We can extend the two dimensional model of the particle on a ring to three dimensions by considering a particle that is constrained to move on the surface of a sphere of radius r. (Clearly this will be useful when it comes to modelling the behaviour of electrons in atoms.)
We have two boundary conditions in this situation – the wavefunction should reproduce itself on each circuit around the equator of the sphere but must also reproduce itself on each successive circuit around the poles. (Together these two conditions ensure that the wavefunction will reproduce itself for any orbit of the sphere.) The presence of two boundary conditions suggests that we will require two quantum numbers to fully describe each state of the system. This does indeed turn out to be the case.
The Hamiltonian for motion in three dimensions is:
The symbol is a shorthand for the sum of the three second derivatives. We set V = 0 on the surface of the sphere, where the particle is free to travel.
It proves easier to express the problem in polar coordinates, as the radius is a constant so the wavefunction reduces to a function of only two variables, the angles θ and φ. (In Cartesian coordinates the surface of the sphere, and hence the wavefunction for the particle on this surface, is a function of the three coordinates x, y and z.) The Schrodinger equation is thus:
where Ψ is a function of θ and Φ, and is expressed in polar coordinates.
The method for solving this partial differential equation is that of separation of variables – the wavefunction is written as a product :
where Θ is a function of θ only, and Φ is a function of φ only.
Since the particle is at constant radius, parts of which involve differentiation with respect to r may immediately be discarded, as they must be equal to zero.
The above substitution is then made for Ψ, and the result may be manipulated to give differential equations for θ and φ.
Though it would be possible to work through the calculation, there is little to be gained from it; we shall merely quote the results that are obtained. The acceptable wavefunctions are specified by two quantum numbers, l and ml , which are restricted in value as follows:
i.e. l may take integer values from 0 upwards. For a given l , ml may take integer values from l to -l. Thus for a given l, there are (2l + 1) values of ml.
It follows from the solutions that the energies are restricted to discrete values:
The energies are quantised, but their value does not depend upon ml. It follows that an energy level with quantum number l is (2l + 1)-fold degenerate.
Each of the quantum numbers also has its own physical interpretation. The value of l determines the magnitude of the angular momentum of the system:
and the value of ml determines the value of the z-component of the angular momentum:
Note that this last result implies that a rotating body cannot take up an arbitrary orientation with respect to some specified axis (e.g. an axis defined by the direction of an externally applied electric field) – its orientation must be such that the z-component of the angular momentum obeys the above relation. This is known as space quantisation.