U(r) = K [(\frac{σ}{r} )^{12} − (\frac{σ}{r} )^{6}]

U(r) = [(\frac{1}{r})^{12} − (\frac{1}{r})^{6}]

f(x) = f(x_{0}) + (x − x_{0})\frac{df}{dx}|_{x_{0}} + \frac{1}{2}(x − x_{0})^{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}} + \frac{1}{6}(x − x_{0})^{3} \frac{d^{3}f}{dx^{3}} |_{x_{0}} + ...

f(x) = f(x_{0}) + ∑_{n=1}^{∞} \frac{1}{n!}(x − x_{0})^{n} \frac{d^{n}f}{dx^{n}} |_{x_{0}}

f(x) = f(x_{0}) + 0 + \frac{1}{2}(x − x_{0})^{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}} + \frac{1}{6}(x − x_{0})^{3} \frac{d^{3}f}{dx^{3}} |_{x_{0}} + ...

f(x) ≃ f(x_{0}) + \frac{1}{2}(x − x_{0})^{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}} + ...

f(x) ≃ f(x_{0}) + [\frac{1}{2} \frac{d^{2}f}{dx^{2}} |_{x_{0}}] (x − x_{0})^{2}

U(x) ≃ U(x_{0}) + [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})^{2}

Force F (x) = − \frac{∂U}{∂x}

\frac{∂U}{∂x} = \frac{∂}{∂x}\{ U(x_{0}) + [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})^{2}\}

= \{ 0 + 2 [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})\}

= [\frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})

(U(x) − U(x_{0})) ≃ [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})^{2}

(\frac{U(x) − U(x_{0})}{x − x_{0}}) ≃ [\frac{1}{2} \frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})

F (x) ≃ − [\frac{∂^{2}U}{∂x^{2}} |_{x_{0}}] (x − x_{0})

F (x) ≃ −K(x − x_{0})

F (x) = −Kx

m a(x) = −Kx

m a = −Kx

m \frac{d^{2}x}{dt^{2}} = −Kx

\frac{d^{2}x}{dt^{2}} = − (\frac{K}{m} ) x

ω ≡ \sqrt{\frac{K}{m}}

\frac{d^{2}x}{dt^{2}} = − ω^{2}x

x(t) = A cos(ωt + ϕ)

x(t) = A sin(ωt) + B cos(ωt)

h(x) = H [1 − cos(\frac{2πx}{L})]