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Kyrios
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Homework Statement
A hydrogen like ion (with one electron and a nucleus of charge Ze) is in the state
[tex] ψ = ψ_{2,0,0} - ψ_{2,1,0} [/tex]
What's the expectation value of \hat{r} (position operator) as a function of Z?
Assuming origin at nucleus.
Homework Equations
for Z=1
[tex] < ψ | \hat{r} | ψ > = -3 \frac{4 π ε_0 \hbar^2}{m e^2} n_z[/tex]
The Attempt at a Solution
Using the values for
[tex] ψ = ψ_{2,0,0} - ψ_{2,1,0} [/tex]
I got
[tex] ψ = \frac{1}{4 \sqrt{2 π a_0 ^3}} e^{- \frac{r}{2 a_0}} ( 2 - \frac{r}{a_0} - \frac{r cos(\theta)}{a_0}) [/tex]
I wouldn't have a clue how to integrate this and I imagine there must be an easier way to find the expectation value.
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