Electronic Potential Behavior and Wave Equations Notes:
One-Dimensional Perfect Crystals and their Energetics
A perfect crystal is a translationally invariant system, where the potential energy U(x) repeats every lattice vector a.
The Schrödinger equation for a perfect crystal is: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi(x) = E\psi(x) $$
Bloch's theorem states that the wave function of a particle in a perfect crystal can be written as: $$ \psi(x) = u(x)e^{ikx} $$ where u(x) has the same periodicity
as the crystal and k is the wave vector.
Forbidden and Allowed Energy Bands in Potential Barriers
In a potential barrier, the energy of the particle E can be greater than, equal to, or less than the potential energy U.
For E > U, there are two classically allowed regions, and the wave function can be described using traveling waves.
For E = U, the particle is trapped in the potential well, and the wave function can be described using standing waves.
For E < U, there are two classically forbidden regions, and the wave function can be described using exponentially decaying or increasing functions.
Allowed and forbidden energy bands arise due to the quantization of the wave vector k in the crystal.
The energy bands are separated by forbidden energy gaps, where no electron states can exist.
The width and location of the energy bands and gaps depend on the details of the potential energy U(x).
One-Dimensional Perfect Crystals and their Energetics
A perfect crystal is a translationally invariant system, where the potential energy U(x) repeats every lattice vector a.
The Schrödinger equation for a perfect crystal is: $$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + U(x)\psi(x) = E\psi(x) $$
Bloch's theorem states that the wave function of a particle in a perfect crystal can be written as: $$ \psi(x) = u(x)e^{ikx} $$ where u(x) has the same periodicity
as the crystal and k is the wave vector.
Forbidden and Allowed Energy Bands in Potential Barriers
In a potential barrier, the energy of the particle E can be greater than, equal to, or less than the potential energy U.
For E > U, there are two classically allowed regions, and the wave function can be described using traveling waves.
For E = U, the particle is trapped in the potential well, and the wave function can be described using standing waves.
For E < U, there are two classically forbidden regions, and the wave function can be described using exponentially decaying or increasing functions.
Allowed and forbidden energy bands arise due to the quantization of the wave vector k in the crystal.
The energy bands are separated by forbidden energy gaps, where no electron states can exist.
The width and location of the energy bands and gaps depend on the details of the potential energy U(x).
