# Homework Help: Finding Twofold Degeneracy

1. Mar 29, 2012

### Rapier

1. The problem statement, all variables and given/known data

A photon is confined by impenetrable barriers to a three-dimensional box of dimensions A = 300 nm by B = 400 nm by C = 500 nm, where A, B, and C are along the x, y, and z-axes, respectively.

d) Let the triplet of integers (nx, ny, nz) denote an energy level. If two triplets give the same energy, then the state is two-fold degenerate. Find the lowest energy which is two fold degenerate.

e) What are the two triplets which give the energy of part d? Enter the triplet with the smaller value of nx here as a three digit integer.

f) Enter the triplet with the larger value of nx here also as a three digit integer.

g) The lowest state that is three-fold degenerate is given by the three triplets, (3,1,10), (3,7,5) and (6,1,5). Find the energy of the state.

2. Relevant equations

E = hc/2 * [ (x/300nm)^2 + (y/400nm)^2 + (z/500nm)^2]^1/2
hc = 1240 eV

3. The attempt at a solution

I need to find where :

E(A,B,C) = E(x,y,z)
hc/2 * [ (x/300nm)^2 + (y/400nm)^2 + (z/500nm)^2]^1/2 = hc/2 * [ (A/300nm)^2 + (B/400nm)^2 + (C/500nm)^2]^1/2
(x/300nm)^2 + (y/400nm)^2 + (z/500nm)^2 = (A/300nm)^2 + (B/400nm)^2 + (C/500nm)^2
400x^2 + 225y^2 + 144z^2 = 400A^2 + 225B^2 + 144C^2

My math is failing me. I know that I can do this with trial and error and I've created a spreadsheet to map out the values, but it is very labour intensive and I'm sure there is an easier method. Any direction would be greatly appreciated.