Thomas-Fermi approximation

The Thomas-Fermi approximation is an approximation used to express the degeneracy of states of a nonrelativistic quantum ideal gas as a differential instead of a discrete sequence. This transforms sums into integrals and allows for analytical computation using, for instance, the partition function of a quantum ensemble.

The approximation holds for a large number of particles and high volume, which means that it automatically applies in the thermodynamic limit.

Description#

A quantum ideal gas is composed of free particles inside of a perfect enclosure of sides $L$ , which can be seen as an infinite square well in some number of dimensions. Since the particles are free, their eigenfunctions are plane waves: $\psi\propto e^{i\mathbf{p}\cdot \mathbf{r}/\hbar}$ . Calling $\mathbf{n}=(n_{x},n_{y},n_{z})$ the direction of the momentum $\mathbf{p}$ , we get (using periodic boundary conditions, they are necessary for this proof!)

\mathbf{p}=\frac{2\pi \hbar}{L} \mathbf{n}

where $\hbar$ is the reduced Planck constant. This is like saying that there is one possible state of $\mathbf{p}$ for every "volume" $\left( \frac{2\pi \hbar}{L} \right)^{3}$ (the volume is in momentum phase space). There is only a discrete number of possible states, enumerated by the components of the direction (in this sense, it is not really a direction as it is not normalized. It's more like a quantum number position vector in phase space). We can use this fact to count quantum states. Call $G(\varepsilon)$ the number of states up to an energy $\varepsilon=p^{2}/2m$ where $p=\lvert \mathbf{p} \rvert$ for simplicity. If the number of particles is very large ( $N\gg 1$ ), the number of states is itself going to be very large. More importantly, the separation between states is going to be so tiny with respect to the internal energy of the system that we can think of $G(\varepsilon)$ as approximately continuous. Because of this, we can allow ourselves to take its derivative

g(\varepsilon)=\frac{ \partial G(\varepsilon) }{ \partial \varepsilon }

This is known as the density of states (DOS) function, and it can be used to turn integrals in three momentum dimensions into integrals of one energy dimension.

A set value of $\varepsilon$ defines a sphere in momentum space (it is the locus of all points for which $p^{2}/2m\leq\varepsilon$ ). All momentum values contained in the sphere are $\varepsilon_{p}<\varepsilon$ . We want to count these. Thankfully, we know both the volume of a sphere and the volume of each state, so the number of states is the ratio of these (i.e. how many state volumes fit inside the sphere volume):

G(\varepsilon)=\frac{\frac{4\pi}{3}p^{3}}{\left( \frac{2\pi \hbar}{L} \right)^{3}}=\frac{\frac{4\pi}{3}(2m\varepsilon)^{3/2}}{(2\pi \hbar)^{3}}V= \frac{\sqrt{ 2 }}{3\pi ^{2}} \frac{(m\varepsilon)^{3/2}}{\hbar ^{3}}V

If we take the derivative of $\varepsilon$ we get

\boxed{g(\varepsilon)=\frac{Vm^{3/2}}{\sqrt{ 2 }\pi ^{2}\hbar ^{3}}\sqrt{ \varepsilon }}

This leads to the conclusion that the density of states in three dimensions is proportional to the square root of energy. This is a fact that is seen in several systems, from atoms to stars.

$N$ dimensions#

More generally, an $N$ dimensional ideal gas behaves like

G(\varepsilon)\propto p^{d}\propto (\sqrt{ \varepsilon })^{d}\propto \varepsilon^{d/2},\qquad g(\varepsilon)\propto \varepsilon^{d/2-1}

This leads to some weird behavior in lower dimensions, where $d=2$ implies that state density is completely independent of energy and in $d=1$ , it is inversely proportional to it.

In general, the DOS function has the form

\boxed{g(\varepsilon)=C_{\alpha}\varepsilon^{\alpha-1}}

where $\alpha=d/2$ is a real number that is half the dimension of the system. For instance, in the 3D gas above we have $\alpha=3/2$ .

Consequences#

The intended use case for this approximation is turning sums over quantum states to integrals over energy. For instance, in a Fermi or Bose gas, the number of particles is

N=\sum_{\varepsilon} \frac{1}{z^{-1}e^{\beta \varepsilon}\pm 1}

In the Thomas-Fermi approximation, this becomes

N=\int_{0}^{\infty} \frac{g(\varepsilon)}{z^{-1}e^{\beta \varepsilon}\pm 1}d\varepsilon=C_{\alpha}\int_{0}^{\infty} \frac{\varepsilon^{\alpha-1}}{z^{-1}e^{\beta \varepsilon}\pm 1}d\varepsilon

This integral can't be solved, but the Sommerfeld expansion provides a strong approximation for high $z$ . They can be analytically written using Fermi and Bose functions. A more general notation could be

\sum_{\varepsilon} \langle n_{\varepsilon} \rangle \to \int_{0}^{\infty}g(\varepsilon)\langle n_{\varepsilon} \rangle d\varepsilon

Thomas-Fermi approximation

Description#

NNN dimensions#

Consequences#

$N$ dimensions#