We shall consider a very simple situation in which a particle of mass m is restricted to moving in a circular path of radius r in the xy plane. (The model is sometimes referred to as a particle on a ring.)

Since there is no variation in the vertical displacement of the particle (the z direction) its potential energy is constant. We may set the value of the potential energy for the system to zero, as we are concerned with variations in the energy of the particle, and not the absolute value of its energy. Thus the total energy is equal to the kinetic energy, and we can use classical mechanics to write E = p²/2m.

We also make use of the classical mechanical expression for the angular momentum, J_z, about the z axis, which is J_z = ± pr, allowing us to manipulate the expression for the energy to give E = J_z²/2mr². Since mr² is the moment of inertia, I, of a particle of mass m rotating around an axis at a distance r, it follows that:

However, it is relatively simple to show that not all values of the angular momentum are permitted in quantum mechanics, and hence the angular momentum and rotational energy of this are both quantised.

We have already used the classical result that J_z = ± pr , and we may substitute for the momentum using the de Broglie relation:

This gives us the following expression for the angular momentum:

The different signs correspond to different directions of rotation. i.e. if the system is viewed from above, the particle may be rotating either clockwise (conventionally a negative angular momentum) or anticlockwise (a positive angular momentum). 

We must now consider the wavelength of the particle, λ.

The circumference of the circle traced out by the particle (the distance that the particle travels) must be an integral number of wavelengths; this means that the wavefunction will reproduce itself on successive orbits.

If the distance travelled by the particle is not an integral number of wavelengths, then the wavefunction becomes out of phase with itself on successive orbits. This is unacceptable for two reasons: Firstly it means that the wavefunction is then multivalued, and secondly it means that over an infinite number of orbits the wavefunction will interfere with itself destructively and the overall wavefunction will cancel to zero, indicating a particle cannot exist under such conditions. We can express this requirement mathematically as:

where m_l can take values 0, ±1, ±2.... This definition of m_l allows us to write:

Using the classical expression for energy derived at the top of the page, we can write:

Notice that in the expression for the energy, m_l occurs as its square. This indicates that the energy of rotation is independent of the direction of rotation, as we would expect. Thus the energy levels in this system are all doubly degenerate (due to the fact that m_l takes both positive and negative integer values) except the level for which m_l = 0, which is non-degenerate. m_l is a quantum number for the system.

Note that since there is one boundary condition, one quantum number is required to describe the system. (The single boundary condition is the cyclic boundary condition that the wavefunction must reproduce itself on successive circuits. This requirement is commonly expressed as Ψ_ml(θ) = Ψ_ml(θ + 2π), i.e. the value of the function must be the same after one complete rotation of 2π for a given value of m_l.)

It will be noted that thus far in the treatment of this system, we have not considered the Schrodinger equation at any point. For this simple situation, it is easier to use the de Broglie relation to manipulate classical results. This course of action even allows us to generate the form of the acceptable energy values for the particle.

Note however that the same results would have been obtained by setting up and solving the Schrodinger equation for the system, and for more complicated systems this more formal approach to the problem is essential. This approach allows us to write down the explicit forms of the wavefunctions for the system, but as there is little to be gained from this we shall not consider them here.