Ethan's Notes

    Infinite Sphere Well

    #Physics

    $$

    V(r)=\begin{cases}
    0, & r\leq a \\\infty, & r>a
    \end{cases}
    $$

    • The potential for the infinite spherical well

    $\displaystyle \psi_{nlm}(r,\theta,\phi)=A_{nl},j_{l}\left( \beta_{Nl} \frac{r}{a} \right)Y_{l}^{m}(\theta,\phi)$

    • $\displaystyle A_{nl}$ is a normalization constant
    • $\displaystyle j_{l}$ is the Bessel function
    • $\displaystyle \beta_{Nl}$ are the $\displaystyle N$th zeroes of the $\displaystyle l$th spherical Bessel function
    • $\displaystyle Y_{l}^{m}(\theta,\phi)$ is the angular wave function
    • $\displaystyle n$ is the principal quantum number which orders the energy levels described by the below equation based on $\displaystyle \beta_{Nl}$
      400
      400

    $\displaystyle E_{Nl}=\frac{\hbar ^{2}}{2ma^{2}}\beta ^{2}_{Nl}$

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