Kinetic Energy - Kinetic Energy in Quantum Mechanics

Kinetic Energy in Quantum Mechanics

Further information: Hamiltonian (quantum mechanics)

In quantum mechanics, observables like kinetic energy are represented as operators. For one particle of mass m, the kinetic energy operator appears as a term in the Hamiltonian and is defined in terms of the more fundamental momentum operator as

Notice that this can be obtained by replacing by in the classical expression for kinetic energy in terms of momentum,

In the Schrödinger picture, takes the form where the derivative is taken with respect to position coordinates and hence

The expectation value of the electron kinetic energy, for a system of N electrons described by the wavefunction is a sum of 1-electron operator expectation values:

where is the mass of the electron and is the Laplacian operator acting upon the coordinates of the ith electron and the summation runs over all electrons.

The density functional formalism of quantum mechanics requires knowledge of the electron density only, i.e., it formally does not require knowledge of the wavefunction. Given an electron density, the exact N-electron kinetic energy functional is unknown; however, for the specific case of a 1-electron system, the kinetic energy can be written as

where is known as the von Weizsäcker kinetic energy functional.

Read more about this topic:  Kinetic Energy

Famous quotes containing the words kinetic, energy, quantum and/or mechanics:

    The poem has a social effect of some kind whether or not the poet wills it to have. It has kinetic force, it sets in motion ... [ellipsis in source] elements in the reader that would otherwise be stagnant.
    Denise Levertov (b. 1923)

    Since the beginning of time, three-quarters of the mental energy and of the lies inspired by vanity have been expended for their inferiors by people who are only abased by such expenditure. And Swann, who was easygoing and unaffected with a duchess, trembled at the thought of being scorned and put on airs when he was with a housemaid.
    Marcel Proust (1871–1922)

    The receipt to make a speaker, and an applauded one too, is short and easy.—Take of common sense quantum sufficit, add a little application to the rules and orders of the House, throw obvious thoughts in a new light, and make up the whole with a large quantity of purity, correctness, and elegancy of style.
    Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)

    It is only the impossible that is possible for God. He has given over the possible to the mechanics of matter and the autonomy of his creatures.
    Simone Weil (1909–1943)