Bound States in One Dimension

Abstract

Before beginning with the problems it is worthwhile to point out some features of one-dimensional bound states. First, there is no complication arising from degenerate states. Degeneracy exits when two or more states have the same energy. Although a very important consideration in two- and three-dimensional problems, there is no degeneracy in one-dimensional problems. We can prove this by assuming that the assertion is not true. We assume that $\psi _{1}\left (x\right )$ and $\psi _{2}\left (x\right )$ are linearly independent eigenfuntions that have the same eigenvalue E and write the TISE (time independent Schrödinger equation) in the form

$$\displaystyle{ \dfrac{1} {\psi \left (x\right )} \dfrac{d^{2}\psi \left (x\right )} {dx^{2}} = \dfrac{2m} {\hslash ^{2}} \left [E - U\left (x\right )\right ] }$$