- #1

- 555

- 19

## Homework Statement

I am trying to calculate the expectation value of ##\hat{P}^3## for the harmonic oscillator in energy eigenstate ##|n\rangle##

## Homework Equations

## The Attempt at a Solution

[/B]

##\hat{P}^3 = (i \sqrt{\frac{\hbar \omega m}{2}} (\hat{a}^\dagger - \hat{a}))^3 = -i(\frac{\hbar \omega m}{2})^{\frac{3}{2}}(\hat{a}^\dagger \hat{a}^\dagger \hat{a}^\dagger + \hat{a} \hat{a}\hat{a}^\dagger - \hat{a} \hat{a}^\dagger \hat{a}^\dagger - \hat{a}^\dagger \hat{a} \hat{a}^\dagger - \hat{a}^\dagger \hat{a}^\dagger \hat{a} - \hat{a} \hat{a} \hat{a} + \hat{a} \hat{a}^\dagger \hat{a} + \hat{a}^\dagger \hat{a} \hat{a})##

I have used the trick of adding a zero such that ##\hat{a} \hat{a} \hat{a}^\dagger = \hat{a} \hat{a} \hat{a}^\dagger - \hat{a} \hat{a}^\dagger \hat{a} + \hat{a} \hat{a}^\dagger \hat{a} = \hat{a} [\hat{a}, \hat{a}^\dagger] + \hat{a} \hat{a}^\dagger \hat{a} = \hat{a} + \hat{a} \hat{a}^\dagger \hat{a}## on the terms ##\hat a \hat a \hat{a}^\dagger, \hat a \hat{a}^\dagger \hat{a}^\dagger, \hat{a}^\dagger \hat{a}^\dagger \hat a, \hat{a}^\dagger \hat a \hat a## and ended up with the expression (dropping the prefactors)

##\hat P^3 \propto (\hat{a}^\dagger \hat{a}^\dagger \hat{a}^\dagger + 3 \hat a \hat{a}^\dagger \hat a - 3 \hat{a}^\dagger \hat a \hat{a}^\dagger + \hat a \hat a \hat a)##

which I don't think can be simplified further using the trick above. Are there any tricks I can use to tackle the central and outside terms?

Although it looks like the expectation value of all that will be zero due to the orthonormality of states...