In a harmonic oscillator, the relationship between momentum (p) and displacement (q) can be understood using the concept of energy conservation.
The total mechanical energy of a harmonic oscillator, which includes both kinetic and potential energy, remains constant throughout the oscillation.
The kinetic energy (KE) of the oscillator is directly proportional to the square of its momentum (KE = 1/2 * m * p²), while the potential energy (PE) is directly proportional to the square of its displacement (PE = 1/2 * k * q²). Here, m represents the mass of the object and k represents the spring constant.
As the total mechanical energy remains constant, changes in momentum and displacement are inversely proportional to each other. This leads to an elliptical relationship between p and q in the graph.
Therefore, the correct option is 4) an ellipse, as an ellipse is a curved shape that perfectly depicts the relationship between momentum and displacement in a harmonic oscillator.