# Degeneracy of each of energies - Quantum

1. Nov 18, 2013

### Dassinia

Hello, I dont understand something in this exercice and i have another question:

1. The problem statement, all variables and given/known data
Use separation of variables in Cartesian coordinates to solve the infinite cubical well (or "particle in a box"):

V (x, y, z) = { O,if x, y, z are all between 0 and a;
∞,otherwise.

(a) Find the stationary state wave functions and the corresponding energies.

(b) Cali the distinct energies E 1, E1, E3, ... , in order of increasing energy. Find E1, E2, E3 , E4 , Es, and E6. Determine the degeneracy of each ofthese energies (that is, the number of different states that share the same energy). Recall (Problem 2.42) that degenerate bound states do not occur in one dimension, but
they are common in three dimensions.

(c) What is the degeneracy of E 14, and why is this case interesting?

II/ I was wondering for the lowering and increasing operators a- and a+
When we have a-a+=1/(hω) H-1/2
H is considered as an operator ?

2. Relevant equations

3. The attempt at a solution

a. The energy : E=h2π2/(2ma2) (nx2+ny2+nz2)
ψ(x,y,z)=(2/a)3/2sin(kxx)sin(kyy)sin(kzz)
with ki=ni2π2/a2

b. The thing that I dont get is that according to the correction that we can find here:
http://physicspages.com/2013/01/05/infinite-square-well-in-three-dimensions
Why the ground state is nx=ny=nz=1 so n=3?
Whyt not for n=nx2+ny2+nz2=1
the combinations 1,0,0 0,1,0 0,0,1 ?

Thanks !

2. Nov 19, 2013

### Staff: Mentor

What happens to the wave function when $n_\alpha=0$ ($\alpha \in \{x,y,z\}$)?

3. Nov 19, 2013

### Dassinia

Hi,
When what ?

4. Nov 19, 2013

### Staff: Mentor

When any of $n_x$, $n_y$ or $n_z$ is equal to zero.

You ask why we can't take, for instance, $n_x = 1$, $n_y = 0$, $n_z = 0$ as the ground state. What does the wave function for that state look like?

5. Nov 19, 2013

### Dassinia

Oh right, it's equal to 0 everywhere, and this is not possible !
Thank you !