Following the Drude model describing the movement of electrons in metals, Sommerfeld developed yet another model for electrons in metals in 1927. This new model would account for electron energy distributions in metals, Pauli’s exclusion principle, and Fermi-Dirac statistics of electrons. This model factors in quantum mechanics and the Schrödinger Equation.

The Somerfeld model’s view of electrons in metals can be taken as an example of a large volume of metal with electrons confined in the volume. We’ll call this a potential well. Inside this volume or potential well, electrons are ‘free’ with zero potential. Outside the potential well, the potential is infinite. The electron states inside this box are governed by the Schroedinger equation.

The quantum state of an electron is generally described by the Schroedinger equation as shown below.

The result of the Schroedinger equation, applying the boundary conditions of the problem with the potential being zero at the boundary, the solution of the wavefunction is below. This solution introduces a concept called the k-space, a 3D grid of allowed quantum states.

The density of grid points in the k-space is related to Lx, Ly, and Lz of the solution. The number of grid points per unit volume or density (i.e. density of states) will be V/(2*pi)^3, where the spacing of points in the 3D k-space are defined as 2pi/Lx, 2pi/Ly, and 2pi/Lz.

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