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only those functions F(x, E) which yield a V independent of E can be considered as solu- tions of a wave-equation. The alternative interpretation suppose that V in Eq. (1) is really independent of E, but dependent on another parameter (call it W) which -happens to be numerically equal to E; but then F(x, E) is not in general a solution of (1), but only if E= W. The origi- nal goal (the construction of a wave-equation, all of whose solutions shall be known) has thus been missed. To put it in another way, F(x, El) and F(x, E2) are solutions of two different wave-equations, corresponding to two physical systems which differ in both their potential and total energies. In order to obtain a complete set of solutions for any one physical problem (single value of the parameter W) it will be necessary to find a function 4,(x, E, W). This function may satisfy the condition 4(x, E, E) = F(x, E), but there are other solutions which do not, since electrons may move in either of two directions along the x-axis. The conclusion is that Wilson has not found a complete set of characteristic func- tions for
Physical Review – American Physical Society (APS)
Published: May 15, 1930
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