- #1

brasidas

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## Homework Statement

Show that, [tex] [a, \hat H] = \hbar\omega, [a^+, \hat H] = -\hbar\omega [/tex]

## Homework Equations

For the SHO Hamiltonian [tex] \hat H = \hbar\omega(a^+a - \frac{\ 1 }{2}) [/tex] with [tex][a^+, a] = 1 [/tex][a, b] = -[b, a]

## The Attempt at a Solution

I have tried the following:

[tex] [a, \hat H] = a\hat H - \hat Ha = \hbar\omega ( (aa^+a - \frac{\ 1 }{2}a) - (a^+a - \frac{\ 1 }{2})a )

= \hbar\omega (aa^+a -a^+aa) = \hbar\omega [a, a^+] a = - \hbar\omega a [/tex]

And this is nothing like [tex] \hbar\omega [/tex] I am supposed to get. Could anyone point out where I have gone wrong?

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