- #1
andre220
- 75
- 1
Homework Statement
For a quantum oscillator find all non-zero matrix elements of the operators ##\hat{x}^3## and ##\hat{x}^4##
Homework Equations
##\hat{x} = \sqrt{\frac{\hbar}{2m\omega}}\left(a+a^\dagger\right)##
##a^\dagger |n\rangle = \sqrt{n+1}|n+1\rangle##
##a |n\rangle = \sqrt{n}|n-1\rangle##
The Attempt at a Solution
Okay so first it is necessary to compute ## \langle n |\hat{x}^3|n'\rangle## and then from that outcome determine the non-zero elements.
$$x^3 = \left(\frac{\hbar}{2m\omega}\right)^{3/2}(a + a^\dagger)^3 =\left(\frac{\hbar}{2m\omega}\right)^{3/2}(a + a^\dagger)(a _ + a^\dagger)(a + a^\dagger) = $$
$$=\left(\frac{\hbar}{2m\omega}\right)^{3/2}(a a a + a^\dagger a a + a a^\dagger a + a^\dagger a^\dagger a + a a a^\dagger + a^\dagger a a^\dagger + a a^\dagger a^\dagger + a^\dagger a^\dagger a^\dagger) $$
So then we compute
$$\langle n|\hat{x}^3|n' \rangle = $$
$$\left(\frac{\hbar}{2m\omega}\right)^{3/2} \langle n|(a a a + a^\dagger a a + a a^\dagger a + a^\dagger a^\dagger a + a a a^\dagger + a^\dagger a a^\dagger + a a^\dagger a^\dagger + a^\dagger a^\dagger a^\dagger)|n'\rangle$$
And then from here I am having trouble evaluating this, however, I am pretty sure that the majority of the terms above go to zero.
Any help would be greatly appreciated.