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 
(b) Calculate (p), (p²), and σ_p.

(c) What’s the classical uncertainty product, σ_x σ_p, 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 hw/2, where is the classical frequency. In that case what can you say about the product σ_x σ_p?

Problem 1.11(b)

(b) Determine Δj for each j, and use Equation 1.11 to compute the standard deviation.

Equation 1.11

Transcribed Image Text:

- √2mE to + √2m E).