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:
“Just as we need to encourage women to test life’s many options, we need to acknowledge real limits of energy and resources. It would be pointless and cruel to prescribe role combination for every woman at each moment of her life. Life has its seasons. There are moments when a woman ought to invest emotionally in many different roles, and other moments when she may need to conserve her psychological energies.”
—Faye J. Crosby (20th century)
“And his wish is intimacy,
Intimater intimacy,
And a stricter privacy;
The impossible shall yet be done,
And, being two, shall still be one.
As the wave breaks to foam on shelves,
Then runs into a wave again,
So lovers melt their sundered selves,
Yet melted would be twain.”
—Ralph Waldo Emerson (1803–1882)
“If the children and youth of a nation are afforded opportunity to develop their capacities to the fullest, if they are given the knowledge to understand the world and the wisdom to change it, then the prospects for the future are bright. In contrast, a society which neglects its children, however well it may function in other respects, risks eventual disorganization and demise.”
—Urie Bronfenbrenner (b. 1917)
“I am a free man. I feel as light as a feather.”
—Javier Pérez De Cuéllar (b. 1920)