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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 :
The Klein–Gordon equation, + =, was the first such equation to be obtained, even before the nonrelativistic one-particle Schrödinger equation, and applies to massive spinless particles. Historically, Dirac obtained the Dirac equation by seeking a differential equation that would be first-order in both time and space, a desirable property for ...
and this is the Schrödinger equation. Note that the normalization of the path integral needs to be fixed in exactly the same way as in the free particle case. An arbitrary continuous potential does not affect the normalization, although singular potentials require careful treatment.
The relativistic calculation of a free particle colliding with a step potential can be obtained using relativistic quantum mechanics. For the case of 1/2 fermions, like electrons and neutrinos , the solutions of the Dirac equation for high energy barriers produce transmission and reflection coefficients that are not bounded.
The solutions of the Schrödinger equation in polar coordinates in vacuum are thus labelled by three quantum numbers: discrete indices ℓ and m, and k varying continuously in [,): = (,) These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves ().
The problem consists of solving the one-dimensional time-independent Schrödinger equation for a particle encountering a rectangular potential energy barrier. It is usually assumed, as here, that a free particle impinges on the barrier from the left.
which is an eigenvalue equation. Very often, only numerical solutions to the Schrödinger equation can be found for a given physical system and its associated potential energy. However, there exists a subset of physical systems for which the form of the eigenfunctions and their associated energies, or eigenvalues, can be found.
In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S 1 {\displaystyle S^{1}} ) is