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.

Classical Free Particle

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

$\backslash mathbf\{p\}=m\backslash mathbf\{v\}$

and the energy by

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

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

Non-Relativistic Quantum Free Particle

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The Schrödinger equation for a free particle is:

$$

- \frac{\hbar^2}{2m} \nabla^2 \ \psi(\mathbf{r}, t) = i\hbar\frac{\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}\cdot\mathbf{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, 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

$$

\langle\mathbf{p}\rangle=\langle \psi |-i\hbar\nabla|\psi\rangle = \hbar\mathbf{k}

The expectation value of the energy E is

$$

\langle E\rangle=\langle \psi |i\hbar\frac{\partial}{\partial t}|\psi\rangle = \hbar\omega

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 \rangle =\frac{\langle p \rangle^2}{2m}

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

$\backslash left.\backslash right.$

v_g= d\omega/dk = dE/dp = v

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

$\backslash left.\backslash right.$

v_p=\omega/k = E/p = p/2m = 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:

$\backslash left.\backslash right.$

\psi(\mathbf{r}, t) = \int A(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} d\mathbf{k}

where the integral is over all k-space.

Relativistic free particle

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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)

Fundamental physics concepts | Classical mechanics | Quantum mechanics

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