Probability density. consider a mass m on a spring with spring constant k, oscillating in simple harmonic fashion about x = 0 between the extremes x = −a and x = a. the motion is entirely classical.

(a) what is the probability density rho(x) for the mass’s position? in other words, find the function rho(x) such that rho(x)dx is the probability of observing the mass between positions x and x + dx when observed at a random moment in time. (griffiths examines a similar problem in example 1.1.)

(b) verify that your result integrates to a total probability of 1.

(c) calculate hxi and hx 2 i, the average values (expectation values) of x and x 2 , respectively, and write down σx, the standard deviation in x. (if you encounter integrals you don’t know how to evaluate, look them up in a table or in a mathematical software package.) how could you have predicted the result for hxi without having to calculate?

(d) sketch rho(x) between x = −a and x = a. where is it smallest, and where is it largest? explain why these results are sensible