1549380323-Statistical Mechanics Theory and Molecular Simulation

406 Quantum ensembles

d. What are the expectation values of the operatorsSˆx,Sˆy, andSˆzat time

t for this case?

e. What is the fluctuation or uncertainty inSˆxat timet? Recall that

∆Sˆx=

√

〈Sˆ^2 x〉−〈Sˆx〉^2

f. Suppose finally that the density matrix is given initially by a canonical

density matrix:

ρ ̃(0) =

e−β

Hˆ

Tr(e−βHˆ)

What is ̃ρ(t)?

g. What are the expectation values ofSˆx,Sˆy, andSˆzat timet?

10.3. Consider a one-dimensional quantum harmonic oscillator of frequencyω, for

which the energy eigenvalues are

En=

(

n+

1

2

)

̄hω n= 0, 1 , 2 ,....

Using the canonical ensemble at temperatureT, calculate〈xˆ^2 〉,〈pˆ^2 〉, and the

uncertainties ∆xand ∆p.

Hint: Might the raising and lowering operators of Section 9.3 be useful?

∗10.4. A weakly anharmonic oscillator of frequencyωhas energy eigenvalues given

by

En=

(

n+

1

2

)

̄hω−κ

(

n+

1

2

) 2

̄hω n= 0, 1 , 2 ,....

Show that, to first order inκand fourth order inr=β ̄hω, the heat capacity

in the canonical ensemble is given by

C

k

=

[(

1 −

r^2

12

+

r^4

240

)

+ 4κ

(

1

r

+

r^3

80

)]

(Pathria, 1972).

10.5. Suppose a quantum system has degenerate eigenvalues.

a. Ifg(En) is the degeneracy of the energy levelEn, show that the expression

for the canonical partition function must be modified to read

Q(N,V,T) =

∑

n

g(En)e−βEn.