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Homework Statement
Suppose the Hamiltonian H for a particular quantum system is a function of some parameter λ. Let En(λ) and ψn(λ) be the eigenvalues and eigenfunctions of H(λ). The Feynman-Hellmann Theorem states that
[tex] \frac{\partial E_n}{\partial \lambda} = \left \langle \psi_n \left | \frac{\partial H}{\partial \lambda} \right | \psi_n \right \rangle [/tex]
assuming that either En is non-degnerate or --- if degenerate, that the ψn's are the "good" linear combinations of the degenerate eigenfunctions.
a) Prove the Feynman-Hellman Theorem Hint: Use Equation 6.9
b) Apply it to the 1D Harmonic Oscillator (i) using λ=ω (this yields a formula for <V>), (ii) using λ = ħ (this yields <T>), and (iii) using λ = m (this yields a relation between <T> and <V>.
Homework Equations
Equation 6.9 is for the first order correction to the energy, given H' is the perturbation:
[tex] E_n^1 = \langle \psi_n^0 | H^\prime | \psi_n^0 \rangle [/tex]
The Attempt at a Solution
I said to represent a perturbation as a small change in λ:
[tex] H(\lambda + \delta \lambda) = H(\lambda) + \frac{\partial H}{\partial \lambda} \delta \lambda [/tex]
to first order.
Similarly
[tex] E_n(\lambda + \delta \lambda) = E_n(\lambda) + \frac{\partial E_n}{\partial \lambda} \delta \lambda [/tex]
So, since the first order term in the expansion of the Hamiltonian above is just H', and the first order term in the expansion of the energy above is just E1n, equation 6.9 becomes:
[tex] \frac{\partial E_n}{\partial \lambda} \delta \lambda = \left \langle \psi_n \left | \frac{\partial H}{\partial \lambda} \delta \lambda \right | \psi_n \right \rangle [/tex]
Then I just divided both sides by δλ to get the result. What I'm wondering is, I just made this up. Is it a valid proof? Wikipedia has another method that makes use of the definition E = <H> and nothing else:
http://en.wikipedia.org/wiki/Hellmann-Feynman_theorem
However, this method doesn't use the given equation 6.9. That's all I'm asking about for now. I have a question about part b as well, but I'll wait until we get part a out of the way.
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