In which of the following cases would you expect degeneracy of ...

[shenderson6]

In which of the following cases would you expect degeneracy of the first excited state? Hint: Make sure to consider all three dimensions.
  a. A neutral helium atom initially in its ground state in a magnetic field.
  b. A particle trapped inside a sphere of radius R.
  c. A particle trapped inside a three dimensional box of lengths L₁ > L₂ > L₃.
  d. Identical particles, with one in an infinite square-well potential and the other in a finite square-well potential with walls of potential V₀ high but of the same width as the infinite potential well.

Question 2

A particle moves along the x-axis in the positive direction with energy E. It encounters a barrier with potential V > E at x = 0 and extending to x = L. Is there any chance for the particle make it to the other side?
 

 

a. Yes. The wave functions allow for a small probability for the particle to pass through the barrier.
  b. Yes, but only if the particle increases in energy so that E > V.

 

c. Yes. The particle will pass by unaffected in both classical and quantum mechanical consideration.
  d. No. An incoming particle would have negative kinetic energy within the potential barrier.
  e. No. The particle will transmit some energy but the particle will always reflect off the barrier.

 

[CourtneyCNorton]

Answer to Question 1

b.
In all three dimensions the lengths of the box are the same, therefore the energies of the first excited state in each dimension are the same. The state is degenerate.

Answer to Question 2

a.
The wave functions for a finite square-well potential allow a small probability for the particle to be in the classically forbidden region where the energy of the particle is not greater than the potential barrier, which is what would be necessary if the particle were to stay in classically defined regions.

[shenderson6]

TYSM

[kilada]

Excellent