Particle in a ring

The particles on the ring is one of the different model systems of quantum mechanics, which leads to the quantization of energy. It is very similar to the particle in the box and is therefore also referred to as "particles in the circular potential well ".

In contrast to the particle in a box the particle does not move on the ring linear but circular floating around a certain point. Therefore it is better to expect polar than with Cartesian coordinates. The particle moves around a certain radius, which is constant. Therefore, the wave function of the particle depends only on the polar angle and not on the distance to the center.

Mathematical viewing

To find the wave functions and the energies of the states of the particle on the ring, it is necessary to solve the stationary Schrödinger equation in the given potential. The potential is given by

The Hamiltonian can be divided into polar coordinates for the relevant area as

Write, which results in the Schrödinger equation to be solved:

It is an ordinary, linear, homogeneous differential equation of 2nd order, which reads for the approach

Substituting into the Schrödinger equation is obtained

In order to solve the differential equation is now clearly still a constraint necessary. After a rotation of the ring, the wave function must be the same again:

Which leads to the following condition:

This condition is satisfied only if is an integer. The energies of the particle on the ring you get now, by simply forming:

Now the wave function must be normalized, what happens by integrating over the absolute square of the wave function from to. Do this by writing the wave function using Euler's identity in

Order. Since the magnitude of a complex number is defined as having obtained

Which arises. Thus, the wave function for a particle is on the ring

Since linear combinations of eigenfunctions are eigenfunctions again, it follows (using the Euler's identity) that one alternative

As degenerate eigenfunctions with eigenvalue can choose. The other factor is due to the necessary normalization of wave functions.

Degeneration

In addition to the quantization of this relatively simple example leads to computing for the first time to the concept of degeneration. Since states in which differs only in sign, though but represent different states because of the same energies, there are here two states with the same energy. The states are 2- fold degenerate.