Expectation values of the quantum harmonic oscillator

[Dean Navels]

Homework Statement

Show the mean position and momentum of a particle in a QHO in the state ψ_γ to be:

<x> = sqrt(2ħ/mω) Re(γ)
<p> = sqrt (2ħmω) Im(γ)

Homework Equations

$\psi_{\gamma} (x) = Dexp((-\frac{mw(x-<x>)^2}{2\hbar})+\frac{i<p>(x-<x>)}{ħ})$

The Attempt at a Solution

I put ψ_γ into the equation

<p> = ∫[ψ(x,t)]* (-iħψ'(x)) dx

Which gave me

(-(mω(x-<x>)/ħ)+((<p>)(i/ħ))*ψ_γ which I can't help but feel is taking me further away from what I'm looking for. Are there easier alternative route to doing this?

Thank you in advance

 

[PeroK]

Homework Statement

Show the mean position and momentum of a particle in a QHO in the state ψ_γ to be:

<x> = sqrt(2ħ/mω) Re(γ)
<p> = sqrt (2ħmω) Im(γ)

Homework Equations

ψ_γ = Dexp(-((mω(x-<x>)^2)/(2ħ))+ ((<p>)*(i/ħ)(x-<x>)))

How is $\psi_{\gamma}$ defined? That expression has no $\gamma$ in it, but it does have the expected values of $x$ and $p$.

It's hard to read your latex. I'll post some you can copy in a minute.

 

[PeroK]

Some latex:

$\psi_0 = (\frac{m \omega}{\pi \hbar})^{\frac14}exp(-\frac{mwx^2}{2\hbar})$

$\psi_{\gamma} = ?$

If you reply to this post, it will give you something to cut and paste.

 

[Dean Navels]

Some latex:

$\psi_0 = (\frac{m \omega}{\pi \hbar})^{\frac14}exp(-\frac{mwx^2}{2\hbar})$

$\psi_{\gamma} = ?$

If you reply to this post, it will give you something to cut and paste.

I've edited it accordingly

 

[PeroK]

$\psi_{\gamma} (x) = Dexp((-\frac{mw(x-<x>)^2}{2\hbar})+\frac{i<p>(x-<x>)}{ħ})$

What's $\gamma$?

 

[Dean Navels]

Some latex:

$\psi_0 = (\frac{m \omega}{\pi \hbar})^{\frac14}exp(-\frac{mwx^2}{2\hbar})$

$\psi_{\gamma} = ?$

If you reply to this post, it will give you something to cut and paste.

What's $\gamma$?

i haven't got that information, ψ_γ represents coherent states.

 

[PeroK]

i haven't got that information, ψ_γ represents coherent states.

Given that the required answer (for the expectation values) depends on $\gamma$, it must be a parameter in the state. You need to check the question.

 

[Dean Navels]

Given that the required answer (for the expectation values) depends on $\gamma$, it must be a parameter in the state. You need to check the question.

Just realized I have missed a little bit out,

γ is a complex parameter and
a_ψ_γ(x) = γψ_γ(x)

That's 100% all the information I have now

 

[PeroK]

Just realized I have missed a little bit out,

γ is a complex parameter and
a_ψ_γ(x) = γψ_γ(x)

That's 100% all the information I have now

That's entirely different. That means that $\psi_{\gamma}$ is an eigenstate of the lowering operator, corresponding to eigenvalue $\gamma$.

The expression you quoted for $\psi_{\gamma}$ before makes no sense.

Hint: can you express the position and momentum operators in terms of the raising and lowering operators?

HInt #2: I suspect you can do this using Linear Algebra, without resorting to integration.