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Chapter 7: Q. 7.15 (page 270)

For a system obeying Boltzmann statistics, we know what μis from Chapter 6. Suppose, though, that you knew the distribution function (equation 7.31) but didn't know μ. You could still determine μ by requiring that the total number of particles, summed over all single-particle states, equal N. Carry out this calculation, to rederive the formula μ=-kTlnZ1/N. (This is normally how μ is determined in quantum statistics, although the math is usually more difficult.)

Short Answer

Expert verified

The formulaμ=-kTlogeZ1Nis derived.

Step by step solution

01

Step 1. Given Information 

We are given a formula,

μ=-kTlnZ1/N

02

Step 2. Deriving the formula

According to Boltzmann statistics, the average number of particles in the single state is determined by Boltzmann distribution function,

n¯Boltzmann=e-(ε-μ)kT

N is the total number of particles, summed over all single-particle states,

N=∑sexp-(ε(s)-μ)kT=∑sexpμkTexp-ε(s)kT=expμkT∑sexp-ε(s)kT=expμkTZ1

As Z1=∑exp-ε(s)kT,

03

Step 3. Simplifying

Simplifying, we get

NZ1=μkTlogNZ1=μkT-logeZ1N=μkTμ=-kTlogeZ1N

Hence, the formula is derived.

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Most popular questions from this chapter

Consider a degenerate electron gas in which essentially all of the electrons are highly relativistic ϵ≫mc2so that their energies are ϵ=pc(where p is the magnitude of the momentum vector).

(a) Modify the derivation given above to show that for a relativistic electron gas at zero temperature, the chemical potential (or Fermi energy) is given by =

μ=hc(3N/8πV)1/3

(b) Find a formula for the total energy of this system in terms of N and μ.

Consider a gas of nidentical spin-0 bosons confined by an isotropic three-dimensional harmonic oscillator potential. (In the rubidium experiment discussed above, the confining potential was actually harmonic, though not isotropic.) The energy levels in this potential are ε=nhf, where nis any nonnegative integer and fis the classical oscillation frequency. The degeneracy of level nis(n+1)(n+2)/2.

(a) Find a formula for the density of states, g(ε), for an atom confined by this potential. (You may assume n>>1.)

(b) Find a formula for the condensation temperature of this system, in terms of the oscillation frequency f.

(c) This potential effectively confines particles inside a volume of roughly the cube of the oscillation amplitude. The oscillation amplitude, in turn, can be estimated by setting the particle's total energy (of order kT) equal to the potential energy of the "spring." Making these associations, and neglecting all factors of 2 and πand so on, show that your answer to part (b) is roughly equivalent to the formula derived in the text for the condensation temperature of bosons confined inside a box with rigid walls.

Suppose you have a "box" in which each particle may occupy any of 10single-particle states. For simplicity, assume that each of these states has energy zero.

(a) What is the partition function of this system if the box contains only one particle?

(b) What is the partition function of this system if the box contains two distinguishable particles?

(c) What is the partition function if the box contains two identical bosons?

(d) What is the partition function if the box contains two identical fermions?

(e) What would be the partition function of this system according to equation 7.16?

(f) What is the probability of finding both particles in the same single particle state, for the three cases of distinguishable particles, identical bosom, and identical fermions?

In addition to the cosmic background radiation of photons, the universe is thought to be permeated with a background radiation of neutrinos (v) and antineutrinos (v-), currently at an effective temperature of 1.95 K. There are three species of neutrinos, each of which has an antiparticle, with only one allowed polarisation state for each particle or antiparticle. For parts (a) through (c) below, assume that all three species are exactly massless

(a) It is reasonable to assume that for each species, the concentration of neutrinos equals the concentration of antineutrinos, so that their chemical potentials are equal: μν=μν¯. Furthermore, neutrinos and antineutrinos can be produced and annihilated in pairs by the reaction

ν+ν¯↔2γ

(where y is a photon). Assuming that this reaction is at equilibrium (as it would have been in the very early universe), prove that u =0 for both the neutrinos and the antineutrinos.

(b) If neutrinos are massless, they must be highly relativistic. They are also fermions: They obey the exclusion principle. Use these facts to derive a formula for the total energy density (energy per unit volume) of the neutrino-antineutrino background radiation. differences between this "neutrino gas" and a photon gas. Antiparticles still have positive energy, so to include the antineutrinos all you need is a factor of 2. To account for the three species, just multiply by 3.) To evaluate the final integral, first change to a dimensionless variable and then use a computer or look it up in a table or consult Appendix B. (Hint: There are very few

(c) Derive a formula for the number of neutrinos per unit volume in the neutrino background radiation. Evaluate your result numerically for the present neutrino temperature of 1.95 K.

d) It is possible that neutrinos have very small, but nonzero, masses. This wouldn't have affected the production of neutrinos in the early universe, when me would have been negligible compared to typical thermal energies. But today, the total mass of all the background neutrinos could be significant. Suppose, then, that just one of the three species of neutrinos (and the corresponding antineutrino) has a nonzero mass m. What would mc2 have to be (in eV), in order for the total mass of neutrinos in the universe to be comparable to the total mass of ordinary matter?

A star that is too heavy to stabilize as a white dwarf can collapse further to form a neutron star: a star made entirely of neutrons, supported against gravitational collapse by degenerate neutron pressure. Repeat the steps of the previous problem for a neutron star, to determine the following: the mass radius relation; the radius, density, Fermi energy, and Fermi temperature of a one-solar-mass neutron star; and the critical mass above which a neutron star becomes relativistic and hence unstable to further collapse.
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