Schrödinger Equation: - $\frac{ħ²}{2m} \frac{d²}{dx²}$ Ψ = iħ $\frac{d}{dt}$ Ψ

Time Independent Schrödinger Equation: - $\frac{ħ²}{2m} \frac{d²}{dx²}$ Ψ = EΨ

Initial Conditions:

Ψ(0) :
Ψ'(0) :

D²Ψ(x) = -*Ψ(x)

s²F(s) - sf(0) - f'(0) = -2mE/ħ² F(s)
s²F(s) + 2mE/ħ²F(s) = sf(0) + f'(0)
F(s) = (sf(0)+f'(0))/(s²+2mE/ħ²)
F(s) = (1) / (s² + 1)

Position, Momentum, and Heisenberg Uncertainty (in Natural Units i.e. ħ=1)
Operator Name	Operator	PlaceValue Operator	Corresponding Wave Function
State Vector Ψ	Ψ	Ψ
Momentum Ψ	pΨ
Position Ψ	xΨ	.0123456 ⊗ .1 * Ψ
Momentum Position Ψ	pxΨ	10 * xΨ
Position Momentum Ψ	xpΨ	.0123456 ⊗ .1 * pΨ
Commutator Ψ	(px-xp)Ψ	pxΨ-xpΨ
Commutator	px-xp	(pxΨ-xpΨ)/Ψ	← Always 1 by Heisenberg Uncertainty