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Particle in a ring
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 S1) is
Under these conditions, the solution to the Schrodinger equation is given by
- , or
This leads to the energy eigenvalues
The full wave functions are, therefore
Except for the case n = 0, there are two quantum states for every value of n (corresponding to ). Therefore there are 2n+1 states with energies less than an energy indexed by the number n.
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.
Interestingly, the statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to Fourier's theorem about the development of any periodic function in a Fourier series.
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