# Is there a mathematical basis for Born rule?

Wave function determines complex amplitudes to possible measurement outcomes. The Born Rule states that the probability of obtaining some measurement outcome is equal to the square of the corresponding amplitude.

How did Born arrive at this revolutionary idea? Was he motivitaed by some mathematical principle? Or was it based on pure experimental evidence?

• When I read Born's paper "Zur Quantenmechanik der Stossvorgaenge", I am having a hard time believing that he had a really good argument in its favor at the time. Just like in the case of Schroedinger's papers about which Feynman wrote "Where did we get that from? It's not possible to derive it from anything you know. It came out of the mind of Schrödinger.", this is simply genius at work guessing its way to reality. And what a good guess it was! In modern terminology one can "derive" it from the density matrix of a measurement, but that's cheating... we know what we want, already. Oct 30, 2015 at 20:17
• @CuriousOne I like your phrase "this is simply genius at work guessing its way to reality". This reminds me of the genius guess of Dirac when he proposed the existence of positive electrons. Oct 30, 2015 at 20:25
• If you like metaphors: that's the difference between the handwaving of amateurs like me and that of professionals: I have giant foam hands and they have fists of steel that pack a lasting punch. :-) I am looking forward to real answers, though. Maybe someone knows another paper or can interpret Born's paper in a way that does not involve the feeling that he was partly blustering his way trough the German language to be vague enough to not be crucified but firm enough to make a non-trivial statement. That's just how the original reads. As a disclaimer, I am still pretty good at German. Oct 30, 2015 at 20:30
• @CuriousOne I thought the very first appearance was in a footnote added as an addition to the final galley proofs. Which means all the original analysis was flawed and the idea for the actual born rule is simply a correction to a bad theory to make it work. So not a mathematical principle, but a correction to a bad theory that didn't fit the data. A correction that has a lot of merit. Oct 31, 2015 at 1:56
• @Timaeus: It's possible. All I can tell you is that the German in the paper sounds awfully wavy. Born, in my opinion, glosses over a lot of details that, at the time, were most likely not clear to him and the other main contributors to QM. Or maybe I just can't tell from what they wrote what they were really thinking... Oct 31, 2015 at 3:34

One motivation comes from looking at light waves and polarisation -- when light passes through some filter, the energy of a light wave is scaled by $$\cos^2\theta$$ -- for a single photon, this means (as you can't have $$\cos^2\theta$$ of a photon) there is a probability of $$\cos^2\theta$$ that the number of photons passing through is "1". This $$\cos\theta$$ is simply the dot product of the "state vector" (polarisation vector) and the eigenvector of the number operator associated with polarisation filter with eigenvalue 1 -- i.e. the probability of observing "1" is $$|\langle\psi|1\rangle|^2$$, and the probability of observing "0" is $$|\langle\psi|0\rangle|^2$$, which is Born's rule.

So if you're motivating the state vector based on the polarisation vector, you can motivate Born's rule from $$E=|A|^2$$, as above.

More abstractly, if you accept the other axioms of quantum mechanics, Born's rule is sort of the "only way" to encode probabilities, as you want probability of the union of disjoint events to be additive (equivalent to the Pythagoras theorem) and the total probability to be one (the length of the state vector is one).

But there is no way to "derive" the Born rule, it is an axiom. Quantum mechanics is fundamentally quite different to e.g. relativity, in the sense that it develops a whole new abstract mathematical theory to connect to the real world. So unlike in relativity, you don't have two axioms that are literally the result of observation and everything is derived from it -- instead, you have an axiomatisation of the mathematical theory, and then a way to connect the theory with observation, which is what Born's rule is. Certainly the motivation for quantum mechanics comes from wave-particle duality, but this is not an axiomatisation.

• The Mackey-Gleason theorem is one way of expressing the idea that "Born's rule is sort of the 'only way' ...". But like you said, this doesn't qualify as a derivation of Born's rule. I would add that the reason it doesn't qualify is because Born's rule only applies to measured observables, and the Mackey-Gleason theorem doesn't know anything about which observables were actually measured. May 31, 2019 at 23:41

Here is an answer to the question posed in the title:

Zurek gave a derivation of Born rule using envariance (invariance due to entanglement).

Also, Saunders gave a derivation of Born rule from operational assumptions.

However, the consensus seems to be that there is no generally accepted derivation of the Born rule.

Update: Born rule, has been derived from simpler physical principles in this paper,The Measurement Postulates of Quantum Mechanics are Redundant. The authors state that " Our result shows that not only is the Born rule a good guess, but it is the only logically consistent guess".

• Your original question (from a couple of years ago) seems to be historical. It asks how Born arrived at the rule. But your self-answer seems to be an answer divorced from historical context. Which did you really have in mind?
– user4552
Jun 8, 2017 at 2:00
• @BenCrowell The core of my question is whether Born rule is derivable via mathematical proof or not. Jun 8, 2017 at 13:25

Born calculated solution to Schroedinger's equation corresponding to electron scattering experiment and what he got was continuous function of scattering angles measured with respect to the original direction of propagation of electrons.

However, in experiment electrons are always detected at definite points of a screen. Clearly, there is no direct match between the $\psi$ function (continuous) and location of detected electrons (discrete). Born realized one way to resolve this mismatch and make use of the calculated function $\psi$ anyway is to assume it gives continuous probability density for angles the electron goes into, or, more generally, probability density for any possible configuration of particles.

• Do you have any reference to support this account? Or is it your own interpretation? Oct 30, 2015 at 22:07
• Yes. But the idea of using $|\Psi|^2$ as a probability density rather than some other function of the continuous $\Psi$ wasn't based on some deep mathematical principle, it was a footnote added at the last second to a prior result that was at odds with experiment. Oct 31, 2015 at 2:30
• Please add the reference to the origins of probabilistic interpretation of Schroedinger's wave function. Oct 31, 2015 at 4:05
• @MohammadAl-Turkistany Do make sure the reference is detailed, since Born's bibliography include two papers published the same year in the same journal with the same title on the same topic. Oct 31, 2015 at 4:38
• @Timaeus: the idea of using |Ψ|2 as a probability density rather than some other function of the continuous Ψ wasn't based on some deep mathematical principle Yes, but $\Psi^2$ has the unique advantage that it gives conservation of probability when the wavefunction evolves according to the Schrodinger equation. Of course you're right that this wasn't a justification that Born had in mind.
– user4552
Jun 8, 2017 at 2:02

If you want to find a probablity in statistics you have to do the same thing but as for wave function no one was able to interpret it, Born suggested that the same thing is also applied to Quantum wavefunction. Before Born, physicists applied the same thing for photons as well.

Like Schrodinger, he wasn't able to visualize the wave function but he still showed his equation for real waves (such as string or any other mechanical waves), and it fitted quantum waves as well.

The Bohr Model is an excellent model for hydrogen in terms of predicting its spectral lines with classical physics, using a ring-source of charge rotating at a specific frequency and radius from the central atom that minimizes the total energy of the configuration, with the boundary condition of coherent constructive interference of the electron wave going once around the ring.

Born's relation is directly related to physical interpretations, and so he squared the wave function. It is like how in sine waves it doesn't show the physical structures, but $\sin^2$ gives the square of the wavefunction of the electrons shot at a screen.