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.
Classical free particleEdit
and the kinetic energy (equal to total energy) by
where m is the mass of the particle and v is the vector velocity of the particle.
Non-relativistic quantum free particleEdit
A free quantum particle is described by the Schrödinger equation:
where ψ is the wavefunction of the particle at position r and time t. The solution for a particle with momentum p or wave vector k, at angular frequency ω or energy E, is given by the complex plane wave:
(similarly for the y and z directions), and the De Broglie relations:
apply. Since the potential energy is (set to) zero, the total energy E is equal to the kinetic energy, which has the same form as in classical physics:
Measurement and calculationsEdit
The integral of the probability density function
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.
where the integral is over all k-space and (to ensure that the wavepacket is a solution of the free particle Schrödinger equation). Here is the value of the wave function at time 0 and is the Fourier transform of .
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
Group velocity and phase velocityEdit
The phase velocity is defined to be the velocity at which a plane wave solution propagates, namely
which agrees with the formula for the classical velocity of the particle. The group velocity is the (approximate) speed at which the whole wave packet propagates, while the phase velocity is the speed at which the individual peaks in the wave packet move.
Relativistic quantum free particleEdit
There are a number of equations describing relativistic particles: see relativistic wave equations.
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