Reading a Potential Energy Curve — Question 3

Question 3

Disconnected allowed regions in a double well

Assume one-dimensional classical motion with $K=E-U(x)$ and $F_x=-dU/dx$, with no dissipation. Let $U(x)=(x^2-1)^2$ J, with x in meters, and E=0.25 J. The particle starts in the left well.

1. Find every allowed x interval.

2. Can it reach the right well without external energy transfer?

3. Locate the stable and unstable equilibria and compare the barrier with E.

Question 3 - Solution

Disconnected allowed regions in a double well

(a) Solve the inequality in two stages.

Solve the allowed-energy inequality in stages: $$(x^2-1)^2\leq.25\Rightarrow|x^2-1|\leq.5\Rightarrow.5\leq x^2\leq1.5.$$ $$\boxed{x\in[-\sqrt{1.5},-\sqrt{.5}]\ \cup\ [\sqrt{.5},\sqrt{1.5}]\ \mathrm m}.$$

(b) Keep the trajectory in a connected allowed region.

The central barrier is forbidden: $$U(0)=1\ \mathrm J>E=.25\ \mathrm J.$$ A left-well trajectory cannot reach the disconnected right allowed interval without additional energy. Allowed endpoint energy alone does not establish reachability.

(c) Read extrema and local force directions.

Stationary points satisfy $$U'=4x(x^2-1)=0\Rightarrow x=0,\pm1,\qquad U''=12x^2-4.$$ Thus $\boxed{x=\pm1\text{ stable};\quad x=0\text{ unstable}}$. The barrier is 0.75 J above the trajectory energy.