Separation of Variables

• In three dimensions (Cartesian coordinates) the TISE is

• If the potential energy separates into the sum of parts

then we can find solutions that factor into parts:

• Substituting into the TISE we find

• Dividing by f_xf_yf_z we have

• Each term in [] depends only on one variable. The only way they can all add up to a constant (E) for all values of x,y,z is for each term in [] to be a constant, with the sum of these constants equal to E. Thus we must have

where

• These are three ordinary differential equations of the eigenvalue type.

• This type of solution is "separated". Not all solutions are of this type, but any solution can be written as a superposition of separated solutions.

• A similar technique can be applied in certain other coordinate systems, including spherical polar coordinates.

Particle in a Cubical Box

• We consider as an example the 3-D version of the particle in a box. The potential is zero inside a cube of side a centered on the origin, and infinite elsewhere. Formally

• We look for the separated solutions. For the x part we have

In the forbidden region the wavefunction vanishes.

• As we found earlier, the solutions for this problem are

• Thus the contribution to the energy from this part is

• The problem for the y and z parts is exactly the same, so the total energy is given by

• The wavefunction is a product of the three parts corresponding to the three quantum numbers n₁, n₂, n₃.

Degeneracy

• We see that there are (in some cases) more than one way to get a given energy level. For example, n₁ = 2, n₂ = 3, n₃ = 1 gives the same energy as n₁= 1, n₂ = 2, n₃ = 3.

• But the wavefunctions for these two choices are different. In the first case we would have

while in the second case we would have

• Thus we have two (or more) different wavefunctions corresponding to the same energy. This energy level is said to be degenerate. The degree of degeneracy is the number of distinct wavefunctions that correspond to the same energy.

• Degeneracy arises when the Hamiltonian exhibits some kind of symmetry. In this case it is unchanged if we interchange, for example, x and y. If the sides of the box were of unequal length, the degeneracy would disappear.

Central Forces

• For a force directed radially from or toward a point (usually taken to the the coordinate origin) and whose strength is independent of direction, the potential energy is a function only of the radial distance. This is a central force. The Coulomb and gravitational forces are important examples.

• It is very convenient in such cases to use spherical polar coordinates, defined by the diagram:

• The transformation from cartesian coordinates is given by:

• The Laplacian operator

becomes

• For a central force the potential energy is U(r). The TISE becomes

• We look for separated solutions as before. Let

• Substitution and division by F/r² gives

• The pieces with [] must separately be the same constant. Call it -K. Then we have

• For the angular part we have

• Note that this does not contain Planck's constant, so it has nothing specifically to do with quantum mechanics. It arises from Laplace's equation in classical physics.

• The solutions Y(q,f) are well known functions, called "spherical harmonics".

• Nevertheless, there is a quantum mechanical importance: these functions are the eigenfunctions of the angular momentum operators in quantum mechanics.

• We will postpone solving this equation until we have looked at the problem of constructing the angular momentum operators.