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Unformatted text preview: 1 APPENDIX 2: BOUND STATES IN ONE DIMENSIONAL PROBLEMS A. Bound state energy levels are non-degenerate in one dimension. Let us consider the hypothesis that there are two linearly independent eigenfunctions u 1 and u 2 corresponding to the same energy eigenvalue E . The energy eigenvalue problem for a particle of mass m moving in a one-dimensional potential V ( x ) leads immediately to- ~ 2 2 mu 1 ( x ) d 2 dx 2 u 1 ( x ) = E- V ( x ) =- ~ 2 2 mu 2 ( x ) d 2 dx 2 u 2 ( x ) . Hence u 2 ( x ) d 2 dx 2 u 1 ( x )- u 1 ( x ) d 2 dx 2 u 2 ( x ) = 0 , which may be written d dx u 2 ( x ) d dx u 1 ( x )- u 1 ( x ) d dx u 2 ( x ) = 0 so that u 2 ( x ) d dx u 1 ( x )- u 1 ( x ) d dx u 2 ( x ) = constant . We can evaluate the LHS at x , and since the bound states u i 0 there, the LHS vanishes and the constant is therefore zero. The resulting equation 1 u 1 ( x ) d dx u 1 ( x ) = 1 u 2 ( x ) d dx u 2 ( x ) has solutions u 1 ( x ) = cu 2 ( x ), where c is a constant. The two solutions are thus physically equivalent and there isis a constant....
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