- #1
zephyr5050
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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?