Wave functions and energy eigenvalues for $E<0$

Our second one-dimensional problem is the motion of a particle in a potential well of finite depth $V$ and width $2a>0$, i.e. a potential

$$V(x)=\left\{ \begin{array}{cc} 0, & -\infty<x\le -a \\ -V<0, & -a <x\le a \\ 0 & a < x < \infty \end{array} \right.$$ (104)

According to our general equation, the wave functions for energies $-\vert V\vert< E =-\vert E\vert<0$ must have the form

$$\psi(x)=\left\{ \begin{array}{cc} a_1e^{\kappa x}+b_1e^{-\kappa x... ...<x\le a \\ a_3e^{\kappa x}+b_3e^{-\kappa x}, & a <x< \infty \end{array} \right.$$ (105)

where

$$k = \sqrt{({2m}/{\hbar^2})\left(-\vert E\vert+V\right)},\quad \kappa = \sqrt{({2m}/{\hbar^2}) \vert E\vert}.$$ (106)

The wave function has to vanish for $x\to \pm \infty$ which can only be fulfilled if $b_1=a_3=0$.