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