Example 11.7: The ergodic hypothesis and the linear harmonic oscillator The classical ergodic hypothesis claims that the phase space trajectory (traced by (q(t), p(t))) of a system passes through each point of the hyperplane E = H(q, p). Show that the classical linear harmonic oscillator with H(q, p) =

p2 2m

+

1 2

mω2q2 (11.77)

satisfies this claim exactly.

[Hint: Show that q =



2E/mω2 sin(ωt + δ), p = √2mE cos(ωt + δ), and that after period 2π/ω

every point of the hyperplane has been passed through].