Showing energy is expectation of the Hamiltonian

[Unredeemed]

Homework Statement

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:

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

Homework Equations

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

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?

Thanks for your help in advance!

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

 

[DrClaude]

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

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.

 

[maajdl]

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

 

[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...

 

[DrClaude]

Have you seen ladder operators?

 

[Unredeemed]

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

 

[DrClaude]

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

 

[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?

 

[DrClaude]

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 $\psi_n$.

 

[Unredeemed]

I've got it! Thanks very much.