Question

A particle performs harmonic oscillations along the x axis according to the law x = a cosωt. Assuming the probability P of the particle to fall within an interval from —a to +a to be equal to unity, find how the probability density dP/dx depends on x. Here dP denotes the probability of the particle falling within an interval from x to x + dx. Plot dP/dx as a function of x.

Answer 1

As the motion is periodic the particle repeatedly passes through any given region in the range - a ≤ x ≤ a. The probability that it lies in the range (x,x + dx) is defined as the fraction Δt/t(as t→∞) where Δt is the time that the particle lies in the range (x, x + dx) out of the total time t. become of periodicity this is

Where the factor 2 is needed to take account of the fact that the particle is in the range (x, x + dx) during both up and down phases of its motion. now in a harmonic oscillator.