . This gives , where is a positive integer, or (1192) These solutions are equivalent to the odd- infinite- depth potential well solutions specified by Equation (1147). Figure : The curves (solid) and Figure : The curves (solid) and (dashed), calculated for . For the case of a totally antisymmetric bound state, similar analysis to the preceding yields (see Exercise 12)
(1193) The solutions of this equation correspond to the intersection of the curve with the curve . Figure 83 shows these two curves plotted for the same value of as that used in Figure 82. In this case, the curves intersect once, indicating the existence of a single totally antisymmetric bound
state in the well. It is, again, apparent, from the figure, that as increases (i.e., as the well becomes deeper)
there are more and more bound states. However, it is also apparent that when
[i.e., becomes sufficiently small then there is no totally antisymmetric bound state. In other words, a very shallow potential well always possesses a totally symmetric bound state, but does not generally possess a totally antisymmetric
bound state. In the limit
(i.e., the limit in which the well becomes very deep), the solutions to
Equation (1193) asymptote to the roots of . This gives , where is a positive integer, or (1194) These solutions are equivalent to...
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