  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
G1 = 2pi/a, where a is the lattice constant. The time-independent Schrodinger
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 ka = n pi
where k is the crystal momentum of the electron. 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 ka = pi/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.
