- #1
dirtyaldante
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Hi. I'm a 3rd year undergraduate studying Applied Physics and I'm having some trouble with a problem concerning the Hydrogen Atom. This is my first post so please forgive the sloppy equations. Not really used to writing this stuff out without an equation editor handy! Anyway, the problem:
1.Determine the expectation value of the potential energy [V(r) = (e^2)/(4.pi.epsilon0.r)] in the 1s (ground state) of the Hydrogen atom.
2.What is the expectation value of r for an electron in the 1s state of the Hydrogen atom?
<V(r)> = INT [PSI*|V(r)^|PSI]
n=1, l=0, ml=0
So the method I used to solve other expectation values (<L^2>,<Lz>,<E> etc...) was to use the appropriate operator upon the square of the wavefunction in question, ie: PSI(n,l,ml).
However I can't find operators for the Potential(V(r)) or r(position I suppose) that are related to the quantum numbers n,l and ml like in the other expectation values I solved. e.g.
L^2^=l(l+1)hbar
E^=13.6/n^2
Lz^=ml.hbar
So then I would square the wavefunctions and their coefficients inside the integral and then use the operators appropriately, in this case on the PSI(1,0,0) for the 1s configuration.
Am I simply missing a fundamental operator here? Or do I have to try arrange the operators myself? Rearranging the Schrodinger equation in terms of V(r)PSI was the only thing I could think of and even then I couldn't navigate through the wavefunctions of the separate parts.
Another idea I had was to use the Laguerre polynomial to redefine the wavefunction, but I'm unsure on how to proceed after that.
Is my method just completely off for these expectation values? I have been scratching my head all day over this and would really appreciate help if anyone can offer it.
Homework Statement
1.Determine the expectation value of the potential energy [V(r) = (e^2)/(4.pi.epsilon0.r)] in the 1s (ground state) of the Hydrogen atom.
2.What is the expectation value of r for an electron in the 1s state of the Hydrogen atom?
Homework Equations
<V(r)> = INT [PSI*|V(r)^|PSI]
n=1, l=0, ml=0
The Attempt at a Solution
So the method I used to solve other expectation values (<L^2>,<Lz>,<E> etc...) was to use the appropriate operator upon the square of the wavefunction in question, ie: PSI(n,l,ml).
However I can't find operators for the Potential(V(r)) or r(position I suppose) that are related to the quantum numbers n,l and ml like in the other expectation values I solved. e.g.
L^2^=l(l+1)hbar
E^=13.6/n^2
Lz^=ml.hbar
So then I would square the wavefunctions and their coefficients inside the integral and then use the operators appropriately, in this case on the PSI(1,0,0) for the 1s configuration.
Am I simply missing a fundamental operator here? Or do I have to try arrange the operators myself? Rearranging the Schrodinger equation in terms of V(r)PSI was the only thing I could think of and even then I couldn't navigate through the wavefunctions of the separate parts.
Another idea I had was to use the Laguerre polynomial to redefine the wavefunction, but I'm unsure on how to proceed after that.
Is my method just completely off for these expectation values? I have been scratching my head all day over this and would really appreciate help if anyone can offer it.