Kinetic Energy: Quantum mechanical kinetic energy of rigid bodies

Quantum mechanical kinetic energy of rigid bodies

In the realm of quantum mechanics, the expectation value of the electron kinetic energy, $\langle\hat{T}\rangle$ , for a system of electrons described by the wavefunction $\vert\psi\rangle$ is a sum of 1-electron operator expectation values:

$\langle\hat{T}\rangle = -\frac{\hbar^2}{2 m_e}\bigg\langle\psi \bigg\vert \sum_{i=1}^N \nabla^2_i \bigg\vert \psi \bigg\rangle$

where

m e

is the mass of the electron and $\nabla^2_i$ is the Laplacian operator acting upon the coordinates of the i^th electron and the summation runs over all electrons. Notice that this is the quantized version of the non-relativistic expression for kinetic energy in terms of momentum:

$E_k = \frac{p^2}{2m}.$

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 $\rho(\mathbf{r})$ , 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

$T[\rho] = \frac{1}{8} \int \frac{ \nabla \rho(\mathbf{r}) \cdot \nabla \rho(\mathbf{r}) }{ \rho(\mathbf{r}) } d^3r$

where

T [ρ]

is known as the von Weizsäcker kinetic energy functional.

Kinetic Energy

Thursday, 14 July 2011

Quantum mechanical kinetic energy of rigid bodies

Quantum mechanical kinetic energy of rigid bodies

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