Who assigned the term ''Born's rule'' to the statement that the measurement of a quantum observable is one of its eigenvalues, with a probability given by the square of the coefficient in the spectral expansion of the system state?

I checked the quantum mechanics books by von Neumann and by Landau and Lifschitz, and didn't find this term.

Edit: In view of the answers and their discussion below, my precise question is when Born's name was first linked (as Born's rule or something similar) to an explicit statement about measuring arbitrary observables.

Edit (March 13,2019): Some of the early history of Born's rule (as I now see it) can now be found in Subsections 3.1 and 3.2 of my paper in arXiv:1902.10778. I am grateful for the help of Francois Ziegler, who had answered below, in the early stages of the literature search.

  • 1
    $\begingroup$ This would be better for hsm.SE. Born's paper (in which he gets the rule wrong and then corrects it in a footnote) was in 1926. A google ngrams search on "Born's rule" shows that the term began to be used in 1946 and "Born rule" not until 1969. $\endgroup$
    – user466
    May 6, 2018 at 20:33
  • $\begingroup$ @BenCrowell: How exactly did you do the search? My attempts gave 1949 as first use. $\endgroup$ May 7, 2018 at 7:04
  • 2
    $\begingroup$ why the downvote? This is a legitimate history question, and the answer is not readily available anywhere. $\endgroup$ May 8, 2018 at 10:18

2 Answers 2


I wonder why the insistence on the (English) word rule, especially as German wikipedia translates / redirects it to interpretation. Isn’t it enough for your purposes to see it stated, named and credited as Born’s Deutung (Jordan 1927, p. 811), assumption (Dirac 1927, p. 257), Interpretation (Hilbert et al. 1928, p. 29), or Auffassung (Schrödinger 1927, p. 967; Handbuch der Physik 1928, p. 589)?

To your edit asking “who first linked Born’s name to an explicit statement about measuring arbitrary observables” (though nothing prevents a hamiltonian from being “arbitrary”...): if not von Neumann (1932, footnotes 8, 118, 122), then I’d guess E. Bauer (1933, p. 42; translation 1962, p. 39) who writes, for a superposition $\psi=\smash{\sum_k\beta_k\psi_k}$ of eigenstates belonging to eigenvalues $\alpha_k$ of a “physical quantity” $A$,

Nous admettrons qu’une expérience faite sur un système à l’état $\psi$ peut nous donner l’une quelconque de ces valeurs $\alpha_k$ avec une probabilité égale à $\overline\beta{}_k\beta{}_k$.

Ce dernier postulat entrevu par Einstein, énoncé en toute précision par Born et développé par Dirac est comme la clef de voûte de l’édifice quantique.

(The mention of Einstein is also in Born (1926, p. 804; 1978, p. 232), Heisenberg (1927, p. 176), Pais (1982, pp. 1196-1197).) Bauer may have the first occurrence of “Born’s rule”, too, in Théorie quantique de la valence. Les liaisons homéopolaires. Bull. Soc. Chim. France. Mém. (5) 1 (1934) 293-347, p. 302:

Multiplions d'autre part $|\psi|^2$ par le volume $4\pi r^2dr$ compris entre les sphères de rayons $r$ et $r + dr$, nous obtenons, d’après la règle de Born, la probabilité de rencontrer l’électron dans la couche sphérique ainsi délimitée, c’est-à-dire à une distance du noyau comprise entre $r$ et $r + dr$.

Finally and perhaps most influentially, I would say that Born’s book Atomic Physics (1935) has everything you asked for (and essentially insisted he never wrote...):

§V.6. The Statistical Interpretation of Wave Mechanics.

We have already mentioned the interpretation of the wave function given by the author (p. 83). Let the proper function corresponding to any state be $\psi_E$; then $\smash{|\psi_E|{}^2dv}$ is the probability that the electron (regarded as a corpuscle) is in the volume element $dv$. (...)

Appendix XXII. The Formalism of Quantum Mechanics, and the Uncertainty Relation. (...)

The statistical interpretation of quantum mechanics consists in the following assumptions: To each physical quantity or “observable” belongs a real operator $A$. The proper functions $\psi_1$, $\psi_2$, ... correspond to the quantised states, for which the operator takes the value $a_1$ or $a_2$ or $a_3$ ... ; any function $\phi$ is a state, which is composed of these states (...) The coefficients $c_n$ of the expansion determine the strength with which the quantum state $n$ occurs in the general state $\phi$. The probability of then finding the proper value $a_n$ in a measurement is given by $\smash{w_n=|c_n|^2}$.

In fact, Born already wrote the same things, slightly less polished, in proceedings of the Fall 1927 conferences in Como (translation p. 15; reprint p. 16) and Brussels (pp. 166–170). There he credits himself (1926, 1927) as well as Dirac (1927), Jordan (1927, 1927) and von Neumann (1927, 1928). This is likely where Bauer got his information.

  • $\begingroup$ I have deleted the (rather large) chain of comments below this answer from this page. They can still be viewed in this chat room. Further discussion (and a large part of the comment chain as well) can be found in this discussion chat room, $\endgroup$
    – Danu
    Sep 2, 2018 at 16:02

Some general remarks first. The answer to the title question is likely "nobody", "coining" often happens without a "coiner". Born's authorship of the rule was acknowledged at least since von Neumann's book. Some might have used "Born's rule" in conversation or in print early meaning by it no more than a rule due to Born. After it was used this way long enough others thought of it as a "term", but to them it was already "coined". This is a typical way expressions spread, not on the authority of the original user ("coiner") but because they seem handy to many, independently so. It makes no difference who happens to be first because there is often no causal link between them and subsequent users, and the original use is certainly not the reason for the eventual adoption, even if some later users are aware of it.

Here is an early occurence in Studia Philosophica, 4 (1949) p. 192, which seems to confirm it:

"This law is linked to the so-called statistical interpretation of quantum mechanics given for the first time for certain particular cases by Born, and afterwards generalized so as to apply to other cases. For brevity's sake we will call it Born's rule."

It comes up as the only match in Google's advanced book search from 1940 to 1955, all other citations fed into Google's Ngram are spurious. However, it is possible that people used it orally and in papers before 1949, which book search may not find, although the context indicates that philosophers did not get it from there. MathSciNet and APS searches do not return anything promising either.

  • $\begingroup$ I have no access to this journal. But Stanford's catalogue says that volume 4 is from 1944. volume 9 from 1949. Thus something is strange about this reference. Volume 9 seems to contain reviews of books on natural philosophy, so this is probably the correct reference. $\endgroup$ May 15, 2018 at 9:22
  • 1
    $\begingroup$ ''It makes no difference who happens to be first''. To me it makes a difference, because I want to point to the first documented use. $\endgroup$ May 15, 2018 at 9:47
  • $\begingroup$ @ArnoldNeumaier You may be able to get it (or scan of the relevant pages) by interlibrary loan, WorldCat shows several US universities as having it $\endgroup$
    – Conifold
    May 16, 2018 at 23:08
  • 1
    $\begingroup$ Well, to get something by interlibrary loan one first needs to have a correct reference. But according to my first remark I don't even know the correct volume or page number.... $\endgroup$ May 18, 2018 at 10:27
  • $\begingroup$ @ArnoldNeumaier Your desired reference is Henryk Mehlberg, The idealistic interpretation of atomic physics, Studia Philosophica (Poznań) 4 (1949-1950, ©1951) 171-235. (So, 15+ years after Bauer.) $\endgroup$ Aug 2, 2018 at 23:23

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.