Hermetian Operators-Simplify

  • Thread starter Biest
  • Start date
  • #1
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Hi,

So for an harmonic oscillator we need to to find the average value for [tex] x^4[/tex], so [tex] <n|x^4|n> [/tex]. We split it up to [tex] \sum_m |<n|x^2|n>|^2 [/tex] and recognize that only m = n+2, m=n and m = n-2 can be used. We find that

m=n

[tex] \frac{\hbar}{2m\omega}<n|\hat{A}\hat{A^\dagger}|n>[/tex]

m= n+2


[tex] \frac{\hbar}{2m\omega}<n+2|\hat{A^\dagger}\hat{A^\dagger}|n>[/tex]

m = n-2

[tex] \frac{\hbar}{2m\omega}<n-2|\hat{A}\hat{A}|n>[/tex]

So we can reduce it all to

[tex] <n|x^4|n> = \frac{1}{4} \hbar^2 \omega^2 (2n+1)^2 + \frac{1}{2} (\frac{\hbar}{m \omega})^2 <n|n>[/tex]


How I simplify the [tex] <n|n> [/tex].

Thanks.

Cheers,

Biest
 
Last edited:

Answers and Replies

  • #2
Tom Mattson
Staff Emeritus
Science Advisor
Gold Member
5,500
8
The basis states are orthonormal, so [itex]<n|m>=\delta_{mn}[/itex].
 

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