- #1
Unkraut
- 30
- 1
Homework Statement
Using the Feynman-Hellman theorem, determine the expectation values of 1/r and 1/r^2 for the hydrogen atom.
Homework Equations
Hamiltonian: H=-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+\frac{\hbar^2}{2m}\frac{l(l+1)}{r^2}-\frac{e^2}{4\pi\epsilon_0}\frac{1}{r}
energy eigenvalues: E_n=-\frac{me^4}{32\pi^2\epsilon_0^2\hbar^2(N+l+1)^2}
N is the largest power of the Laguerre polynomial associated with the solution of u_{nl} (and I have actually no idea what that is and I hope I don't need it), l is the azimuthal quantum number and n the principal quantum number.
n=N+l+1
Feynman-Gellman theorem: \frac{\partial E_n}{\partial \lambda}=\langle \psi_n | \frac{\partial H}{\partial \lambda} | \psi_n \rangle for a Hamiltonian which depends on a parameter \lambda
The Attempt at a Solution
For <1/r^2>:
Take the derivative of the Hamiltonian with respect to l:
\frac{\partial H}{\partial l}=\frac{\hbar^2}{2m}\frac{2l+1}{r^2}
\Rightarrow \frac{1}{r^2}=\frac{2m}{\hbar^2(2l+1)}\frac{\partial H}{\partial l}
\Rightarrow \langle\frac{1}{r^2}\rangle=\frac{2m}{\hbar^2(2l+1)}\langle\frac{\partial H}{\partial l}\rangle
=\frac{2m}{\hbar^2(2l+1)}\frac{\partial E_n}{\partial l} (Feynman-Gellman theorem used here)
=... ... straightforward direct calculation which is missing now because texing is so tedious and I think maybe I have made an error in my notes which I must check later.
So, now I have the expectation value of 1/r^2.
But I can't do the same for 1/r. I cannot express 1/r in terms of the derivative of the Hamiltonian with respect to some parameter because in the term \frac{e^2}{4\pi\epsilon_0}\frac{1}{r} there is no parameter.
Seems I need the expectation value of \frac{d^2}{dr^2}, then I can calculate the expectation value of 1/r from the Hamiltonian. My uneducated intuition tells me that maybe that is zero, but that is not a good argument and often my intuition is wrong. I have no idea. Any hint appreciated.