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

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

This is an ordinary 2nd-order differential equation. Thus, for each value $E\; \backslash in\; \backslash mathbb\{R\}$ 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 $\backslash hbar\; \backslash omega$.

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 $\backslash psi\; (x)$ must decay to 0 at infinity (technically known as the "boundary conditions").

For an analytical-yet-accessible derivation of the solutions see [https://www.lancaster.ac.uk/staff/schomeru/lecturenotes/Quantum%20Mechanics/S6.html here].

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