Change for position to energy basis

[lrf]

Homework Statement

Give expressions for computing the matrix elements X_mn of the matrix X representing the position operator X in the energy basis (using eigenvectors of the Harmiltonian operator)

Also told to consider the example of the harmonic oscillator where energy eigenvalues are E_n=(1/2+n)hω

Homework Equations

X_mn=<e_m|X|e_n>

H|e_n>=E_n|e_n>

The Attempt at a Solution

I'm thrown off a bit by how X_mn is defined here - if it is originally in the |x> basis, why is X_mn defined using |e_m> and |e_n>. Shouldn't these be inserted using the completeness relation to convert the matrix into the energy basis representation?

Here goes...
X_mn=<e_m|X|e_n>
X_mn=ƩƩ<e_m|x><x|X|x'><x'|e_n>
X_mn=ƩƩe_m(x)X(x,x')e_n(x')

 

[dextercioby]

You can use the algebraic formulation of the harmonic oscilator and write X in terms of a and a^dagger. The action of a and a^dagger on the standard basis (eigenvectors of N and H) is already known, so...

 

[lrf]

You can use the algebraic formulation of the harmonic oscilator and write X in terms of a and a^dagger.

Sorry, still very confused!

So use -h²/2m d²ψ/dx²+1/2mω²x²ψ=Eψ how?

 

[lrf]

or use 1/2P²+1/2m²X²=H ?

 

[dextercioby]

No, X and P need to be replaced by the raising and the lowering ladder operators, a and $a^{\dagger}$. You should be familiar with them, I hope...

 

[lrf]

yes, got it now, thank you for the push in the right direction!