Minimum Mechanical Energy for Unbounded Motion

The minimum mechanical energy above which the motion of the particle cannot be bounded for any given initial condition in the potential V(x)=x²-x⁴ can be determined by analyzing the behavior of the potential function and considering the conditions for unbounded motion.

Understanding the Potential Function
The potential function V(x)=x²-x⁴ represents a one-dimensional potential energy landscape. It is a downward-opening parabola with a maximum value at the origin (x=0) and two local minima at x=-1 and x=1. The potential function approaches positive infinity as x approaches negative or positive infinity.

Determining the Boundaries of Motion
To determine the boundaries of motion, we need to consider the turning points of the potential function. These turning points occur where the derivative of the potential function is equal to zero.

Taking the derivative of the potential function with respect to x, we get:
V'(x) = 2x - 4x³

Setting V'(x) equal to zero and solving for x, we find the turning points:
2x - 4x³ = 0
2x(1 - 2x²) = 0
x = 0 or x = ±1/√2

Therefore, the turning points are located at x = 0, x = 1/√2, and x = -1/√2.

Analyzing the Energy Levels
To understand the behavior of the particle for different energy levels, we need to consider the total mechanical energy of the system. The total mechanical energy (E) is given by the sum of the kinetic energy (K) and the potential energy (V). Since the mass of the particle is unity, the kinetic energy is given by K = p²/2m = p²/2.

For bounded motion, the total mechanical energy of the particle must be less than or equal to the maximum value of the potential energy. In this case, the maximum value of the potential energy occurs at the origin, V(0) = 0.

Minimum Mechanical Energy for Unbounded Motion
For unbounded motion, the total mechanical energy of the particle must be greater than the maximum value of the potential energy. Since the potential energy approaches positive infinity as x approaches negative or positive infinity, there is no upper limit on the potential energy.

Therefore, the minimum mechanical energy above which the motion of the particle cannot be bounded for any given initial condition is infinity.

Conclusion
In the potential V(x)=x²-x⁴, the minimum mechanical energy above which the motion of the particle cannot be bounded for any given initial condition is infinity. This is because the potential energy approaches positive infinity as x approaches negative or positive infinity, making it impossible to confine the particle within any finite energy level.