QM: Prove Dirac Eigenstates of SHO are Orthonormal

[hanks721]

Homework Statement

Show that the eigenstates of the simple harmonic oscillator using Dirac notations are orthonormal.

Homework Equations

The Attempt at a Solution

 

[malawi_glenn]

attempt? ...

 

[hanks721]

Sorry malawi_glenn. I didn't post the attempt because I don't know how to write equations like everybody else does!
But I'll try to explain.
The first thing that came up to my mind is, to prove orthonormality I have to show,
<psi_m|psi_n>=0
and that is,

<psi_0|[(a^dagger)^m]/sqrt(m!)*[(a^dagger)^n]/sqrt(n!)|psi_0>=0

and then, I don't know how to evaluate further.

Once again, I'm sorry. I hope you don't get annoyed by my poor explanation.
Thanks

 

[malawi_glenn]

we can read it, do your best effort

now the dual correspondance to (a^dagger |psi>) is (<psi| a)

So, start over again.

 

[malawi_glenn]

also "show that"... is quite unprecise, since you construct states |n> according to (a^dagger)^n]/sqrt(n!)|psi_0> in order to make orthonormality.

What may be used? The canonical commutator relation only? Or what? Boring excerice I would say ;-)