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

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

This is an ordinary 2nd-order differential equation. Thus, for each value ${\displaystyle E\in \mathbb {R} }$ 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 ${\displaystyle E}$ is scanned in steps of 1/40 in units of ${\displaystyle \hbar \omega }$.

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 ${\displaystyle E}$ 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 ${\displaystyle \psi (x)}$ 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).