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The Born rule is a postulate of quantum mechanics that gives the probability that a measurement of a quantum system will yield a given result. In one commonly used application, it states that the probability density for finding a particle at a given position is proportional to the square of the amplitude of the system's wavefunction at that position.
Quantum mechanics is intrinsically indeterministic. The correspondence principle: in the appropriate limit, quantum theory comes to resemble classical physics and reproduces the classical predictions. The Born rule: the wave function of a system yields probabilities for the outcomes of measurements upon that system.
This difference is called the measurement problem of quantum mechanics. To predict measurement outcomes from quantum solutions, the orthodox interpretation of quantum theory postulates wave function collapse and uses the Born rule to compute the probable outcomes. [9]
The old quantum theory is a collection of results from the years 1900–1925 [23] which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. [24] The theory is now understood as a semi-classical approximation [25] to modern quantum mechanics. [26]
The inner product of two wave functions is a measure of the overlap between the corresponding physical states and is used in the foundational probabilistic interpretation of quantum mechanics, the Born rule, relating transition probabilities to inner products.
Quantum mechanics is a fundamental theory that describes the behavior of nature at and below the scale of atoms. [2]: 1.1 It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot.
The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures.
In consequence, quantum theory is "a tighter package than one might have first thought". [24]: 94–95 Various approaches to rederiving the quantum formalism from alternative axioms have, accordingly, employed Gleason's theorem as a key step, bridging the gap between the structure of Hilbert space and the Born rule. [c]