# Change for position to energy basis

1. Nov 29, 2011

### lrf

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

Give expressions for computing the matrix elements Xmn 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 En=(1/2+n)hω

2. Relevant equations

Xmn=<em|X|en>

H|en>=En|en>

3. The attempt at a solution

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

Here goes...
Xmn=<em|X|en>
Xmn=ƩƩ<em|x><x|X|x'><x'|en>
Xmn=ƩƩem(x)X(x,x')en(x')

2. Nov 29, 2011

### 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...

3. Nov 30, 2011

### lrf

Sorry, still very confused!

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

4. Nov 30, 2011

### lrf

or use 1/2P2+1/2m2X2=H ?

5. Nov 30, 2011

### 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...

6. Nov 30, 2011

### lrf

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