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The case of a particle in a one-dimensional ring is an instructive example when studying the quantization of angular momentum for, say, an electron orbiting the nucleus. The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring.
Working from the definition, a partial solution for a wavefunction of a particle with a constant energy can be constructed. If the wavefunction is assumed to be separable, then the time dependence can be stated as e − i E t / ℏ {\displaystyle e^{-iEt/\hbar }} , where E is the constant energy.
the mass–energy equivalence formula which gives the energy in terms of the momentum and the rest mass of a particle. The equation for the mass shell is also often written in terms of the four-momentum ; in Einstein notation with metric signature (+,−,−,−) and units where the speed of light c = 1 {\displaystyle c=1} , as p μ p μ ≡ p ...
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 :
If a particle is confined to the motion of an entire ring ranging from 0 to , the particle is subject only to a periodic boundary condition (see particle in a ring). If a particle is confined to the motion of − π 2 {\textstyle -{\frac {\pi }{2}}} to π 2 {\textstyle {\frac {\pi }{2}}} , the issue of even and odd parity becomes important.
The kinetic energy term for a free particle in the absence of an electromagnetic field is just where is the kinetic momentum, while in the presence of an electromagnetic field it involves the minimal coupling =, where now is the kinetic momentum and is the canonical momentum.
The first term rotates the phase of ψ(x) locally by an amount proportional to the potential energy. The second term is the free particle propagator, corresponding to i times a diffusion process. To lowest order in ε they are additive; in any case one has with :
Source: [1] The potential splits the space in two parts (x < 0 and x > 0).In each of these parts the potential is zero, and the Schrödinger equation reduces to =; this is a linear differential equation with constant coefficients, whose solutions are linear combinations of e ikx and e −ikx, where the wave number k is related to the energy by =.