I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method.
If we suppose that the solutions have the form $$\Psi(x,t)=T(t)\psi(x)$$ we obtain two equations, the first one give the time evolution phase factor $T_n(t)=e^{-iE_nt/\hbar}$ and the other one the "spatial wave function" $\psi_n(x)$.
So all the separable solutions have the form $$\Psi_n(x,t)=e^{-iE_nt/\hbar}\psi_n(x)$$ and these represents the stationary states.
If we sum these solutions we can obtain even non-separable solutions $$\Psi=\sum C_n\psi_nT_n.$$
But I can't find any postulate or theorem which states that every solution of the Schroedinger equation can be expressed in this form.
Are all the possible solutions of the equation expressible by (infinite) sum of separable solutions?
If I recall correctly math courses this can be expressed asking if the Hamiltonian operator eigenvectors are a complete set (basis) of the Hilbert space.
And in the case of continuous spectrum?
Best Answer
This is indeed possible only in some situations, e.g. when the continuous spectrum is absent (it may also consist of a single point, see Valter Moretti's comment below). A sufficient condition for that to be true is that either the Hamiltonian is compact or it has compact resolvent.
Sadly, very few interesting Hamiltonians satisfy that property (an example being the harmonic oscillator). In general, the solution of the Schrödinger equation exists for any initial condition $\psi_0\in\mathscr{H}$ (the Hilbert space), using the unitary one-parameter strongly continuous group $e^{-it H}$ associated to the self-adjoint Hamiltonian $H$ by Stone's theorem. The solution at any time $t\in \mathbb{R}$ is then simply written $$\psi(t)=e^{-itH}\psi_0\; .$$ Such solution is continuous in time, and with respect to initial data, but it is differentiable in time only if $\psi_0\in D(H)$, where $D(H)$ is the domain of the self-adjoint operator $H$. In the language of analysis of PDEs, that means that the Schrödinger equations, for self-adjoint Hamiltonians, are globally well posed on $\mathscr{H}$.