# Expectation value in the ground state

1. May 19, 2013

### Roodles01

1. The problem statement, all variables and given/known data
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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 px4 in ground state of harmonic oscilator can be;
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<px4> = [∫ 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
<px4> = 3*hbar / 4a4

2. Relevant equations
commutation formula AA†− A†A=1,
Normal order A†A

3. The attempt at a solution
See attached

#### Attached Files:

• ###### expectation value.jpg
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Last edited: May 19, 2013