1549380323-Statistical Mechanics Theory and Molecular Simulation

408 Quantum ensembles

Hint: You might find the Ritz variational principle of quantum mechanics

helpful. The Ritz principle states that for an arbitrary wave function|Ψ〉, the

ground-state energyE 0 obeys the inequality

E 0 ≤〈Ψ|Hˆ|Ψ〉

where equality only holds if|Ψ〉is the ground state wave function ofHˆ.

10.8. Prove the following inequality: IfA 1 andA 2 are the Helmholtz free energies

for systems with HamiltoniansHˆ 1 andHˆ 2 , respectively, then

A 1 ≤A 2 +〈Hˆ 1 −Hˆ 2 〉 2

where〈···〉 2 indicates an ensemble average calculated with respect to the

density matrix of system 2. This inequality is known as the Gibbs–Bogliubov

inequality (Feynman, 1998).

∗10.9. A simple model of a one-dimensional classical polymer consists of assigning

discrete energy states to different configurations of the polymer. Suppose the

polymer consists of flat, elliptical disc-shaped molecules that can align either

along their long axis (length 2a) or short axis (lengtha). The energy of a

monomer aligned along its short axis is higher by an amountεso that the

total energy of the molecule isE=nε, wherenis the number of monomers

aligned along the short axis.

a. Calculate the canonical partition functionQ(N,T) for such a polymer

consisting ofNmonomers.

b. What is the average length of the polymer?