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)
“But often the presence of mind and energy of a person remote from the spotlight decide the course of history for centuries to come.”
—Stefan Zweig (18811942)
“A personality is an indefinite quantum of traits which is subject to constant flux, change, and growth from the birth of the individual in the world to his death. A character, on the other hand, is a fixed and definite quantum of traits which, though it may be interpreted with slight differences from age to age and actor to actor, is nevertheless in its essentials forever fixed.”
—Hubert C. Heffner (1901–1985)
“the moderate Aristotelian city
Of darning and the Eight-Fifteen, where Euclid’s geometry
And Newton’s mechanics would account for our experience,
And the kitchen table exists because I scrub it.”
—W.H. (Wystan Hugh)