Free particle

In physics, a free particle is a particle that, in some sense, is not bound. In classical physics, this means the particle is present in a "field-free" space.

Classical Free Particle

The classical free particle is characterized simply by a fixed velocity. The momentum isgiven by

:$mathbf\{p\}=mmathbf\{v\}$

and the energy by

:$E=frac\{1\}\{2\}mv^2$

where m is the mass of the particle and v is the vector velocity of the particle.

Non-Relativistic Quantum Free Particle

The Schrödinger equation for a free particle is:

:$-\; frac\{hbar^2\}\{2m\}\; abla^2\; psi(mathbf\{r\},\; t)\; =\; ihbarfrac\{partial\}\{partial\; t\}\; psi\; (mathbf\{r\},\; t)$

The solution for a particular momentum is given by a plane wave:

:$psi(mathbf\{r\},\; t)\; =\; e^\{i(mathbf\{k\}cdotmathbf\{r\}-omega\; t)\}$

with the constraint

:$frac\{hbar^2\; k^2\}\{2m\}=hbar\; omega$

where r is the position vector, t is time, k is the wave vector, and ω is the angular frequency. Since the integral of ψψ^* over all space must be unity, the wave function must first be normalized. This is not a problem for general free particles somewhat localized in momentum and position. (See particle in a box for a further discussion.)

The expectation value of the momentum p is

:$langlemathbf\{p\}\; angle=langle\; psi\; |-ihbar\; abla|psi\; angle\; =\; hbarmathbf\{k\}$

The expectation value of the energy E is

:$langle\; E\; angle=langle\; psi\; |ihbarfrac\{partial\}\{partial\; t\}|psi\; angle\; =\; hbaromega$

Solving for k and ω and substituting into the constraint equation yields the familiar relationship between energy and momentum for non-relativistic massive particles

:$langle\; E\; angle\; =frac\{langle\; p\; angle^2\}\{2m\}$

where p=|p|. The group velocity of the wave is defined as

:$left.\; ight.v\_g=\; frac\{domega\}\{dk\}\; =\; frac\{dE\}\{dp\}\; =\; v$

where v is the classical velocity of the particle.The phase velocity of the wave is defined as

:$left.\; ight.v\_p=frac\{omega\}\{k\}\; =\; frac\{E\}\{p\}\; =\; frac\{p\}\{2m\}\; =\; frac\{v\}\{2\}$

A general free particle need not have a specific momentum or energy. In this case, the free particle wavefunction may be represented by a superposition of free particle momentum eigenfunctions:

:$left.\; ight.psi(mathbf\{r\},\; t)\; =\; intA(mathbf\{k\})e^\{i(mathbf\{k\}cdotmathbf\{r\}-omega\; t)\}dmathbf\{k\}$

where the integral is over all k-space.

Relativistic free particle

There are a number of equations describing relativistic particles. For a description of the free particle solutions, see the individual articles.

* The Klein-Gordon equation describes charge-neutral, spinless, relativistic quantum particles

* The Dirac equation describes the relativistic electron (charged, spin 1/2)

ee also

:Particle in a box:Finite square well:Delta potential well:Wave packet