a particle that moves in three dimensions is trapped in a deep spherically symme

Question

a particle that moves in three dimensions is trapped in a deep spherically symmetric potential V(r): V(r)=0 at r<r0 and V(r) approaches infinity at r is greater than or equal to r0, where r0 is a posotive constant. The ground state wave function is a spherically symmetric, so the radial wave function u(r) satisfies the one-dimensional Schrodinger energy eigenvalue equation, with the boundary condition u(0)-0. Explain why, in the potential well the wave function is forced to vanish at r=r0. Using the known boundary conditions on the radial wave function u(r) at r=0 and r=r0, find the ground state energy of the particle in this potential well.

Explanation / Answer

A) the question is why the wavefunction must vanish (the angular parts are irrelevant; only u(r) really need vanish) at r=r0 where the potential jumps to be unboundedly large. The wavefunction does not "seep into" the region of infinite potential. The reason this must be can be seen in Schroedinger's equation: What happens when we try to put in V=infinity? What would d^2/dr^2*u or E have to be to satisfy the equation, and why would this be physically ruled out? On the other hand, if we just demand u=0 starting at r0, then the equation does not even need to be solved in a non-trivial way...it is trivially solved already.

B), this kind of reasoning is similar in many 1-dimensional problems. First think of just the well region where V=0. Then in that region you'd get free particle plane waves as solutions to the equation. But of course the particle isn't free because the V=0 region is not all there is to this particle's world; it sees the V=infinity jump. But we know that the conditions on the function u(r) are: u(r=0)=0=u(r=r0). So that just restricts the types of wavefunctions u(r) can be: Not just any exp(ikr) type plane wave, but a subset of them...what type of waveforms? Remember that the waveform you write down must be chosen so that the boundary conditions u(r=0)=0=u(r=r0) could be satisfied. You will also have to write down conditions on k such that the conditions are satisfied, just like in other 1-dimensional problems

If V is at infinity, then C= (2m/h-bar^2)*(V(x)-E)=(1/phi)*(d^2/dx^2) approaches infinity as well which means that wave function bends far away from the axis and the ground state function approaches zero.=> u=0, since u=r*phi_0

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