# Showing energy is expectation of the Hamiltonian

1. Jan 10, 2014

### Unredeemed

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

The vector $\psi =\psi_{n}$ is a normalized eigenvector for the energy level $E=E_{n}=(n+\frac{1}{2})\hbar\omega$ of the harmonic oscillator with Hamiltonian $H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}$. Show that:

$E=\frac{\mathbb{E}_{\psi}[P^{2}]}{2m}+\frac{1}{2}m\omega^{2}\mathbb{E}_{\psi}[X^{2}]$

2. Relevant equations

Time independent Schrodinger: $H\psi=E\psi$

3. The attempt at a solution

Am I wrong in thinking it's as simple as taking the expectation of both sides? I feel like I must be as that gives $\mathbb{E}_{\psi}[E]=\frac{\mathbb{E}_{\psi}[P^{2}]}{2m}+\frac{1}{2}m\omega^{2}\mathbb{E}_{\psi}[X^{2}]$ which isn't quite right.

But I can't see how else I'd do it?

EDIT: I believe the extra information is needed to answer the rest of the question (this is just part 1).

2. Jan 10, 2014

### Staff: Mentor

I guess you mean (using your notation)
$$\mathbb{E}_{\psi}[H]=\frac{\mathbb{E}_{\psi}[P^{2}]}{2m}+\frac{1}{2}m\omega^{2}\mathbb{E}_{\psi}[X^{2}]$$

I would start from $E = \mathbb{E}_{\psi}[H]$ and develop the right-hand-side until the desired result (the equation just above) is obtained.

3. Jan 10, 2014

### maajdl

Indeed, taking the EV of both sides is the solution.
Disturbingly simple ;>) .

4. Jan 10, 2014

### Unredeemed

We're then asked to show by considering $<\psi\mid (P\pm im\omega X)^{k}\psi>$ for k=1,2 and using orthogonality properties of eigenvectors that:

$\mathbb{E}_{\psi}(P)=0=\mathbb{E}_{\psi}(X)$ and $\mathbb{E}_{\psi}(P^{2})=m^{2}\omega^{2}\mathbb{E}_{\psi}(X^{2})=mE$

I've shown that $(P-im\omega X)*=P+im\omega X$ and I can see that $<\psi\mid (P- im\omega X)\psi>=\mathbb{E}_{\psi}(P)-im\omega\mathbb{E}_{\psi}(X)$ etc.

But I really can't see how to get to the answer?

Am I supposed to show that $<\psi\mid (P- im\omega X)\psi>=0=<\psi\mid (P+ im\omega X)\psi>$? Because I can't seem to do that...

5. Jan 10, 2014

### Staff: Mentor

6. Jan 10, 2014

### Unredeemed

Not yet. But they do come in the next chapter of the notes?

7. Jan 10, 2014

### Staff: Mentor

Then I don't get how you're supposed to solve the problem. Which textbook are you using?

8. Jan 10, 2014

### Unredeemed

It's my university notes - we could well have to use ladder operators to solve this. In the past we've occasionally had questions that can only be covered with material from the next chapter. What in specific should I be looking for with regard to ladder operators?

9. Jan 10, 2014

### Staff: Mentor

You should be able to express $P \pm i m \omega X$ in terms of the ladder operators. Once this is done, you can figure out what $(P \pm i m \omega X) \left| \psi \right\rangle$ does, keeping in mind that $\psi$ is actually $\psi_n$.

10. Jan 10, 2014

### Unredeemed

I've got it! Thanks very much.