What if we were interested in the distribution of momenta (p=mv), for the classical harmonic oscillator (Problem 1.11(b)).
(a) Find the classical probability distribution ρ(p) (note that p ranges from −2mE to +2mE).
(b) Calculate ⟨p⟩, ⟨p2⟩, and σp.
(c) What's the classical uncertainty product, σxσp, for this system? Notice that this product can be as small as you like, classically, simply by sending E→0. But in quantum mechanics, as we shall see in Chapter 2, the energy of a simple harmonic oscillator cannot be less than ℏω/2, where ω=k/m is the classical frequency. In that case what can you say about the product σxσp?