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## Animation of a quantum wave packet to explain the energy band gap in solid crystals

The energy band gap in semiconductors is the most important concept in all of our modern
electronic devices. Semiconductors have band gaps narrow enough to allow some thermally generated electrons to cross the band gap.
Semiconductors can be formed into pn junctions by doping the top with positive ions p and the bottom with negative ions n.
Semiconductor pn junctions with direct band gaps become light emitting diodes with photon energy equal to the band gap energy when a voltage is applied.
Conversely, semiconductor junctions provide electrical current and power when light with photon energy greater than the band gap irradiates them.
If narrow band gap materials are fabricated into npn or pnp junction devices, current flow can be manipulated by applying bias voltages to the center region
and this is the basis of transistors.
Since the concept of direct or indirect gap depends on the sameness or difference of orientation of the reflecting ion lattice planes,
in the one dimensional simulation described here, the gap is direct.
The physical effect of an energy band gap is that it is impossible to accelerate (or decelerate) a particle to acheive a kinetic energy
that is inside the energy band gap.
The reason is that the potential distribution due to atom charges in a crystal host is such that the quantum wave packet (particle)
gets reflected again and again by the periodic potential distribution of the host material. If the host is polycrystalline then the periodic
potential gets washed out and the band gap edges get poorly defined and the application of the host as an electronic device is very limited.
The reflection phenomenon of a crystal is what will be animated here for many variations of the potential, `V`, and initial wave packet.
Plots of reflectivity and transmissivity Vs wave packet momentum and energy can also be done. The interesting
features of these plots are the ranges over which the reflectivity is large which correspond to the momentum band
gap and the energy band gap. The steepness of the band edges is also important.
The wave packet's motion is computed using the finite difference expression for the Hamiltonian operator, `H`, associated with the spatial part
of the Schrodinger equation. The time dependence of the wave packet, `Psi(x,t)`, is obtained by a simple integration of Schrodinger Hamiltonian equation:
`i*(dPsi)/dt=H*Psi(x,t)`. For this simulation both the particle mass and Planck constant, `"hbar"=h/(2pi)`, are taken to be 1. The speed of the calculation
it greatly aided by using Visscher's Advance where the imaginary part of the wave function is updated first and then
the real part is computed using the combination of the old real part and the new imaginary part. This removes a lot of error terms which
would not otherwise keep the integral of `Psi^("*")Psi` constant.
The most important parameters are the spatial wave vector, `k_V`, of the potential, `V`, distribution and the spatial wave vector, `kPsi`, of the wave packet
The highest reflectivity occurs when `2*kPsi=k_V` which means that the wavelength of the wave packet is twice the wavelength
(period) of the potential variation. Therefore, the calculations are done using only the ratio `(2*k)/(k_V)Psi`.
For single propagations the ratio of the potential amplitude to the kinetic energy `V/(KE(1))` for the ratio `(2*k)/(k_V)Psi` is adjustable.
Of course, when the stop bands are computed, the value of the ratio `(2*k)/(k_V)Psi` varies from adjustable ratios, `r_("Start")` to `r_("Stop")`,
so the kinetic energy of the
packet at small values of the ratio is much lower than at large values. This means that, at small `(2*k)/(k_V)Psi` ratios
and values of `V/KE(1)` are far greater than 1 the packet is reflected even when the value of the `(2*k)/(k_V)Psi` ratio is far from 1. It also means
that when values of `V/(KE(1))` are far less than 1, the packet will usually "float over" the top of the potential and not be reflected at all.
Note that, to save space, all plots and their legends are color coded,