Imagine a particle of mass m and energy E in a potential well , 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:

This is perhaps the closest classical analog to |Ψ|².
(a) Use conservation of energy to express v(x) in terms of E and V(x).
(b) As an example, find ρ(x) for the simple harmonic oscillator, V(x) = kx²/2. Plot ρ(x), and check that it is correctly normalized.
(c) For the classical harmonic oscillator in part (b), find (x), (x²), and σ_x.

Figure 1.10: Classical particle in a potential well.

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p(x) dx = T = where U (x) is the speed, and ( 2 (1) Thus dt p(x) = dt = (dt/dx) dx T 1 v(x) T dx. = 1 v (x) T dx, (1.41) (1.42) (1.43)