Can Degenerate States be Expressed as a Linear Combination of Orthogonal States?

[quantumfireball]

Homework Statement

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>

Homework Equations

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>?

 

[borgwal]

[tex]<n',l',m'|n,l,m>=\delta_{n'n}\delta_{l'l}\delta_{m'm}[/tex]

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

 

[quantumfireball]

[tex]<n',l',m'|n,l,m>=\delta_{n'n}\delta_{l'l}\delta_{m'm}[/tex]

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

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.

 

[borgwal]

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