"The Darwin term" is one of the three contributions to the fine structure of hydrogen (and other atoms). It is a perturbation to the hydrogen hamiltonian, which gives rise to a change in energy levels of $O(\alpha^4m_ec^2)$ of the form (SI units): $$ H_D(\mathbf{r})=\frac{\hbar^2e^2}{8m_e^2c^2\epsilon_0}\delta^3(\mathbf{r}) $$

The original explanation of the Darwin term, and indeed some contemporary descriptions of it, describes it as the result of a "fast quantum fluctuation" in the position of the electron on a length scale $\hbar /m_e c=10^{-13}\text{ m}$. However, the only actual derivation from first principles I've ever seen is from the Foldy-Wouthuysian transform of the Dirac equation, where it doesn't seem to arise from anything like a fast fluctuating quantum motion.

And this kind of "quantum fluctuation" doesn't seem like it's a real phenomenon predicted in quantum mechanics. When an electron is in the groundstate of the hydrogen atom, its position isn't "quantum fluctuating around the proton." It's in a quantum state that can be described by a stationary wavefunction.

Is there genuinely some kind of way of looking at a solution to the Dirac equation and coming to the conclusion that the electron does some kind of rapid quantum motion of a length scale given by $\hbar /m_e c=10^{-13}\text{ m}$? (and presumably a timescale $\hbar /m_e c^2$) Or is my general impression right, that this is a historical description that needs to be thrown away because it just confuses students about the nature of quantum mechanics?

I am aware of this question, which is similar. But the question really just wants to know how zitterbewegung can be used to derive the form of the Darwin term, which is a perfectly straightforward derivation if you accept that zitterbewegung is a real thing. And I'm looking for a different kind of answer. All the accepted answer says is that "zitterbewegung is also correct, at least in the heuristic level," and I am explicitly wanting to know if this description of the Darwin term is valid beyond the heuristic level. Also note this great answer talks about what quantum fluctuations are, but none of these considerations seem to apply here, to a perturbation to the groundstate of a system.

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    $\begingroup$ I think the question would benefit from giving more background about what Darwin term is. $\endgroup$
    – Roger V.
    Apr 25, 2023 at 14:49
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    $\begingroup$ @RogerVadim done. $\endgroup$
    – AXensen
    Apr 25, 2023 at 14:52
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    $\begingroup$ I generally wonder where the term "quantum fluctuations" even came from. It does not fit either observations or theory. What we are observing are shifts in energy levels, not some kind of random noise phenomena. In non-relativistic perturbation theory these can be identified as fairly trivial summations over the states of the unperturbed system and the perturbation Hamiltonian, if I remember correctly and it's probably not much different in QED/QFT. If Breit and Schroedinger came up with the "Zitterbewegung" nonsense, then it's time to put an end to it. $\endgroup$ Apr 25, 2023 at 16:22

1 Answer 1


The main thing is to say is that you are correct about the extreme looseness of the talk of 'quantum fluctuations'.

All the so-called 'relativistic corrections' which make hydrogen fine structure are a result of doing a calculation incorrectly in the first instance and then fixing it up afterwards. That type of method is ok if and only if the sequence of approximation is well-defined. For example it is ok to do standard perturbation theory using Schrodinger's eqn when the non-relativistic theory is accurate enough but the Hamiltonian is complicated. But in the case of hydrogen fine structure the problem is not just a complicated Hamiltonian. The problem is that in order to get the kind of accuracy one is looking for, the single-particle approach to quantum theory is not up to the job. To be sure of getting it right we have to go via quantum field theory. This means that if we start out from single-particle non-relativistic QM then the situation is not a well-defined sequence of approximations as in perturbation theory. It is a case of a whole conceptual apparatus that is not up to the job being tweaked post-hoc.

The Darwin term can be justified post-hoc once one already knows what the answer is. But I think any attempt to argue it up front without doing the QED is bound to be unconvincing.

There is a reasonable presentation in Shankar's textbook on quantum mechanics I think.

We can get a pretty good calculation of hydrogen fine structure using the Dirac equation as you are probably aware. This method is intermediate between single-particle qm and the full apparatus of QED.


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