Problem 1.12 – Griffith’s Intro to QM

Problem 1.12

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 $\rho (p)$ (note that $p$ ranges from $- \sqrt{2mE}$ to $+ \sqrt{2mE}$ ).
(b) Calculate $\langle p \rangle$ , $\langle p^2 \rangle$ , and $\sigma_p$ .
(c) What’s the classical uncertainty product, $\sigma_x \sigma_p$ , for this system? Notice that this product can be as small as you like, classically, simply by sending $E \rightarrow 0$ . But in quantum mechanics, as we shall see in Chapter 2, the energy of a simple harmonic oscillator cannot be less than $\hbar \omega /2$ , where $\omega = \sqrt{k/m}$ is the classical frequency. In that case what can you say about the product $\sigma_x \sigma_p$ ?