- #1
atomicpedals
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Homework Statement
I'm re-hashing a problem from my notes; given the wave function
[tex]\psi(x)=Ne^{-(x-x_0)/2k^2}[/tex]
Find the expectation value <x>.
Homework Equations
The normalization constant N for this is in my notes as [tex]N^2=1/\sqrt{2 \pi k^2}[/tex] [tex]N=1/(2\pi k^2)^{(1/4)}[/tex] It should be solved through a u substitution.
The Attempt at a Solution
[tex]<x>=\int \psi(x)^{*}x \psi(x) dx=\int x |\psi(x)|^{2} dx[/tex]
[tex]=\int x N^2 e^{-(x-x_0)^2/k^2} dx[/tex]
[tex]=N^2 \int x e^{-(x-x_0)^2/k^2} dx[/tex]
[tex]u=x-x_0, du=x dx[/tex]
[tex]<x>=N^2 \int e^{-u^2/k^2}du[/tex]
This is where I stumble. Assuming I wrote the problem down correctly the expectation value should come out to <x>=x0. But I'm just not seeing it which leads me to believe that I'm not solving the final integral correctly.
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