A particle of mass $m$ moves in a potential field $U(x)=a{x}^2-b{x}^4$, where $a$ and $b$ are positive constants. What is the minimum speed required for the particle to escape to infinity starting from the origin?

Solution

1. The potential energy is $U(x)=a{x}^2-b{x}^4$.
2. As $x\to \infty$, $U(x)\to -\infty$.
3. The particle encounters a potential barrier at $x=\sqrt{a/(2b)}$ with height $U_{max}=a^2/(4b)$.
4. Since the potential decreases indefinitely for large $x$, the particle is trapped in a local well and cannot reach infinity.
5. Hence the answer is (A).