The free particle

Shabnam Siddiqui

This chapter discusses several applications of the formalism of theory of quantum mechanics. The time-independent Schrodinger equation is applied to understand simple quantum systems such as a free particle, a particle in an infinite square well, step potentials, potential barrier and finite square well. The chapter deals with the simplest case, a free particle. It is the most elementary application of the formalism. A free particle means that no forces are acting on the particle, and hence, there is no interaction between the particle and its environment. According to classical mechanics, a free particle moves with a constant velocity along a straight line. The energy of the wave associated with the free quantum particle is same as that of the classical free particle. The chapter describes a quantum free particle as a wave packet traveling in space and time. It discusses a quantum particle under the influence of a force.

Jiang Xue

First the Dirac equation of free particle is solved under static condition. Then the change of state vector space is used to gain plane wave solutions of the free particle that spins and moves along any divection. A simple way of solving the free particle Dirac equation is given.

Alessandro Teta

Denys I. Bondar, Robert R. Lompay, Wing-Ki Liu

Based on the assumption that the probability density of finding a free particle is independent of position, we infer the form of the eigenfunction for the free particle, ⟨x|p⟩=exp(ipx/ℏ)/2πℏ. The canonical commutation relation between the momentum and position operators and the Ehrenfest theorem in the free particle case are derived solely from differentiation of the delta function and the form of ⟨x|p⟩.