# Degeneracy and orthogonality

1. Nov 5, 2008

### quantumfireball

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

Are all electronis states orthonormal?
I mean the degenerate states ie [n,l,m>states corresponding to same energy
for example can one write
[2,0,0>=a[2,1,-1>+b[2,1,0>+c[2,1,+1>

2. Relevant equations

3. The attempt at a solution

for example can one write
[2,0,0>=a[2,1,-1>+b[2,1,0>+c[2,1,+1>???

2. Nov 6, 2008

### borgwal

$$<n',l',m'|n,l,m>=\delta_{n'n}\delta_{l'l}\delta_{m'm}$$

So you cannot write |2,0,0> as a superposition of the three l=1 states.

3. Nov 6, 2008

### quantumfireball

Fine but how do you go about the proof???

forget about in the wavemechanics
just a general proof in bra-ket notation,showing that degenerate states are orthonormal.

4. Nov 6, 2008

### borgwal

They are eigenstates of hermitian operators (namely, angular momentum) with different eigenvalues.

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