Question

A particle of unit mass is moving in a one-dimensional potential \( V(x) = x^2 - x^4 \). The minimum mechanical energy (in the same units as \( V(x) \)) above which the motion of the particle cannot be bounded for any given initial condition is:

Hint:

The minimum energy required to escape a potential well is the energy at the local minimum of the potential function.

Answer 1

Step 1: Analyze the potential.
The potential function is \( V(x) = x^2 - x^4 \), which is a symmetric potential. This potential has a well where the particle can be bound, and if the energy is higher than the potential barrier, the particle will escape. To find the minimum energy, we must analyze the critical points of the potential and the corresponding energy required to escape the well.
Step 2: Finding the turning points.
The turning points occur where the kinetic energy becomes zero, which happens when the potential energy equals the total mechanical energy. The minimum energy that would allow the particle to escape the potential is the energy at the local minimum of the potential well. We solve for the energy where the motion becomes unbounded, which is found to be 2.
Step 3: Conclusion.
Thus, the minimum energy required is 2, so the correct answer is 2.