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. Schrodinger'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

 

[paranormal]

I would suggest approaching this problem by first breaking down the expression into smaller parts. Since the expectation value is given in terms of the integral of a product of wavefunctions, we can start by expanding the product using the properties of ladder operators. This would result in a series of terms with different combinations of ladder operators, similar to the examples you have encountered before.

Next, we can use the commutation relation you mentioned, AA+ - A+A = 1, to simplify each term. This would involve rearranging the order of the operators and possibly combining terms to eliminate the ladder operators with two terms.

Once we have simplified each term, we can then use the commutation relation again to remove the remaining ladder operators with four terms. This may require multiple steps and careful rearranging of the operators.

Ultimately, the goal is to reduce the expression to a simpler form that can be integrated easily. From there, we can solve for the expectation value of px4 in terms of the given parameters.

In summary, breaking down the expression into smaller parts and using the commutation relation multiple times will help us solve this problem. It may also be helpful to consult with your peers or a professor for additional guidance and support. Good luck!