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

​
energy
allowed energy values (even parity)
0.5
2.5
4.5
6.5
allowed energy values (odd parity)
1.5
3.5
5.5
7.5
parity
even
odd
This illustrates the quantized solutions of the Schrödinger equation for the one-dimensional harmonic oscillator:
-
1
2
2
d
2
dx
ψ(x)+
1
2
2
x
ψ(x)=Eψ(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
ψ0
as
x∞
and normalization is possible when
∞
∫
-∞
2
ψ
(x)x
exists.
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