F⃗ (x) = −k x⃗

U(x) = \frac{1}{2} k x^{2}

\frac{−h̄^{2}}{2m} \frac{d^{2}Ψ}{dx^{2}} + UΨ = EΨ

\frac{−h̄^{2}}{2m} \frac{d^{2}Ψ}{dx^{2}} + \frac{1}{2}kx^{2}Ψ = EΨ

\frac{d^{2}Ψ}{dx^{2}} + −\frac{2m}{h̄^{2}} \frac{1}{2}kx^{2}Ψ = −\frac{2m}{h̄^{2}}EΨ

\frac{d^{2}Ψ}{dx^{2}} = \frac{m}{h̄^{2}} kx^{2}Ψ − \frac{2m}{h̄^{2}}EΨ

\frac{d^{2}Ψ}{dx^{2}} = (\frac{km}{h̄^{2}})x^{2}Ψ − (\frac{2mE}{h̄^{2}} )Ψ

Ψ(x) = Ae^{−αx^{2}}

\frac{dΨ(x)}{dx} =

\frac{dΨ(x)}{dx} = −2αxAe^{−αx^{2}} = −2xαΨ

\frac{d^{2}Ψ}{dx^{2}} = −2αAe^{−αx^{2}} + −2αxA(−2αx)e^{−αx^{2}}

\frac{d^{2}Ψ}{dx^{2}} = −2αAe^{−αx^{2}} + 4α^{2}x^{2}Ae^{−αx^{2}}

\frac{d^{2}Ψ}{dx^{2}} = −2αΨ + 4α^{2}x^{2}Ψ

x(t) = A cos(ωt)

E = \frac{1}{2}kA^{2}

ω = \sqrt{\frac{k}{m}}

\frac{dΨ}{dx} = (...)x Ψ

α = \frac{\sqrt{km}}{2h̄}

E = \frac{1}{2}h̄\sqrt{\frac{k}{m}}

L ∼ \frac{1}{\sqrt{α}}

E = \frac{1}{2}h̄ ω

E = h ν

Ψ_{0}(x) = Ae^{−αx^{2}}

Ψ_{1}(x) = Bxe^{−αx^{2}}

E_{0} = \frac{1}{2}h̄ ω

E_{1} = \frac{3}{2}h̄ ω

E_{n} = \frac{(2n + 1)}{2} h̄ ω