## 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.