Particle in a Sphere *

This is like the particle in a box except now the particle is confined to the inside of a sphere of radius \bgroup\color{black}$a$\egroup. Inside the sphere, the solution is a Bessel function. Outside the sphere, the wavefunction is zero. The boundary condition is that the wave function go to zero on the sphere.

\begin{displaymath}\bgroup\color{black} j_\ell(ka)=0\egroup\end{displaymath}

There are an infinite number of solutions for each \bgroup\color{black}$\ell$\egroup. We only need to find the zeros of the Bessel functions. The table below gives the lowest values of \bgroup\color{black}$ka=\sqrt{2ma^2E\over\hbar^2}$\egroup which satisfy the boundary condition.

\bgroup\color{black}$\ell$\egroup \bgroup\color{black}$n=1$\egroup \bgroup\color{black}$n=2$\egroup \bgroup\color{black}$n=3$\egroup
0 3.14 6.28 9.42
1 4.49 7.73
2 5.72 9.10
3 6.99 10.42
4 8.18
5 9.32
We can see both angular and radial excitations.

Jim Branson 2013-04-22