Kinetic energy in 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 ${\displaystyle \stackrel{^<!--\; ^\; -->}{p}}$. The kinetic energy operator in the non-relativistic case can be written as

${\displaystyle \stackrel{^<!--\; ^\; -->}{T}=\frac{{\stackrel{^<!--\; ^\; -->}{p}}^2}{2m}.}$

Notice that this can be obtained by replacing ${\displaystyle p}$ by ${\displaystyle \stackrel{^<!--\; ^\; -->}{p}}$ in the classical expression for kinetic energy in terms of momentum,

${\displaystyle E_{\text{k}}=\frac{p^2}{2m}.}$

In the Schrödinger picture, ${\displaystyle \stackrel{^<!--\; ^\; -->}{p}}$ takes the form ${\displaystyle -<!--\; -\; -->i\mathrm{\hslash <!--\; \hslash \; -->}\mathrm{\nabla <!--\; \nabla \; -->}}$ where the derivative is taken with respect to position coordinates and hence

${\displaystyle \stackrel{^<!--\; ^\; -->}{T}=-<!--\; -\; -->\frac{{\mathrm{\hslash <!--\; \hslash \; -->}}^2}{2m}{\mathrm{\nabla <!--\; \nabla \; -->}}^2.}$

The expectation value of the electron kinetic energy, ${\displaystyle ⟨\stackrel{^<!--\; ^\; -->}{T}⟩}$, for a system of N electrons described by the wavefunction ${\displaystyle |\mathrm{\psi <!--\; \psi \; -->}⟩<!--\; ⟩\; -->}$ is a sum of 1-electron operator expectation values:

${\displaystyle ⟨\stackrel{^<!--\; ^\; -->}{T}⟩=⟨\mathrm{\psi <!--\; \psi \; -->}\left|\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}\frac{-<!--\; -\; -->{\mathrm{\hslash <!--\; \hslash \; -->}}^2}{2m_{\text{e}}}{\mathrm{\nabla <!--\; \nabla \; -->}}_i^2\right|\mathrm{\psi <!--\; \psi \; -->}⟩=-<!--\; -\; -->\frac{{\mathrm{\hslash <!--\; \hslash \; -->}}^2}{2m_{\text{e}}}\underset{i=1}{\overset{N}{\sum <!--\; \sum \; -->}}⟨\mathrm{\psi <!--\; \psi \; -->}\left|{\mathrm{\nabla <!--\; \nabla \; -->}}_i^2\right|\mathrm{\psi <!--\; \psi \; -->}⟩}$

where ${\displaystyle m_{\text{e}}}$ is the mass of the electron and ${\displaystyle {\mathrm{\nabla <!--\; \nabla \; -->}}_i^2}$ 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 ${\displaystyle \mathrm{\rho <!--\; \rho \; -->}(\mathbf{r})}$, 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

${\displaystyle T\mathrm{\rho <!--\; \rho \; -->}=\frac{1}{8}\int <!--\; \int \; -->\frac{\mathrm{\nabla <!--\; \nabla \; -->}\mathrm{\rho <!--\; \rho \; -->}(\mathbf{r})\cdot <!--\; \cdot \; -->\mathrm{\nabla <!--\; \nabla \; -->}\mathrm{\rho <!--\; \rho \; -->}(\mathbf{r})}{\mathrm{\rho <!--\; \rho \; -->}(\mathbf{r})}{d}^{3}r}$

where ${\displaystyle T\mathrm{\rho <!--\; \rho \; -->}}$ is known as the von Weizsäcker kinetic energy functional.