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