Results of the quantization of the Schrödinger equation for free fermions in 1, 2, and 3 dimensions.

A simple model for metals is the free electron model where the potential energy of the electrons is zero and the electron-electron interactions are ignored. This is equivalent to any system of noninteracting fermions with zero potential energy. In this model the thermodynamic properties only depend on one parameter, the particle density n. In the table below, n denotes the number of particles per meter in one-dimension, the number of particle per square meter in two-dimensions, and the number of particles per cubic meter in three dimensions. In the table many quantities are given twice: once in terms of the density of states at the Fermi energy D(EF) and once in terms of the electron density n. The expressions with D(EF) were derived using the Sommerfeld expansion and should be more accurate than the simple free electron model which only depends on the electron density n. Derivations (in German) of these results can be downloaded for 1-D, 2-D, and 3-D.

1-D Schrödinger equation for a free particle

2-D Schrödinger equation for a free particle

3-D Schrödinger equation for a free particle

Eigenfunction solutions

Eigenvalues of the translation operator

Dispersion relation

Density of states

Density of states

Fermi energy EF

Chemical potential μ

Internal energy distribution

Internal energy

Specific heat


Helmholtz free energy


Bulk modulus

\begin{align} h &= u + P \end{align}
\begin{align} h &\approx \left(nE_F+\frac{\pi^2D(E_F)}{2}(k_BT)^2\right) \quad \mathrm{J\,m^{-1}}\\ &\approx \left(\frac{\pi^2 \hbar^2 n^3}{8m}+\frac{2m}{\hbar^2 n}(k_BT)^2\right)\quad\mathrm{J\,m^{-1}} \end{align} \begin{align} h &\approx \left(nE_F+\frac{\pi^2D(E_F)}{3}(k_BT)^2\right) \quad \mathrm{J\,m^{-2}}\\ &\approx \left(\frac{\pi \hbar^2 n^2}{m}+\frac{\pi m}{3\hbar^2}(k_BT)^2\right) \quad\mathrm{J\,m^{-2}} \end{align} \begin{align} h &\approx \left(nE_F+\frac{5\pi^2D(E_F)}{18}(k_BT)^2\right) \quad \mathrm{J\,m^{-3}}\\ &\approx \left(\frac{\hbar^2}{2m}(3\pi^2)^{\frac{2}{3}}n^{\frac{5}{3}}+\frac{5 m (3\pi^2n)^{\frac{1}{3}}}{18\hbar^2}(k_BT)^2\right) \quad\mathrm{J\,m^{-3}} \end{align}
Specific heat
\begin{align} c_p &= \left(\frac{\partial h}{\partial T}\right)_p \end{align}
\begin{align} c_p &\approx \pi^2 D(E_F)k_B^2 T \qquad\mathrm{J\,K^{-1}\,m^{-1}} \\ &\approx \frac{4m}{\hbar^2 n}k_B^2 T \qquad\mathrm{J\,K^{-1}m^{-1}} \end{align} \begin{align} c_p &\approx \frac{2\pi^2 D(E_F)}{3}k_B^2 T \qquad\mathrm{J\,K^{-1}m^{-2}} \\ &\approx \frac{2\pi m}{3\hbar^2}k_B^2 T \qquad\mathrm{J\,K^{-1}m^{-2}} \end{align} \begin{align} c_p &\approx \left(\frac{5\pi^2D(E_F)}{9}k_B^2 T\right) \qquad \mathrm{J\,K^{-1}m^{-3}}\\ &\approx \left(\frac{5 m (3\pi^2n)^{\frac{1}{3}}}{9\hbar^2}k_B^2 T\right)\qquad\mathrm{J\,K^{-1}m^{-3}} \end{align}
Gibbs energy
\begin{align} g &= u + P - Ts \\ &= h - Ts \notag \end{align}
\begin{align} g &\approx \left(nE_F+\frac{\pi^2D(E_F)}{6}(k_BT)^2\right) \quad \mathrm{J\,m^{-1}}\\ &\approx \left(\frac{\pi^2 \hbar^2 n^3}{8m}+\frac{2m}{3\hbar^2 n}(k_BT)^2\right)\quad\mathrm{J\,m^{-1}} \end{align} \begin{align} g &\approx nE_F \qquad \mathrm{J\,m^{-2}}\\ &\approx \frac{\pi \hbar^2 n^2}{m}\qquad\mathrm{J\,m^{-2}} \end{align} \begin{align} g &\approx \left(nE_F-\frac{\pi^2 D(E_F)}{18}(k_BT)^2\right) \quad \mathrm{J\,m^{-3}}\\ &\approx \left(\frac{\hbar^2}{2m}(3\pi^2)^{\frac{2}{3}}n^{\frac{5}{3}}-\frac{m (3\pi^2n)^{\frac{1}{3}}}{18\hbar^2}(k_BT)^2\right)\quad\mathrm{J\,m^{-3}} \end{align}