Problem 2.44: If two (or more) distinct solutions to the (time-independent) Schrödinger equation have the same energy E, these states are said to be degenerate. For example, the free particle states are doubly degenerate, with one solution representing motion to the right and the other representing motion to the left. However, we have never encountered normalizable degenerate solutions, and this is no accident. Prove the following theorem: In one dimension (-∞ < x < ∞), there are no degenerate bound states. [Hint: Suppose there are two solutions, ψ1 and ψ2, with the same energy E. Multiply the Schrödinger equation for ψ1 by ψ2, and the Schrödinger equation for ψ2 by ψ1, and subtract them to show that (ψ2 dψ1/dx - ψ1 dψ2/dx) is a constant. Use the fact that for normalizable solutions, ψ approaches 0 as x approaches ±∞, to demonstrate that this constant is in fact zero. Conclude that ψ2 is a multiple of ψ1 and hence the two solutions are not distinct.]