FREE PARTICLE

In physics, a 'free particle' is a particle that, in some sense, is not bound. In the classical case, this is represented with the particle not being influenced by any external force.

Contents

Classical Free Particle

Non-Relativistic Quantum Free Particle

Relativistic free particle

Classical Free Particle

The classical free particle is characterized simply by a fixed velocity. The momentum is
given by
:$mathbf\{p\}=mmathbf\{v\}$
and the energy by
:
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:
:$$
- rac{hbar^2}{2m}
abla^2 psi(mathbf{r}, t) =
ihbarrac{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
:$$
rac{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, there will be a problem normalizing this momentum eigenfunction. This will not be a problem for a general free particle which is 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 |ihbarrac{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 =rac{langle p
angle^2}{2m}

where p=|'p'|. The group velocity of the wave is defined as
:$left.$
ight.
v_g= rac{domega}{dk} = rac{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=rac{omega}{k} = rac{E}{p} = rac{p}{2m} = rac{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) = int
A(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)