Example 3.8: System of linear oscillators Show that for a system of one-dimensional linear harmonic oscillators with coordinate q given by q = Acos(ωt + θ), the phase θ in the range 0 ≤ θ ≤ 2π, the a priori probability P(q)dq is proportional to dq/



A2 − q2, and when normalized to 1 as a proper probability it has P(q) =

1/π



A2 − q2. [Hint: P(θ)dθ = dθ/2π].