Question

What is the shape of the phase space trajectory for a harmonic oscillator?

• A. A straight line
• B. A circle
• C. An ellipse
• D. A parabola

Hint:

Conserved energy \(E = p^2/2m + \tfrac12 m\omega^2 x^2\) is the equation of an ellipse in the \((x,p)\) plane.

Answer 1

The Correct Option is C

Step 1: For a one-dimensional simple harmonic oscillator of mass \(m\) and angular frequency \(\omega\), the total energy is conserved: \[E = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2.\]

Step 2: Rewrite this equation in the phase plane with coordinates \((x, p)\): \[\frac{p^2}{2mE} + \frac{x^2}{\,2E/(m\omega^2)\,} = 1.\]

Step 3: This is the standard form \(\dfrac{p^2}{b^2} + \dfrac{x^2}{a^2} = 1\), the equation of an ellipse with semi-axes \(a = \sqrt{2E/(m\omega^2)}\) along \(x\) and \(b = \sqrt{2mE}\) along \(p\).

Step 4: Since the two semi-axes are generally unequal, the trajectory is an ellipse (a circle only in special scaled units where \(a=b\)).\[\boxed{\text{An ellipse}}\]