Uncertainty of coherent states of QHO

[BiotFartLaw]

Homework Statement

What is the uncertainty in x and p of the coherent state
$|z> = e^{-|z|^2/2} \sum\frac{z^{n}}{\sqrt{n!}} |n>$

Homework Equations

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The Attempt at a Solution

This seems pretty straight-forward to me. You just find the expected value of (let's say) x:
$<z|x|z>$, and use the relation between <x> and uncertainty. And you use the ladder operator definition of x:
$\hat{x}= \sqrt{ \frac{\hbar}{m \omega}}( \hat{a} + \hat{a}^{+})$

When I work out the math using the fact that $\hat{a}|n> = \sqrt{n}|n-1>$ and $\hat{a^{+}}|n> = \sqrt{n+1}|n+1>$
I run in to:
A*<n-1|n> + B*<n|n+1>
(Where A and B are summations).

So my question is: what are <n-1|n> and <n|n+1>? They are energy eigenkets for the QHO But are they orthogonal? And if they are...what do? (Since everything then goes to 0). And if they aren't, what is their inner product?

Thanks.

 

[fzero]

When I work out the math using the fact that $\hat{a}|n> = \sqrt{n}|n-1>$ and $\hat{a^{+}}|n> = \sqrt{n+1}|n+1>$
I run in to:
A*<n-1|n> + B*<n|n+1>
(Where A and B are summations).

So my question is: what are <n-1|n> and <n|n+1>? They are energy eigenkets for the QHO But are they orthogonal? And if they are...what do? (Since everything then goes to 0). And if they aren't, what is their inner product?

Thanks.

Indeed $\langle n | n\pm 1\rangle =0$. However, you've oversimplified things. What you should really find is an expression like

$$\sum_{n, n'} \left( A_{nn'} \langle n' | n -1 \rangle + B_{nn'} \langle n' | n +1 \rangle \right).$$

There are nonvanishing terms appearing in the sum which you can evaluate using $\langle n | n' \rangle = \delta_{nn'}.$