(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Hello everyone...

I'm kinda stuck with a problem I'm trying to do.

The problem states:

Express the operator [itex]\hat{x}[/itex] by the ladder operators [itex]a_{+}[/itex] and [itex]a_{-}[/itex], and determine the mean of the position [itex]\left\langle x \right\rangle[/itex] in the state [itex]\left| \psi \right\rangle[/itex].

2. Relevant equations

[tex]\left| \psi \right\rangle = \frac{1}{\sqrt{2}}(\left| 3 \right\rangle + \left| 2 \right\rangle)[/tex]

Hamiltonian for a one dimensional harmonic oscillator:

[tex]\hat{H} = \frac{\hat{p}^{2}}{2m}+\frac{1}{2}mw^{2}\hat{x}^{2},[/tex]

where [itex]w[/itex] is the oscillators frequency, and [itex]x[/itex] and [itex]p[/itex] are the operators for position and momentum. The normalized energy eigenfunctions for [itex]H[/itex] is denoted [itex]\left| n \right\rangle[/itex], where [itex]n = 0,1,2,....[/itex] so that:

[tex]\hat{H}\left| n \right\rangle = (n + \frac{1}{2})\hbarw\left| \psi \right\rangle[/tex]

3. The attempt at a solution

The first is easy, since:

[tex]\hat{x} = \sqrt{\frac{\hbar}{2mw}}(a_{+} + a_{-}).[/tex]

My problem is finding the mean of the position.

I tried to do it like this:

[tex]\left\langle x \right\rangle = \sqrt{\frac{\hbar}{2mw}}\int \psi^{*}_{n}(a_{+} + a_{-})\psi_{n} dx[/tex]

And that didn't go well. It got very confusing, so I was not sure if I was on the right track or not. So here I am.

I know the answer should be:

[tex]\left\langle x \right\rangle = \sqrt{\frac{3}{2}}\sqrt{\frac{\hbar}{mw}},[/tex]

but again, I'm kinda lost atm.

So I was hoping any of you could give me a clue. Something in my heads tells me it's pretty simple, but I really can't figure it out right now, so :)

Regards

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# Homework Help: Determine the mean of the position <x> in a state

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