Born rule

The Born rule (also called the Born law, Born's rule, or Born's law) is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of quantum mechanics.

Quotes

 * If one translates this result into terms of particles, only one interpretation is possible: $$\Phi_{n,m} (\alpha, \beta, \gamma)$$[* addition in proof: More careful consideration shows that the probability is proportional to the square of the quantity $$\Phi_{n,m}$$] gives the probability for the electron, arriving from the z-direction, …
 * Max Born, "On the Quantum Mechanics of Collisions" (1926)


 * One of the most profound and mysterious principles in all of physics is the Born Rule, named after Max Born. In quantum mechanics, particles don’t have classical properties like “position” or “momentum”; rather, there is a wave function that assigns a (complex) number, called the “amplitude,” to each possible measurement outcome. The Born Rule is then very simple: it says that the probability of obtaining any possible measurement outcome is equal to the square of the corresponding amplitude. (The wave function is just the set of all the amplitudes.) … The Born Rule is certainly correct, as far as all of our experimental efforts have been able to discern. But why? … The status of the Born Rule depends greatly on one’s preferred formulation of quantum mechanics. … Everett, … is claiming that all the weird stuff about “measurement” and “wave function collapse” in the conventional way of thinking about quantum mechanics isn’t something we need to add on; it comes out automatically from the formalism. The trickiest thing to extract from the formalism is the Born Rule.
 * Sean M. Carroll, "Why Probability in Quantum Mechanics is Given by the Wave Function Squared" (July 24, 2014)


 * ... the Born rule just comes out of the blue without being derived from the time-dependent Schrödinger equation — which is supposed to account for everything.
 * Steven Weinberg: The Problems of Quantum Mechanics: Steven Weinberg (July 17, 2018) YouTube video at 39:42 of 45:42


 * And Po'mi has just asked: "Why should the subjective probability of finding ourselves in a side of the split world, be exactly proportional to the square of the thickness of that side?" When the initial hubbub quiets down, the respected Nharglane of Ebbore asks: "Po'mi, what is it exactly that you found?" "Using instruments of the type we are all familiar with," Po'mi explains, "I determined when a splitting of the world was about to take place, and in what proportions the world would split.  I found that I could not predict exactly which world I would find myself in—" "Of course not," interrupts De'da, "you found yourself in both worlds, every time -" "—but I could predict probabilistically which world I would find myself in.  Out of all the times the world was about to split 2:1, into a side of two-thirds width and a side of one-third width, I found myself on the thicker side around 4 times out of 5, and on the thinner side around 1 time out of 5.  When the world was about to split 3:1, I found myself on the thicker side 9 times out of 10, and on the thinner side 1 time out of 10."
 * Eliezer Yudkowsky, "Where Experience Confuses Physicists" (April 25, 2008)


 * One serious mystery of decoherence is where the Born probabilities come from, or even what they are probabilities of. What does the integral over the squared modulus of the amplitude density have to do with anything? … So what could it mean, to associate a "subjective probability" with a component of one factor of a combined amplitude distribution that happens to factorize? … But what does the integral over squared moduli have to do with anything? On a straight reading of the data, you would always find yourself in both blobs, every time. How can you find yourself in one blob with greater probability? What are the Born probabilities, probabilities of? Here's the map—where's the territory? I don't know. It's an open problem. This problem is even worse than it looks, because the squared-modulus business is the only non-linear rule in all of quantum mechanics.
 * Eliezer Yudkowsky, "The Born Probabilities" (May 1, 2008)