The Laplacian in spherical coordinates can be derived from eq87, eq88 and eq89. Substituting eq78 through eq86 in eq87, eq88 and eq89, adding the resulting three equations and simplifying, we have

or

It can be easily shown that eq47 is equivalent to eq46 by computing the derivatives in eq47. Hence, eq45 in spherical coordinates is

For the case of a particle confined to a spherical surface of zero relative potential, is constant and . The first term of the Laplacian operating on a wavefunction will return a result of zero. We can therefore discard it and eq48 becomes:

Substitute eq96 in the above equation

where .

Hence, an eigenvalue of is the energy associated with the angular motion of a particle. This implies that the linear (or radial) part of the Hamiltonian is:

Similarly, an eigenvalue of is the energy associated with the linear motion of a particle. We can, therefore, rewrite eq48 as: