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Expectation value of position of wavepacket

by Werbel22
Tags: expectation, position, wavepacket
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Werbel22
#1
Mar23-10, 04:18 AM
P: 8
Hello, this is just a general question, how is <x^2> evaluated, if

<x> = triple integral of psi*(r,t).x.psi(r,t).dr (this is the expectation value of position of wavepacket)

Is it possible to square a triple integral? Is <x^2> the same as <x>^2 ?

I'm only wondering how the squared works in this situation, I would understand how to use <x> if the square wasn't there.

Thank you!
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nickjer
#2
Mar23-10, 09:37 AM
P: 675
You cannot square the integral. The way it is written is (in 1 dimension):

[tex]\left<x^2\right> = \int \psi(x)^{\dagger}x^2\psi(x) dx[/tex]

It will be different in most cases from <x>^2. For example,

[tex]\int x^2 dx = \tfrac13 x^3 \neq \left(\int x dx\right)^2 = \tfrac14 x^4[/tex]

So you are unable to take the square outside of the integral.
Werbel22
#3
Mar23-10, 11:23 PM
P: 8
Got it, thank you very much!


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