1549380323-Statistical Mechanics Theory and Molecular Simulation

Problems 445

Hˆ|ψ−〉=E−|ψ−〉.

Here,E+< E−, and

E±=E∓

1

2

∆.

a. Show that if ∆/kT <<1 for an ensemble of such systems at temperature

T, then ∆ is given approximately by

∆≈ 2 kT

〈ψ 1 |ρˆ|ψ 2 〉

〈ψ 1 |ρˆ|ψ 1 〉

,

where ˆρis the canonical density matrix of the full HamiltonianHˆ.

b. For a system ofNnoninteracting Boltzmann particles with allowed en-

ergiesE+andE−, calculate the canonical partition function, average

energy, and chemical potential.

c. For a system ofNnoninteracting bosons with allowed energyE+andE−,

calculate the canonical partition function, average energy, and chemical

potential.

11.11. Consider an ideal boson gas and letν=−lnζ. Nearζ= 1, it can be shown
that that following expansion is valid (Huang, 1963):

g 5 / 2 (ζ) =aν^3 /^2 +b+cν+dν^2 +···,

wherea= 2.36,b= 1.342,c=− 2 .612,d=− 0 .730. Using the recursion

formulagn− 1 =−dgn/dν, show that the heat capacity exhibits a discontinuity

given by

lim

T→Tc+

(

∂

∂T

CV

Nk

)

− lim

T→Tc−

(

∂

∂T

CV

Nk

)

=

λ

Tc

and derive an approximate numerical value forλ.

11.12. Reproduce the plots in Figs. 11.3, 11.5, and 11.6.