Expectation value of position of wavepacket

In summary, the expectation value of the squared position of a wavepacket, <x^2>, cannot be evaluated by simply squaring the expectation value of the position, <x>. This is because the integral cannot be squared and the results will be different.
  • #1
Werbel22
8
0
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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  • #2
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.
 
  • #3
Got it, thank you very much!
 

What is the expectation value of position of a wavepacket?

The expectation value of position of a wavepacket is the average value of the position of the wavepacket over time. It is a measure of the most likely position of the wavepacket.

How is the expectation value of position of a wavepacket calculated?

The expectation value of position of a wavepacket is calculated by multiplying the position of each point in the wavepacket by its probability and summing all of these values.

What does the expectation value of position of a wavepacket tell us about the wavepacket?

The expectation value of position of a wavepacket gives us information about the most probable position of the wavepacket. It also tells us about the spread or uncertainty of the wavepacket's position.

Can the expectation value of position of a wavepacket change over time?

Yes, the expectation value of position of a wavepacket can change over time as the wavepacket evolves and spreads out. As the wavepacket spreads, the expectation value of position will also shift.

How is the expectation value of position of a wavepacket related to the uncertainty principle?

The expectation value of position of a wavepacket is related to the uncertainty principle in that it gives us information about the uncertainty or spread of the wavepacket's position. According to the uncertainty principle, the more precisely we know the position of a wavepacket, the less certain we are about its momentum, and vice versa.

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