Expectation value of position of wavepacket

[Werbel22]

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!

 

[nickjer]

You cannot square the integral. The way it is written is (in 1 dimension):

$$\left<x^2\right> = \int \psi(x)^{\dagger}x^2\psi(x) dx$$

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

$$\int x^2 dx = \tfrac13 x^3 \neq \left(\int x dx\right)^2 = \tfrac14 x^4$$

So you are unable to take the square outside of the integral.

 

[Werbel22]

Got it, thank you very much!