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. Classical free particle 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. Quantum free particle Mathematical description 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 the complex plane wave: with amplitude A and restricted to: 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. Measurement and calculations 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. Fourier decomposition The free particle wave function may be represented