Consider a quantum mechanical particle of mass $m$ moving in one dimension, in a potential well

$$V(x)=\left\{\begin{array}{cr} \infty, & x<0 \\ 0, & 0<x<a \\ V_{0}, & x>a \end{array}\right.$$

Sketch the ground state energy eigenfunction $\chi(x)$ and show that its energy is $E=\frac{\hbar^{2} k^{2}}{2 m}$, where $k$ satisfies

$$\tan k a=-\frac{k}{\sqrt{\frac{2 m V_{0}}{\hbar^{2}}-k^{2}}} .$$

[Hint: You may assume that $\chi(0)=0 .]$