Editing Kinetic energy (section)

==Kinetic energy in quantum mechanics==
{{Further|Hamiltonian (quantum mechanics)}}

In [[quantum mechanics]], observables like kinetic energy are represented as [[Operator (physics)|operators]]. For one particle of mass ''m'', the kinetic energy operator appears as a term in the [[Hamiltonian (quantum mechanics)|Hamiltonian]] and is defined in terms of the more fundamental momentum operator <math>\hat p</math>. The kinetic energy operator in the [[Relativistic quantum mechanics#Non-relativistic and relativistic Hamiltonians|non-relativistic]] case can be written as

<math display="block">\hat T = \frac{\hat p^2}{2m}.</math>

Notice that this can be obtained by replacing <math>p</math> by <math>\hat p</math> in the classical expression for kinetic energy in terms of [[momentum]],

<math display="block">E_\text{k} = \frac{p^2}{2m}.</math>

In the [[Schrödinger picture]], <math>\hat p</math> takes the form <math>-i\hbar\nabla </math> where the derivative is taken with respect to position coordinates and hence

<math display="block">\hat T = -\frac{\hbar^2}{2m}\nabla^2.</math>

The expectation value of the electron kinetic energy, <math>\left\langle\hat{T}\right\rangle</math>, for a system of ''N'' electrons described by the [[Wave function|wavefunction]] <math>\vert\psi\rangle</math> is a sum of 1-electron operator expectation values:

<math display="block">\left\langle\hat{T}\right\rangle =
   \left\langle \psi \left\vert \sum_{i=1}^N \frac{-\hbar^2}{2m_\text{e}} \nabla^2_i \right\vert \psi \right\rangle =
  -\frac{\hbar^2}{2m_\text{e}} \sum_{i=1}^N \left\langle \psi \left\vert \nabla^2_i \right\vert \psi \right\rangle
</math>

where <math>m_\text{e}</math> is the mass of the electron and <math>\nabla^2_i</math> is the [[Laplacian]] operator acting upon the coordinates of the ''i''<sup>th</sup> electron and the summation runs over all electrons.

The [[Density functional theory|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 <math>\rho(\mathbf{r})</math>, 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

<math display="block"> T[\rho] = \frac{1}{8} \int \frac{ \nabla \rho(\mathbf{r}) \cdot \nabla \rho(\mathbf{r}) }{ \rho(\mathbf{r}) } d^3r </math>

where <math>T[\rho]</math> is known as the [[Carl Friedrich von Weizsäcker|von Weizsäcker]] kinetic energy functional.