Consider a particle in a time-independent scalar potential.
Is separation of variables possible?
Assume solutions of the form Ψ(r,t) = Φ(r)ζ(t) exist.
Inserting Ψ(r,t) = Φ(r)ζ(t) into the Schroedinger equation we obtain
(iħ/ζ(t)) ∂ζ(t)/∂t = (1/Φ(r))[-(ħ2/(2m))∇2Φ(r) ] + U(r)Φ(r).
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
iħ∂ζ(t))/∂t = Eζ(t). ζ(t) = A exp(-iEt/ħ) = A exp(-iωt).
Here ω 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 H Φ(r) = E Φ(r), with H = -(ħ2/(2m))∇2 + U(r) being a differential operator, or
∇2Φ(r) + (2m/ħ2)(E - U(r))Φ(r) = 0.
If solutions Φn(r) can
be found, for different values of En, then the linear combination
Ψ(r,t) = ∑n cnΦn(r)exp(-iEnt/ħ)
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.