How to Calculate the Expectation Value of H'?

[Denver Dang]

Homework Statement

Calculate the expectation value of $$\hat{H}'$$ in the state $$\psi(x,t=0)$$.

$$\hat{H}'=k(\hat{x}\hat{p}+\hat{p}\hat{x})$$

$$\psi(x,t=0)=A(\sqrt{3}i\varphi_{1}(x)+\varphi_{3}(x))$$,
where $A=\frac{1}{2}$

Homework Equations

The Attempt at a Solution

I know it's found by:
$$\left\langle\psi,\hat{H}'\psi\right\rangle$$,
but it's been so long since I calculated this, so I'm not quite sure how to tackle/calculate it to be honest.

So I hope you might be able to give me some pointers.Regards.

 

[diazona]

To start with, you can see that the state ψ is given in terms of two basis states φ₁ and φ₃. What are the properties of those basis states? Specifically, do you know the expectation values of certain operators in those basis states?

 

[Denver Dang]

To start with, you can see that the state ψ is given in terms of two basis states φ₁ and φ₃. What are the properties of those basis states? Specifically, do you know the expectation values of certain operators in those basis states?

Ehhh, should I know any ? If so, I'm kinda blank. As I said, it's been a while since I did QM calculations, so I'm not really in the game atm.

 

[diazona]

You have to be given some information about the states φ₁ and φ₃. It's impossible to do the problem if you don't know what those states are.

Typically, φ_n is defined to be an energy eigenstate (i.e. an eigenstate of the Hamiltonian), and the number n is assigned so that either φ₁ or φ₀ is the eigenstate with the lowest energy. Maybe you're supposed to assume that. I guess if you don't have any other information, try making that assumption.

Anyway, here's something you can do, even without knowing about the basis states: you know (correctly) that the expectation value is computed as
$$\langle\psi\vert H'\psi\rangle$$
Plug in the definition you're given for ψ and use the distributive property. You should be able to reduce it to a sum of terms of the form
$$\langle\varphi_m\vert H'\varphi_n\rangle$$
where m and n are integers, either 1 or 3.

 

[Denver Dang]

You're right about them being energy eigenstates. Just forgot to mention.
And I think I got it now :)

Thank you very much.