Expectation value in the ground state

[Roodles01]

Homework Statement

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Hi, could someone look at the attachment & comment on whether I'm anywhere near getting the expectation value correct, please.

In the grnd state;
1. terms such as AA†A†A, with lowering operator on RHS has zero expectation value,
2. terms such as AA†A†A† with uneven numbers of lowering or raising op's has zero expectation value.

Noting these points, the expectation value of p_x⁴ in ground state of harmonic oscilator can be;
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<p_x⁴> = [∫ ψ₀^*(x) (AAA†A† + AA†AA† + A†AAA†) ψ₀(x) dx]
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ψ₀(x) is grnd-state energy eigen-function.
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Show that in grnd-state
<p_x⁴> = 3*hbar / 4a⁴

Homework Equations

commutation formula AA†− A†A=1,
Normal order A†A

The Attempt at a Solution

See attached

 

[mark726]

document
Thank you for your post. Your approach to finding the expectation value of px4 in the ground state of a harmonic oscillator is correct. Your use of the commutation formula and normal order is appropriate and leads to the correct result of <px4> = 3*hbar / 4a4.

One suggestion I have is to provide a bit more explanation for each step in your solution. This will help readers understand your thought process and reasoning behind each step. Additionally, it would be helpful to include a brief summary of the main points of your solution at the end, to tie everything together and make it easier for readers to follow.

Overall, your solution is clear and correct. Keep up the good work!