A particle in a time-independent scalar potential

The Schroedinger equation is

.

Is separation of variables possible?

Let us try

y(r,t)=f(r)c(t)

This yields

.

The left hand side of the equation is a function of t, independent of r, and the right hand side is a function of r, independent of t.  Both sides must therefore be equal to a constant, which we call E.  The left hand side then yields

.

Here w is the frequency of the wave function solution, and the de Broglie relation identifies E as the energy of the particle.  The right hand side yields Hf(r)=Ef(r), with

being a differential operator, or

.

Summary

If V=V(r), then solutions of the type f(r)c(t) exist.  They are called stationary solutions or stationary states.  They correspond to states in which the particle has a well defined energy E=hw.  f(r) is a solution of Hf(r)=Ef(r) the time-independent Schroedinger equation.  The time-independent Schroedinger equation is an eigenvalue equation of the linear operator H.  The allowed energies are called the eigenvalues of H.

If solutions f_n(r) can be found, for different values of E_n, then the linear combination 

is also a solution of the Schroedinger equation, because the Schroedinger equation is a linear equation.  However, such a linear combination does not represent a particle with a well defined energy.