Noninteracting electrons in a prototypical one-dimensional sinusoidal potential

A prototypical model of a one-dimensional metallic monatomic solid containing noninteracting electrons is studied, where the argument of the cosine potential energy, periodic with the lattice, contains the first reciprocal lattice vector $G 1 = 2 π / a$⁠, where a is the lattice constant. The time-independent Schrödinger equation can be written in reduced variables as a Mathieu equation for which numerically exact solutions for the band structure and wave functions are obtained. The band structure has band gaps that increase with increasing amplitude q of the cosine potential. In the extended-zone scheme, the energy gaps decrease with increasing index n of the Brillouin-zone boundary $k a = n π$⁠, where k is the crystal momentum of the electron. The wave functions of the band electron are derived for various combinations of k and q as complex combinations of the real Mathieu functions with even and odd parity, and the normalization factor is discussed. The wave functions at the bottoms and tops of the bands are found to be real or imaginary, respectively, corresponding to standing waves at these energies. Irrespective of the wave vector k within the first Brillouin zone, the electron probability density is found to be periodic with the lattice. The Fourier components of the wave functions are derived versus q, which reveal multiple reciprocal-lattice-vector components with variable amplitudes in the wave functions unless q = 0. The magnitudes of the Fourier components are found to decrease exponentially as a power of n for $n ∼ 3$ to 45 for $k a = π / 2$ and q = 2, and a precise fit is obtained to the data. The probability densities and probability currents obtained from the wave functions are also discussed. The probability currents are found to be zero for crystal momenta at the tops and bottoms of the energy bands, because the wave functions for these crystal momenta are standing waves. Finally, the band structure is calculated from the central equation and compared to the numerically exact band structure.