How do I simplify <n|n> for Hermetian Operators?

[Biest]

Hi,

So for an harmonic oscillator we need to to find the average value for x^4, so &lt;n|x^4|n&gt;. We split it up to \sum_m |&lt;n|x^2|n&gt;|^2 and recognize that only m = n+2, m=n and m = n-2 can be used. We find that

m=n

\frac{\hbar}{2m\omega}&lt;n|\hat{A}\hat{A^\dagger}|n&gt;

m= n+2 \frac{\hbar}{2m\omega}&lt;n+2|\hat{A^\dagger}\hat{A^\dagger}|n&gt;

m = n-2

\frac{\hbar}{2m\omega}&lt;n-2|\hat{A}\hat{A}|n&gt;

So we can reduce it all to

&lt;n|x^4|n&gt; = \frac{1}{4} \hbar^2 \omega^2 (2n+1)^2 + \frac{1}{2} (\frac{\hbar}{m \omega})^2 &lt;n|n&gt;How I simplify the &lt;n|n&gt;.

Thanks.

Cheers,

Biest

 

[quantumdude]

The basis states are orthonormal, so &lt;n|m&gt;=\delta_{mn}.