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General solutions of the 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, the solution space is 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 ${\displaystyle \hbar \omega }$.

The normalisation postulate of quantum mechanics requires us to select only those solutions that are normalisable. Then, each normalised function (also called a Stationary_state or simply "standing wave") acquires the physical meaning of "probability amplitude density" of Wave_function, and the associated value of E acquires the meaning of "Energy eigenvalue". The rest of the solutions are unphysical, and 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").