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

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:

-ψ(x)+ψ(x)=Eψ(x)

1

2

2

d

2

dx

1

2

2

x

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 (x)x exists.

ψ0

x∞

∞

∫

-∞

2

ψ