File:SolveTimeIndepSchroedingerEqQuantumHarmonicOsc.gif

Description
General solutions of the time-independent 1D Schrödinger (differential) equation with the harmonic oscillator potential

$$ E\psi(x) = -\frac{\hbar^2}{2m}\frac{d^2}{d x^2}\psi(x) + \frac{1}{2} m\omega^2 x^2\psi(x) $$

This is an ordinary 2nd-order differential equation. Thus, for each value $$ E \in \mathbb{R} $$ the set of solutions forms a vector space spanned by two linearly-independent basis functions. I have chosen the even-odd basis functions because the parity operator commutes with the Hamiltonian. The value $$ E $$ is scanned in steps of 1/40 in units of $$ \hbar \omega $$.

The normalisation postulate of quantum mechanics requires us to select only those solutions that are normalisable, as these will be the only feasible results obtained from a measurement. Then, each normalised function (also called a Stationary_state or simply "standing wave") acquires the physical meaning of "probability amplitude" or Wave_function, and the associated eigenvalue $$ E $$ acquires the meaning of "(eigen)energy". The rest of the solutions are unphysical, and thus discarded. Typically, this results in a discrete distribution of energies, which puts the "quantum" in quantum mechanics. In this case, the normalisation condition is equivalent to saying that the solutions $$ \psi (x) $$ must decay to 0 at infinity (technically known as the "boundary conditions").

For an analytical-yet-accessible derivation of the solutions see here.

The energy levels are non-degenerate (see Griffiths, for example).