I have the following problem that I was unable to solve for class, but I had a couple first steps that I started with that I am unable to finish. I know I can't get this since it's already been turned in, but I would like to see if this was even a viable option for doing this proof in the first place.
``Given an orthogonal set of states, {ϕn}, and a Hamiltonian, ˆH, show that the partition function, Qβ, satisfies the following Qβ≥∑nexp{−β⟨ϕn|ˆH|ϕn⟩}
I started by dropping in the identity in the exponential (as eigenstates of the Hamiltonian) ∑nexp{−β∑k⟨ϕn|ψk⟩⟨ψk|ˆH|ϕn⟩}=∑nexp{−β∑kEk|cnk|2}
I realize I didn't make it very far, so this might not be the best way to show this, but it seems manifestly true just by looking at it, but I can't actually show it. Does anyone have any hints about how to continue with this?
Answer
Here is a sketched proof of the inequality. The problem is to show that
∑n⟨ϕn|e−βˆH|ϕn⟩ ?≥ ∑ne−β⟨ϕn|ˆH|ϕn⟩,(1)
where the Hamiltonian ˆH is a selfadjoint operator, and |ϕn⟩ denote orthonormal basis vectors in the Hilbert space of states. The lhs. of eq. (1) is the partition function Tr(e−βˆH). By scaling ˆH, we may assume that β=−1. The inequality (1) would follow if we can show the inequality for each and every summand
⟨ϕn|eˆH|ϕn⟩ ?≥ e⟨ϕn|ˆH|ϕn⟩,(2)
or equivalently, in a simplified notation for fixed n,
⟨eˆH⟩ ?≥ e⟨ˆH⟩.(3)
But eq. (3) is just Jensen's inequality for a convex function (with the exponential function playing the role of the convex function).
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