Physics Friday 89

Continuing from last week’s discussion of the thermodynamics of the ideal Fermi fluid, we now explore the classical limit.

First, I introduce the physical property known as fugacity. Fugacity is defined as , and can be seen as an alternative parameter to μ. Note that constant ξ is equivalent to holding the product βμ constant. In this post on the grand canonical ensemble, I showed that , where ; thus, using the chain rule to rewrite in terms of fugacity,
.

The key difference between the ideal Fermi fluid and the classical ideal gas is that a given fermion particle is not free to occupy an arbitrarily chosen state, as the Pauli exclusion principle forbids it from states that are already occupied. At high temperature or low density, though, the probability of occupation for each orbital state becomes small, thus reducing the effect of the Pauli exclusion principle; so we see the gas becomes classical when the density is sufficiently low or temperature sufficiently high, so that the occupancy for each state is small; this happens when . For this to happen for all energies, we need , or in other terms, ; the classical regime occurs when the fugacity is small. (Since β>0, we see that in the classical regime, the Fermi level must be at very negative values). In this range, we see that since ,
.
Note that the energy dependence in this classical approximation is the Boltzmann factor, confirming that in the classical limit, we approach the Maxwell-Boltzmann distribution of the classical ideal gas.

Now, we find the needed physical conditions (density and temperature) so that the fugacity is small. We found last week that the number of particles is
.
In our classical limit of small fugacity, we see that . Thus, we see that
,
where is a quantity with units of length, and g=2S+1 is the number of different spin orientations, here equal to 2.
Similarly,
,
which is the classical result for an ideal gas (with only translational modes for the gas molecules), and which combines with our to give the ideal gas law .

Solving our particle number formula for the fugacity, we see: , where is the particle density. Thus, the classical condition means that , and the classical quantum boundary occurs when .

Now, let us examine the physical interpretation of the quantity . This is called the “thermal de Broglie wavelength,” and is equivalent to the de Broglie wavelength of a particle of mass m and velocity , roughly that of the average particle of an ideal gas at temperature T. The thermal de Broglie wavelength decreases with increasing temperature.
Note that for a gas of particle density n, the average interparticle distance will be approximately , and so the classical condition is equivalent to saying that the thermal de Broglie wavelength must be significantly smaller than the average interparticle distance, again confirming the need for high temperature or low density.

Next week, we will explore what happens when an ideal Fermi fluid is firmly in the quantum regime.

Physics Friday 44

Quantum Mechanics and Momentum
Part 1: The De Broglie Relations

Planck’s relation tells us that the energy of a photon is proportional to the frequency: E=hν, where h is Planck’s constant. In terms of angular frequency ω, we use ω=2πν to find that , where is called the reduced Planck constant, or the Dirac constant.
Classical electrodynamics of an electromagnetic wave and the relativistic energy-momentum relation for a particle with zero rest mass both give the same result for the momentum of a photon: that , and so the momentum is
, where λ is the wavelength. Solving for wavelength, we have . This wavelength can be found for any particle, not just photons, and is called the de Broglie wavelength (see here for details on the historical context and experimental support for this).

Now, let us consider the corresponding angular wavenumber . Then we have

or .

The two equations, and , are known as the de Broglie relations, and the latter one will be important in the later parts of this series.