= Harmonic optical trap
{title2=$U(x)=kx^2/2,\ F(x)=-kx$}
A harmonic <optical trap> approximates one coordinate near a stable <equilibrium> by a quadratic <potential energy>. Measure <displacement> $x$ from that <equilibrium>, set its <potential energy> to zero there, and take a positive effective <spring constant> $k$, also called the trap stiffness:
$$
U(x)=\frac12kx^2,\qquad F(x)=-\frac{dU}{dx}=-kx,\qquad k>0.
$$
The <restoring force> points left when $x>0$ and right when $x<0$. This is a local model: a <Taylor series> of a smooth <potential energy> about a nondegenerate <local minimum> gives $k=U''(0)>0$ as its leading curvature. A real <optical trap> has a finite capture region; the endlessly rising parabola does not describe its escape boundary. A three-dimensional <optical trap> can have different stiffnesses along different axes, and nonconservative optical <forces> can limit the <potential energy> description.
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