Infinite Sphere Well

$$V(r)=\{\begin{array}{ll}0,& r\le a\\ \infty ,& r>a\end{array}$$

• The potential for the infinite spherical well

${\psi }_{nlm}(r,\theta ,\varphi )=A_{nl}{j}_l\left({\beta }_{Nl}\frac{r}{a}\right)Y_l^m(\theta ,\varphi )$

• $A_{nl}$ is a normalization constant
• $j_l$ is the Bessel function
• ${\beta }_{Nl}$ are the $N$th zeroes of the $l$th spherical Bessel function
• $Y_l^m(\theta ,\varphi )$ is the angular wave function
• $n$ is the principal quantum number which orders the energy levels described by the below equation based on ${\beta }_{Nl}$
  
  

$E_{Nl}=\frac{{\hslash }^2}{2m{a}^2}{\beta }_{Nl}^2$

• Energy for $N$th zero of the $l$th spherical Bessel function