F⃗ = ∫ −kx⃗

U = \frac{1}{2}kx^{2}

\begin{aligned}Ψ(x) & = Ae^{−αx} \\ Ψ(x) & = Axe^{αx} \\ Ψ(x) & = Ae^{−αx^{2}} \end{aligned}

\frac{−h̄^{2}}{2m} \frac{∂^{2}Ψ}{∂x^{2}} + UΨ = EΨ

\frac{−h̄^{2}}{2m} \frac{∂^{2}Ψ}{∂x^{2}} + \frac{1}{2}kx^{2}Ψ = EΨ

\frac{∂^{2}Ψ}{∂x^{2}} = −(\frac{2m}{h̄^{2}})(E − \frac{1}{2}kx^{2})Ψ

\frac{∂^{2}Ψ}{∂x^{2}} = (\frac{2m}{h̄^{2}})(\frac{1}{2}kx^{2} − E)Ψ

\frac{∂^{2}Ψ}{∂x^{2}} = (\frac{m}{h̄^{2}} )kx^{2}Ψ − (\frac{2m}{h̄^{2}})EΨ

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

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