- 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 theangular 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. (Seeparticle 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
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