In physics, a free particle is a particle that, in some sense, is not bound by an external force, or equivalently not in a region where its potential energy varies. In classical physics, this means the particle is present in a "field-free" space. In quantum mechanics, it means a region of uniform potential, usually set to zero in the region of interest since potential can be arbitrarily set to zero at any point (or surface in three dimensions) in space.
Propagation of de Broglie waves in 1d - real part of the complex amplitude is blue, imaginary part is green. The probability (shown as the colour opacity) of finding the particle at a given point x is spread out like a waveform, there is no definite position of the particle. As the amplitude increases above zero the curvature decreases, so the decreases again, and vice versa - the result is an alternating amplitude: a wave. Top: Plane wave. Bottom: Wave packet.
where * denotes complex conjugate, over all space is the probability of finding the particle in all space, which must be unity if the particle exists:
This is the normalization condition for the wave function. The wavefunction is not normalizable for a plane wave, but is for a wavepacket.
Increasing amounts of wavepacket localization, meaning the particle becomes more localized.
In the limit ħ → 0, the particle's position and momentum become known exactly.
Interpretation of wave function for one spin-0 particle in one dimension. The wavefunctions shown are continuous, finite, single-valued and normalized. The colour opacity (%) of the particles corresponds to the probability density (which can measure in %) of finding the particle at the points on the x-axis.
In this case, the free particle wavefunction may be represented by a superposition of free particle momentum eigenfunctions ϕ(k), the Fourier transform of the momentum space wavefunction:
where the integral is over all k-space, and and (to ensure that the wavepacket is a solution of the free particle Schrödinger equation). Note that here we abuse notation and denote and with the same symbol, when we should denote , where is the p-space and the k-space function.
The expectation value of the momentum p for the complex plane wave is
and for the general wavepacket it is
The expectation value of the energy E is (for both plane wave and general wave packet; here one can observe the special status of time and hence energy in quantum mechanics as opposed to space and momentum)
For the plane wave, solving for k and ω and substituting into the constraint equation yields the familiar relationship between energy and momentum for non-relativistic massive particles
In general, the identity holds in the form
where p = |p| is the magnitude of the momentum vector.