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Concept
Free particle
Summary
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 the particle is in a region of uniform potential, usually set to zero in the region of interest since the potential can be arbitrarily set to zero at any point in space.
The classical free particle is characterized 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.
Schrödinger equation and Matter wave
A free particle with mass in non-relativistic quantum mechanics is described by the free 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 a complex plane wave:
with amplitude A and has two different rules according to its mass:
if the particle has mass : (or equivalent ).
if the particle is a massless particle: .
The eigenvalue spectrum is infinitely degenerate since for each eigenvalue E>0, there corresponds an infinite number of eigenfunctions corresponding to different directions of .
The De Broglie relations: , apply. Since the potential energy is (stated to be) zero, the total energy E is equal to the kinetic energy, which has the same form as in classical physics:
As for all quantum particles free or bound, the Heisenberg uncertainty principles apply. It is clear that since the plane wave has definite momentum (definite energy), the probability of finding the particle's location is uniform and negligible all over the space. In other words, the wave function is not normalizable in a Euclidean space, these stationary states can not correspond to physical realizable states.
Official source
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