In physics a free particle is a particle that is never under the influence of an external force

Classical Free Particle

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

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 Schroedinger equation for a free particle is:

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

with the constraint

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

The expectation value of the energy E is

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

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

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

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:

where the integral is over all k-space.

Relativistic free particle (Klein-Gordon equation)

If the particle is charge-neutral and spinless, and relativistic effects cannot be ignored, we may use the Klein-Gordon equation to describe the wave function. The Klein-Gordon equation for a free particle is written

with the same solution as in the non-relativistic case:

except with the constraint

Just as with the non-relativistic particle, we have for energy and momentum:

Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles:

For massless particles, we may set m=0 in the above equations. We then recover the relationship between energy and momentum for massless particles: