Free particle

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.

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

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.

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:

with amplitude A. As for all quantum particles free or bound, the Heisenberg uncertainty principles

(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:

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.

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