How Do You Determine the Ground State Energy in a Spherical Infinite Well?

[zephyr5050]

Homework Statement

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.

Homework Equations

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

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?

 

[Simon Bridge]

however, my problem comes from the fact that I don't know how to get anything out of these boundary conditions.

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