Quantized Solutions of the 1D Schrödinger Equation for a Harmonic Oscillator

This illustrates the quantized solutions of the Schrödinger equation for the one-dimensional harmonic oscillator:

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As you vary the energy, the normalization and boundary conditions (for even or odd parity) are only satisfied at discrete energy values of the solution of the second-order ordinary differential equation. Boundary conditions are met when as and normalization is possible when exists.