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Some key results for QPM. Microscopic definition of temperature. Distribution of microstates. Microscopic definition of entropy. This applies to an isolated system for which all the microstates are equally probable. Systems in contact with a heat bath. Boltzmann distribution.

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microscopic definition of temperature

Some key results for QPM

Microscopic definition of temperature

Distribution of microstates

Microscopic definition of entropy

This applies to an isolated system for which all the microstates are equally probable

systems in contact with a heat bath
Systems in contact with a heat bath

Boltzmann distribution

Partition Function

This gives the normalised probability of finding the system in a state with energy Ei when in thermal equilibrium at temperature T.

Calculating thermal averages

Canonical Entropy

connection to thermodynamics
Connection to thermodynamics

Starting from the results

leads to a connection between the Helmholtz Free Energy and the partition function:

It follows that

We have a route for calculating these from Z

k states in three dimensions
k-states in three dimensions

Standing wave solutions of the form

so k= (kx,ky,kz) with each component quantised:

Constant energy surface

This gives a cubic mesh with mesh size π/L .

density of states for em modes
Density of states for EM modes

In the case of electromagnetic radiation we have a linear “dispersion relation”, since E = ħω = ħck.

with k= E/ħc and dk= dE/ħc:

Density of single particle states (3d)

Suppose we are dealing with non-relativistic particles where

distribution of speeds for atom in a gas
Distribution of speeds for atom in a gas

Since ħk = mv, we can define the number of states that have speeds in the range v to v+dv as:

The probability that an atom has a speed in the range v to v+dv equals the number of states in that range times the Boltzmann probability of having the corresponding energy:

Substituting the result for Z we get:

maxwell boltzmann distribution
Maxwell-Boltzmann distribution

Velocity distribution for a nitrogen molecule at three temperatures

the planck distribution

Quantum Harmonic Oscillator

The Planck Distribution

The mean number of oscillators is:

The mean energy per oscillator is:

(ignoring zero point energy)

This applies to any system that can be regarded as a collection of QHOs, e.g. photons in a cavity (black body radiation), lattice vibrations, etc.

planck s black body radiation law

Stefan’s Law

Planck’s Black Body Radiation Law

The energy density of EM radiation in a cavity in the frequency range ωtoω+dω is

  • Wien’s law ωmax ≈ T
  • The area under the curve increases asT 4 .
  • Power radiated by a black body is

 = 5.6710-8 Wm-2K-4


Debye approx. for lattice vibrations





Molar heat capacity of copper compared to Debye theory



The dispersion surface ωj(k) (where j=1,2,3 labels the branches) is replaced by the simple linear dispersion ω = ck, where c is an average sound velocity.

Debye introduced a cut-off wavevector kD which is chosen so that there are exactly 3N normal modes. The cut-off in frequency is then ωD = ckD,

  • CV is proportional to T 3 as T→0
  • CV →3NkB for kBT>> ħωD
chemical potential

Systems with variable numbers of particles

Chemical Potential

The Grand (Gibbs) Distribution


This is the equivalent of the Boltzmann distribution for the situation where the number of particles in the system is not fixed.

bose einstein distribution
Bose-Einstein Distribution













Fermi-Dirac Distribution


Bose Einstein Condensation

To calculate the chemical potential for a Bose gas:

We find that μ→ 0 at a finite temperature. This leads to a macroscopic number of particles in the ground state. These particles cannot contribute to CV or the pressure, since they have zero momentum.

In real BEC systems the condensate forms a macroscopic quantum state that displays superfluidity etc.

the fermi energy f t 0


The Fermi energy: εF = μ(T=0)

We need N/2 k states to accommodate N fermions. The total number of fermions is the integral of the occupancy over all the states:

Ground state energy of the fermi gas

Using our result for the Fermi energy:

we find for the ground state energy:

degeneracy pressure of fermi gas

This is called the degeneracy pressure.

Degeneracy pressure of fermi gas

It contributes to the bulk modulus of metals and is the main source of stability of white dwarf and neutron stars.

Heat capacity of fermi gas

  • Heat capacity is linear as long as T << TF.
  • At high temperatures CV approaches the classical value (3/2)NkB above T ~ 2TF.
the classical limit

Expect <n (ε)> << 1

The Classical Limit

In the limit where , we may ignore the 1 in the denominator, compared to the exponential. This leads to:

Classical Distribution Function:



We find that for both BE and FD gases the chemical potential becomes large and negative in the high temperature limit. This is the criterion that defines the classical limit.