Note 06 Back to Ch. 04 · Electron Diffraction
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A caveat The mean of the field and the field at the mean position

For agreement with Newton’s equation, in the chapter we require that

d2⟨x⟩/dt2=q⟨E(x)⟩/md^2\langle x\rangle/dt^2=q\langle E(x)\rangle/m

On the right-hand side there is the mean of the field, computed over the probability distribution of position:

⟨E(x)⟩=∫−∞+∞p(x)E(x)  dx\langle E(x)\rangle=\int_{-\infty}^{+\infty}p(x)E(x)\;dx

It is not the same as writing E(⟨x⟩)E(\langle x\rangle), the field evaluated at the mean position. The latter is the quantity Newton’s equation speaks of, since it describes a particle at a point. The two agree if E(x)E(x) depends linearly on xx, and in general they do not: the mean of a function is not the function of the mean.

Let us see by how much they differ. We expand the field about the mean position:

E(x)=E(⟨x⟩)+dEdx∣⟨x⟩(x−⟨x⟩)+12d2Edx2∣⟨x⟩(x−⟨x⟩)2+…E(x)=E(\langle x\rangle)+\left.\frac{dE}{dx}\right|_{\langle x\rangle}(x-\langle x\rangle)+\frac{1}{2}\left.\frac{d^2E}{dx^2}\right|_{\langle x\rangle}{(x-\langle x\rangle)}^2+\dots

Taking the mean term by term, the first term is already a number, the second vanishes because ⟨x−⟨x⟩⟩=0\langle x-\langle x\rangle\rangle=0, and there remains

⟨E(x)⟩=E(⟨x⟩)+12d2Edx2∣⟨x⟩⟨(x−⟨x⟩)2⟩+…\langle E(x)\rangle=E(\langle x\rangle)+\frac{1}{2}\left.\frac{d^2E}{dx^2}\right|_{\langle x\rangle}\langle{(x-\langle x\rangle)}^2\rangle+\dots

The discrepancy is governed by the variance of the position multiplied by the second derivative of the field. It is negligible when the probability distribution is narrow compared with the scale over which the field varies.

Why this is enough. It is the condition set by the fourth principle stated in the introduction: agreement with Newton is required under the conditions in which Newton’s laws are borne out by experience, and those are precisely the conditions in which the particle is well localised compared with the way the field varies. When the distribution is broad, the mean position follows no Newtonian trajectory at all — and that is where the theory says something classical mechanics does not.

The equations for the mean values obtained in the chapter are known as Ehrenfest’s theorem (1927); the condition just seen is the one the theorem requires for the mean motion to be Newtonian.

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La Quantistica · Note No. 06 · Rev. 2026 F. Palma