Quantum Mechanics (1) · Problem 3.1

The density matrix formulation

Physics — Quantum Mechanics · central-force-problem-spherical-well

Problem

We have discussed the ``central force problem''. Consider a particle of mass

m

under the influence of the following potential:

V(r) = \begin{cases} V_0, & 0 \leq r \leq a \\ 0, & a < r, \end{cases}

where

V_0

is a constant. Write down the Schr\"odinger equation for the wave function

\psi(\mathbf{x})

. Consider solutions which are simultaneous eigenvectors of

H

\mathbf{L}^2

, and

L_z

. Solve the angular dependence, and reduce the remaining problem to a problem in one variable. Let

E

be the eigenvalue of the Hamiltonian,

H

. Consider the case where

E > V_0

. Solve the Schr\"odinger equation for eigenstates

\psi(\mathbf{x})

. It will probably be convenient to use the quantity

k = \sqrt{2m(E - V_0)}

. Consider the limit as

r \to \infty

for your solutions, and give an interpretation in terms of spherical waves. Repeat the solution for the case where

E < V_0

. It will probably be convenient to use the quantity

K = \sqrt{2m(V_0 - E)}

. Again, consider the limit as

r \to \infty

and give an interpretation, contrasting with the previous case. Hint: You will probably benefit by thinking about solutions in the form of spherical Bessel/Neumann functions, and/or spherical Hankel functions.

Related theory

Central Force Motion Kepler Orbits

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