# Spherical Infinite Well

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1. Sep 29, 2014

### zephyr5050

1. The problem statement, all variables and given/known data
A particle of mass $m$ is constrained to move between two concentric hard spheres of radii $r = a$ and $r = b$. There is no potential between the spheres. Find the ground state energy and wave function.

2. Relevant equations
$$\frac{-\hbar^2}{2m} \frac{d^2 u}{dr^2} + [V(r) + \frac{-\hbar^2}{2m} \frac{\ell (\ell + 1)}{r^2}]u = Eu$$

3. The attempt at a solution
The relevant equation here is the radial equation component of the time independent schroedinger equation for a central potential, where $u(r) \equiv rR(r)$. Effectively, this is an infinite square well potential such that inside the concentric spheres the potential is $0$ and in the ground state $\ell = 0$ so our effective differential equation becomes
$$\frac{d^2 u}{dr^2} = -\frac{2mE}{\hbar^2}u \equiv -k^2 u$$
with the solution
$$u(r) = rR(r) = A sin(kr) + B cos(kr)$$
We can apply the boundary conditions that $R(a) = 0$ and $R(b) = 0$. However, my problem comes from the fact that I don't know how to get anything out of these boundary conditions. Most of the time, the problem is that $a = 0$ and that boundary condition gives the quantization of $k$, but here I don't see how to pull out that quantization. Is there some part of this problem that I'm missing?

2. Sep 29, 2014

### Simon Bridge

If you just look at it you can see the shape the solutions have to have. Sketch the first few on your diagram.