Problem 1.11 – Griffith’s Intro to QM

Problem 1.11

[This problem generalizes Example 1.2.] Imagine a particle of mass $m$ and energy $E$ in a potential well $V(x)$ , sliding frictionlessly back and forth between the classical turning points ( $a$ and $b$ in Figure 1.10). Classically, the probability of finding the particle in the range $dx$ (if, for example, you took a snapshot at a random time $t$ ) is equal to the fraction of the time $T$ it takes to get from $a$ to $b$ that it spends in the interval $dx$ :

$\rho (x) dx = \dfrac{dt}{T} = \dfrac{(dt/dx)dx}{T} = \dfrac{1}{v(x)T}dx$ ,

where $v(x)$ is the speed, and $T = \int_{0}^{T} \mathrm{d}t = \int_{a}^{b} \dfrac{1}{v(x)} \mathrm{d}x$ .

Thus $\rho (x) = \dfrac{1}{v(x) T}$ . This is perhaps the closest classical analog to $|\Psi|^2$ .

(a) Use conservation of energy to express $v(x)$ in terms of $E$ and $V(x)$ .
(b) As an example, find $\rho (x)$ for the simple harmonic oscillator, $V(x) = kx^2/2$ . Plot $\rho (x)$ , and check that it is correctly normalized.
(c) For the classical harmonic oscillator in part (b), find $\langle x \rangle$ , $\langle x^2 \rangle$ , and $\sigma_x$ .