1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 407

b. A harmonic oscillator of frequencyωinddimensions has energy eigen-

values given by

En=

(

n+

d

2

)

̄hω,

but the energy levels become degenerate. The degeneracy of each level is

g(En) =

(n+d−1)!

n!(d−1)!

.

Calculate the canonical partition function, free energy, total energy, and

heat capacity in this case.

10.6. The Hamiltonian for a free particle in one dimension is

Hˆ= pˆ

2

2 m

.

a. Using the free particle eigenfunctions, show that the canonicaldensity

matrix is given by

〈x|e−β

Hˆ

|x′〉=

(

m

2 πβ ̄h^2

) 1 / 2

exp

[

−

m

2 β ̄h^2

(x−x′)

2

]

b. Recall that an operatorAˆin the Heisenberg picture evolves in time ac-

cording to

Aˆ(t) = eiHˆt/ ̄hAˆe−iHˆt/ ̄h.

Now consider a transformation from real timetto an imaginary time

variableτviat=−iτ ̄h. In imaginary time, the evolution of an operator

becomes

Aˆ(τ) = eτHˆAˆe−τHˆ

Using this evolution, derive an expression for the imaginary-time mean-

square displacement of a free particle defined to be

R^2 (τ) =〈[ˆx(0)−xˆ(τ)]^2 〉.

Assume the particle is a one-dimensional box of lengthL. This function

can be used to quantify the quantum delocalization of a particle at tem-

peratureT.

10.7. The following theorem is due to Peierls (1938): Let{|φn〉}be an arbitrary set
of orthonormal functions on the Hilbert space of a quantum system whose
Hamiltonian isHˆ. The functions{|φn〉}are assumed to satisfy the same
boundary and symmetry conditions of the physical system. It follows that
the canonical partition functionQ(N,V,T) satisfies the inequality

Q(N,V,T)≥

∑

n

e−β〈φn|

Hˆ|φn〉

,

where equality holds only if{|φn〉}are the eigenfunctions ofHˆ. Prove this

theorem.