Question

A classical particle confined to the positive \(x\) - cxis experiences a force whose potential energy is $$ U(x)=\frac{1}{x^{2}}-\frac{2}{x}+1 \quad(\mathrm{~S} 1 \text { units }) $$ (a) By finding its minimum value and determining its behaviors at \(x=0\) and \(x=\) toe, sketch this potential energy. (b) Suppose the particle has an energy of 0.5 . Find any turning points. Would the particle be bound? (c) Suppose the particle has an energy of \(2.0 \mathrm{~J}\). Find any turning points. Would the particle be bound?

Short answer

The minimum value of the potential energy function \(U(x)\) is \(-1\) at \(x=1\), the value of \(U(x)\) tends to infinity as \(x\) goes to zero, and \(U(x)\) tends to 1 as \(x\) tends to infinity. When the energy level is 0.5J, there are two turning points \(x=2\) and \(x=0.5\), and the particle is bound. With the energy level at 2.0J, there are no real turning points resulting in the particle being unbound.

Step-by-step solution

Sketching the Potential Energy Function

To achieve this, we need to find the minimum value of the potential energy function. This can be done by setting its derivative equal to zero and solving for x. The minimum value we get is \(-1\) at \(x=1\). At \(x=0\) the potential is infinity since there is an \(x^{2}\) in the denominator and as \(x\) approaches infinity, \(U(x)\) approaches 1. You can plot these values on a graph to visualize the function.

Calculate Turning Points At Energy = 0.5

Potential energy equals total energy when kinetic energy is zero, which happens at the turning points of the motion. Thus set \( U(x) = 0.5 \), and solve the resulting equation for \(x\). This gives two solutions, namely \(x=2\) and \(x=0.5\), meaning there are two turning points. The particle will remain within these points and hence will be bound.

Calculate Turning Points At Energy = 2.0

Proceed in a similar way by setting \( U(x) = 2.0 \) and solving for \(x\). The only possible solution to this results in a complex number, indicating that real turning points do not exist in this case. Without real turning points, the particle is not bound.

Key concepts

Classical Particle Dynamics

In classical mechanics, the motion of a particle can be described using Newton's laws of motion, which relate the movement of a body to the forces acting upon it. For a classical particle moving along one dimension, its dynamics are often understood through its potential energy function, denoted as \( U(x) \).

The potential energy is a scalar quantity, which means it's a single value that does not have a direction associated with it, unlike vector quantities like force. This energy is dependent on the position of the particle within a force field and provides a convenient way to calculate the forces acting on the particle, as force is the negative gradient of potential energy.

A key aspect of analyzing such a system is understanding that the particle's total mechanical energy, which is the sum of its kinetic energy (energy due to motion) and potential energy (energy due to position), is conserved if the system is closed and conservative. For instance, if a particle has a total energy of 0.5 Joules, that includes both its kinetic and potential energy at any given point during its motion.

Turning Points in Physics

Turning points occur when the kinetic energy of a particle is zero. In the context of a particle's motion, this is significant because it represents the locations where the particle changes direction. These points are where the potential energy equals the total mechanical energy of the particle.

To find these turning points, one sets the potential energy function equal to the particle's total energy. By solving this equation, one can determine the positions along the particle's path where it momentarily stops before reversing its direction of motion. For instance, if a classical particle has an energy of 0.5 Joules, locating the turning points involves solving the equation \( U(x) = 0.5 \).

Understanding turning points is crucial for visualizing the particle's motion within a potential well. It allows us to delineate the boundaries within which the particle is confined, marking the extremities of its motion. This can also tell us whether a particle is bound within a potential well or not, a fundamental concept in both classical and quantum mechanics.

Bound States in Potential Wells

The concept of bound states is essential when discussing the motion of particles in potential wells. A potential well is a region in space where a particle's potential energy is lower than at surrounding points, resembling a valley in a potential energy diagram.

A bound state refers to a particle's condition when its total mechanical energy is less than the potential energy at infinity. In such states, a particle is confined to the well due to its inability to acquire enough energy to 'climb out'. When analyzing the potential energy function \( U(x) = \frac{1}{x^2} - \frac{2}{x} + 1 \), we can determine whether a particle with a given energy is bound by comparing that energy to the potential outside the well.

When a particle has total energy less than the potential energy at infinity, real turning points exist, between which the particle oscillates. However, if the energy is equal to or greater than the potential energy far away from the well, the particle is free and not bound, as indicated by the absence of real turning points at an energy level of 2.0 Joules in our example.