\[ U(x)= \begin{cases} \infty, & |x| > \frac{a}{2} \\ 0, & \frac{-a}{2} < x < \frac{a}{2} \end{cases} \] 

 \[ \varepsilon_n = \mbox{energy eigenvalues} \] 

 \[ \varepsilon_n = \frac{h^2 n^2}{8ma^2},n=1,2,... \] 

 \[ U(x) = \frac{1}{2} k x^2 \] 

 \[ \varepsilon_n = (n + \frac{1}{2})\hbar \omega(k), n = 0, 1, 2, ... \] 

 \[ 0 \le l \le n-1 \] 

 \[ -l \le m \le l \] 

 \[ 3d_{x^2-y^2}, 3d_{z^2-r^2} \] 

 \[ n_x,n_y,n_z=1, 2, 3 ,... \] 

 \[ n_x^2+n_y^2+n_z^2 = R^2 \] 

 \[ R^2 = \frac{8ma^2 \varepsilon}{h^2} \] 

 \[ \varepsilon > 0 \] 

 \[ \varepsilon = \varepsilon_x + \varepsilon_y + \varepsilon_z \] 

 \[ \phi (\epsilon) = \frac{\pi}{6} \left ( \frac{8 m a^2 \epsilon}{h^2} \right )^{\frac{3}{2}} \] 

 \[ w( \epsilon, \Delta \epsilon ) = \frac {\pi}{6} \left ( \frac{m a^2}{h^2} \right )^{\frac{3}{2}} \left ( \left (\epsilon + \Delta \epsilon \right )^{\frac{3}{2}} - \epsilon^{\frac{3}{2}} \right ) \]

 \[ w( \epsilon, \Delta \epsilon ) = \frac{\pi}{6}\left ( \frac{8 m a^2 \epsilon}{h^2} \right )^{\frac{3}{2}} \left ( \left (1 + \frac{\Delta \epsilon}{\epsilon} \right )^{\frac{3}{2}} - 1 \right ) \] 

 \[ w( \epsilon, \Delta \epsilon ) = \frac{\pi}{4} \left ( \frac{8 m a^2 \epsilon}{h^2} \right )^{\frac{3}{2}} \epsilon^{\frac{3}{2}} \Delta \epsilon \] 

 \[ m=10^{-22} g \] 

 \[ a=10 cm \] 

 \[ \Delta \epsilon = 0.01 \epsilon \] 

 \[ w( \epsilon, \Delta \epsilon ) \approx 10^{28} \] 

 \[ \varepsilon > 0 \] 

 \[ \varepsilon = \varepsilon_x + \varepsilon_y + \varepsilon_z \]