A comparison of quantum & classical trajectories

Link to download talk notebook.

Darrell F. Schroeter
Reed College
AAPT Winter Meeting 2026

Ehrenfest’s Theorem

In a typical introductory quantum mechanics course (Chap 1 of G&S), students will be introduced to Ehrenfest’s theorem,

(d/dt)⟨x⟩=(1/m)⟨p⟩                (d/dt)⟨p⟩=⟨F[x]⟩               F[x]=-(dV/dx) ,

as a way of touching base with Newton’s 2nd Law:

(d/dt)x=(1/m)p                         (d/dt)p=F[x]                        F[x]=-(dV/dx) .

Despite the formal similarity, ⟨x⟩ for a quantum particle and x(t) for a classical particle are not the same functions of time since ⟨F[x]⟩≠F([x]).

[see Styer, AJP 58 (8) for application of Ehrenfest to oscillator where ⟨F[x]⟩ = F ([x])]
[see Mukherjee et al, AJP 55 (9) for an example where the trajectories agree in the large-n limit]

Equations and identities

While the classical statement is a system of differential equations that can be solved for x[t], the Ehrenfest version...

(d/dt)⟨x⟩=(1/m)⟨p⟩                (d/dt)⟨p⟩=⟨F[x]⟩               F[x]=-(dV/dx)

...is not. The only way to find ⟨x⟩ for a quantum system is to solve the time-dependent Schrödinger equation for Ψ(x,t) and then calculate ⟨x⟩ from Ψ:

i ℏ(d/dt)Ψ[x,t]=(OverHat[H]) Ψ[x,t]             ⟨x⟩ = ∫(SuperStar[Ψ[x,t]]) x Ψ[x,t]dx

[an excellent paper on solving the time-dependent Schrödinger equation numerically is Schwarz et al, AJP 35 (3)]

Comparisons of trajectories

In this talk we are going to compare classical and quantum trajectories:

x[t] for a classical particle

(Subscript[⟨x⟩, Q])[t] for a quantum particle

(Subscript[⟨x⟩, C])[t] for a classical ensemble

We will do this with the most familiar system for which it is a meaningful comparison : Gaussian initial wave functions in an infinite square well.

There’s no new physics in this talk!  This and other results can be found in Ballentine et al, PRA 50 (4), 2854.

[if you’re concerned about F[x] for an infinite square well, see Rokhsar AJP 64(11)]

Classical trajectory of a particle in a box

A classical particle in a box simply bounces back and forth.

Show[xcpPlot]

Quantum dynamics of a particle in a box

Given an initial wave packet we can solve for Ψ[x,t]

s0 = (1/2); σs = (1/16);v0 = 40;Ψ0[s_]:= Exp[-(((((s-s0)/(2 σs))))^2)] Exp[(π/4)v0 I s];

ListAnimate[ΨTable]

Quantum vs. classical particle trajectories

Once you have Ψ[x,t] it is simple to calculate ⟨x⟩ as a function of t. We can then compare this with the classical trajectory.

Show[xqpPlot,xcpPlot]

Note that the correspondence at short times is nearly perfect!

Why don’t they agree?

Because of the Heisenberg uncertainty relation there is uncertainty in the quantum particle’s position and momentum.  But this isn’t uniquely quantum. An ensemble of classical particles also has this property!

Because of the wave nature of the quantum particle, it interferes with itself after bouncing off the wall.

Definition of classical ensemble

Now we consider a collection of classical particles where

ρ(x,p,t) dx dp = probability of finding a particle with position within dx of x and momentum within dp of p at time t .

We can construct a classical ensemble that has the same position and momentum distributions as our quantum particle,

ρ(x,p,t)  = ((Abs[Ψ[x,t]])^2) ((Abs[Φ[p,t]])^2) ,

where Φ(p,t) is the momentum-space wave function.

Dynamics of classical ensemble

The code on this page calculates the evolution of a classical ensemble in an infinite square well in time.

ListAnimate[ρTable]

Quantum vs . classical particle vs. classical ensemble trajectories

Once we have ρ[t], it is simply to calculate its centroid: (Subscript[⟨x⟩, c])(t).

Show[xqpPlot,xcpPlot,xcePlot]

For short times, the trajectories of the classical ensemble and quantum particle are remarkably similar.  (They diverge for longer times.)

How to use in a quantum class ...

show comparison of quantum particle to the classical particle after discussing infinite square well

show comparison of quantum particle to the classical ensemble after discussing the momentum-space wave function

makes great student projects!  students could investigate what happens as you increase the particle’s energy or what happens with other potentials