Solving Ladder Operator Problem w/ 4 Terms

[Sheepattack]

Homework Statement

I have been given the following problem -
the expectation value of p_x⁴ in the ground state of a harmonic oscillator can be expressed as
<p_x⁴> = h⁴/4a⁴ {integral(-infin to +infin w₀^*(x) (AAA+A+ + AA+AA+ + A+AAA+) w₀ dx}

I think I know how to proceed on other examples, but my given examples only have ladder operators with two terms, i.e. AA+ or A+A.
I can then use the commutation relation, AA+ - A+A =1 to remove them.

what has stuck me here is the four ladder operators in a term. I'm totally unsure on how to proceed!

any advice would be greatly appreciated!

 

[ideasrule]

Do you know what A and A+ do to eigenstates? A₊ψ_n = sqrt(n+1)ψ_n+1 and Aψ_n=sqrt(n)*ψ_n-1, so you can just keep applying the appropriate operator until you get down to a constant multiple of the eigenfunction.

 

[praharmitra]

I'd like to suggest a completely different method. Use two things here

1. Schrödinger's equation, which can be rearranged to form
$$p^2\psi = c (E-V)\psi$$ where c is some constant

2. The fact the p is hermitian

$$\left<p^4\right> = \left<\psi |p^4|\psi\right> = \left<p^2\psi|p^2\psi\right> = c^2 \left<(E-V)^2\right>$$

The last part of the above integral can be greatly simplified using the evenness or oddness of the functions $$\psi$$ and V. Try it.

 

[Sheepattack]

Many thanks for both your replies.

I think ideasrule's method was the one I am supposed to follow - managed to get the right answer! A few more clouds lifted...
thanks again