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Spherical Infinite Well

  1. Sep 29, 2014 #1
    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. jcsd
  3. Sep 29, 2014 #2

    Simon Bridge

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