(84) Particle in Finite Square Potential Well

In the limit becomes ie ie the limit in which the

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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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This note was uploaded on 08/25/2013 for the course PHY 315 taught by Professor Staff during the Fall '08 term at University of Texas at Austin.

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