Nearly free electron model
In solid-state physics, the nearly free electron model (or NFE model) is a quantum mechanical model of physical properties of electrons that can move almost freely through the crystal lattice of a solid. The model is closely related to the more conceptual Empty Lattice Approximation. The model enables understanding and calculating the electronic band structure of especially metals.
Free electrons are traveling plane waves. Generally the time independent part of their wave function is expressed as
These plane wave solutions have an energy of
The expression of the plane wave as a complex exponential function can also be written as the sum of two periodic functions which are mutually shifted a quarter of a period.
In this light the wave function of a free electron can be viewed as the sum of two plane waves. Sine and cosine functions can also be expressed as sums or differences of plane waves moving in opposite directions
Assume that there is only one kind of atom present in the lattice and that the atoms are located at the lattice points. The potential of the atoms is attractive (negative) and concentrated near the lattice points. In the remainder of the cell the potential is close to zero.
The Hamiltonian is expressed as
in which is the kinetic and is the potential energy. From this expression the energy expectation value, or the statistical average, of the energy of the electron can be calculated with
...
Wikipedia