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1.4 Ehrenfest’s Theorem and the Classical Limit

Summary: The form of classical mechanics which inspired Heisenberg’s formulation of Classical Mechanics allows us to see when particles should behave classically.

Using iℏ d d t ψ(t) = Ĥ ψ(t) and hence − iℏ dd t ψ(t) = ψ(t) Ĥ, and writing ⟨Ω ̂⟩≡⟨ψ(t)|Ω|ψ(t)⟩, we have Ehrenfest’s Theorem

d dt⟨Ω ̂⟩ = 1 iℏ⟨[Ω ̂,Ĥ]⟩ + ⟨∂Ω ̂ ∂t ⟩

The second term disappears if Ω ̂ is a time-independent operator (like momentum, spin...). Note we are distinguishing between intrinsic time-dependence of an operator, and the time-dependence of its expectation value in a given state.

This is very reminiscent of a result which follows from Hamilton’s equations in classical mechanics, for a function of position, momentum (and possibly time explicitly) Ω(p, x, t)

d dtΩ(p,x,t) = ∂Ω ∂x dx dt + ∂Ω ∂p dp dt + ∂Ω ∂t = ∂Ω ∂x ∂H ∂p −∂Ω ∂p ∂H ∂x + ∂Ω ∂t ≡ {Ω,H} + ∂Ω ∂t

where the notation {Ω,H} is called the Poisson bracket of Ω and H , and is simply defined in terms of the expression on the line above which it replaced. (For Ω = x and Ω = p we can in fact recover Hamilton’s equations for ṗ and ẋ from this more general expression.)

In fact for Ĥ = p̂2 2m + V (x̂), we can further show that

d dt⟨x̂⟩ = ⟨p̂ m ⟩and d dt⟨p̂⟩ = −⟨dV (x̂) dx̂ ⟩

which looks very close to Newton’s laws. Note though that ⟨d V (x ̂ )∕dx̂⟩≠d⟨V (x̂)⟩∕d⟨x̂⟩ in general.

This correspondence is not just a coincidence, in the sense that Heisenberg was influenced by it in coming up with his formulation of quantum mechanics. It confirms that it is the expectation value of an operator, rather than the operator itself, which is closer to the classical concept of the time evolution of some quantity as a particle moves along a trajectory.

Similarity of formalism is not the same as identity of concepts though. Ehrenfest’s Theorem does not say that the expectation value of a quantity follows a classical trajectory in general. What it does ensure is that if the uncertainty in the quantity is sufficiently small, in other words if Δx and Δp are both small (in relative terms) then the quantum motion will aproximate the classical path. Of course because of the uncertainty principle, if Δx is small then Δp is large, and it can only be relatively small if p itself is really large—ie if the particle’s mass is macroscopic. More specifically, we can say that we will be in the classical regime if the de Broglie wavelength is much less that the (experimental) uncertainty in x. (In the Stern-Gerlach experiment the atoms are heavy enough that (for a given component of their magnetic moment) they follow approximately classical trajectories through the inhomogeneous magnetic field.)

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