Free Electron Model - Energy and Wave Function of A Free Electron

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

Famous quotes containing the words energy, wave, function and/or free:

    The scholar may be sure that he writes the tougher truth for the calluses on his palms. They give firmness to the sentence. Indeed, the mind never makes a great and successful effort, without a corresponding energy of the body.
    Henry David Thoreau (1817–1862)

    The history of any nation follows an undulatory course. In the trough of the wave we find more or less complete anarchy; but the crest is not more or less complete Utopia, but only, at best, a tolerably humane, partially free and fairly just society that invariably carries within itself the seeds of its own decadence.
    Aldous Huxley (1894–1963)

    The intension of a proposition comprises whatever the proposition entails: and it includes nothing else.... The connotation or intension of a function comprises all that attribution of this predicate to anything entails as also predicable to that thing.
    Clarence Lewis (1883–1964)

    No man hath any quarrel to me. My remembrance is very free and clear from any image of offence done to any man.
    William Shakespeare (1564–1616)