A classical particle of mass m moves in 1 dimension and is attached to an ideal spring of spring constant mw_0^2 . Ignore friction. Take the equilibrium position of the particle to be the origin of the particle’s position x.
a) Calculate the probability density p(x,E) , such that p(x,E)dx is the time- averaged probability of finding the particle between x and x+dx for infinitesimal dx, when the particle has energy E. Those words mean p(x,E)dx is the time a particle with energy E spends between x and x+dx divided by the total time, for total time going to infinity.
b) Calculate the average value of x, x^2 and the standard deviation.