Square potentials (cube and sphere)

Rectangular box

Problem:

An electron is trapped in a two-dimensional, infinite rectangular well of width Lx = 800 pm and Ly = 1200 pm.  What is the electron's ground state energy?

Solution:

Problem:

Find the eigenvalues and eigenfunctions for a particle in a box with sides a, b, c by solving the time independent Schroedinger equation (-ħ2/(2m))∇2Φ(r) + U(r)Φ(r)  = EΦ(r).  Do not just write down your answer but derive it.

Solution:

Problem:

A particle of mass m is in a cubical well of side L, corresponding to the potential energy function
U(r) = 0 if |x|, |y| and |z| < L/2,  U(r) = ∞ otherwise.
(a)  What is its ground-state energy?
(b)  What is the energy of the first excited state?  Is this energy level degenerate or non-degenerate?  Explain!
(c)  Suppose 20 identical, non-interacting particles of mass m and spin ½ are in this well.  What is the ground state energy of this system?  Is this ground state degenerate or non-degenerate?  Explain!

Solution:


Spherical box

Problem:

(a)  Determine the energy levels and normalized wave functions ψ(r) of a particle with zero angular momentum in a spherical "potential well" U(r) = 0, (r < a), U(r) = ∞, (r > a).
(b)  Determine the average value of r and r2 for each of the energy eigenstates with l = 0.

Solution:

Problem:

Assume that the potential energy of the deuteron is given by
U(r) = -U0, r < r0; U(r) = 0, r ≥ r0.
(a)  Show that the ground state of the deuteron possesses zero orbital angular momentum (l = 0).  Since this is true for any central potential, you may not need the detailed nature of the square well potential.
(b)  Assume that l = 0 and estimate the value of U0 under the additional condition that the value of the binding energy is much smaller than U0.

Solution:

Problem:

Let us represent the interaction between two He atoms by the sum of a short-range repulsive part and a long-range attractive part.
U(r) = + ∞   for r < a,
U(r) = -|U0|   for a < r < b,
U(r) = 0   for r > b.
(a)  Find the condition satisfied by a, b, and U0 for a bound state to exist in the relative motion of two He atoms.
(b)  Experimental evidence indicates that He does not solidify at atmospheric pressure, even at ~0 K.  What do you think this implies about the effective helium-helium interaction?

Solution: