Energy and Wave Function of A Free Electron
For a free particle the potential is . The Schrödinger equation for such a particle, like the free electron, is
The wave function can be split into a solution of a time dependent and a solution of a time independent equation. The solution of the time dependent equation is
with energy
The solution of the time independent equation is
with a wave vector . is the volume of space where the electron can be found. The electron has a kinetic energy
The plane wave solution of this Schrödinger equation is
For solid state and condensed matter physics the time independent solution is of major interest. It is the basis of electronic band structure models that are widely used in solid-state physics for model Hamiltonians like the nearly free electron model and the Tight binding model and different models that use a Muffin-tin approximation. The eigenfunctions of these Hamiltonians are Bloch waves which are modulated plane waves.
Read more about this topic: Free Electron Model
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