If two (or more) distinct solutions to the (time-independent) Schrodinger equation have the same energy E, these states are said to be degenerate. For example, the free particle states are doubly degenerate—one solution representing motion to the right, and the other motion to the left. But we have never encountered normalizable degenerate solutions, and this is no accident. Prove the following theorem: In one dimension there are no degenerate bound states. Hint: Suppose there are two solutions, ψ1 and ψ2, with the same energy E. Multiply the Schrodinger equation for ψ1 by ψ1, and the Schrodinger equation for ψ2 by ψ1, and subtract, to show that (ψ1dψ2 – ψ1ψ2/dx) is a constant. Use the fact that for normalizable solutions i/r 0 at ± ∞ to demonstrate that this constant is in fact zero. Conclude that ψ2 is a multiple of ψ1, and hence that the two solutions are not distinct.
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