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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 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.
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]
The Cauchy–Born rule or Cauchy–Born approximation is a basic hypothesis used in the mathematical formulation of solid mechanics which relates the movement of atoms in a crystal to the overall deformation of the bulk solid.
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] Despite the widespread quantitative success of these postulates scientists remain dissatisfied and have sought more detailed physical models.
Nowadays Bohm's theory is considered to be one of many interpretations of quantum mechanics. Some consider it the simplest theory to explain quantum phenomena. [33] Nevertheless, it is a hidden-variable theory, and necessarily so. [34] The major reference for Bohm's theory today is his book with Basil Hiley, published posthumously. [35]
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.
In quantum mechanics, every state is described as a vector in Hilbert space. When a measurement is performed, it is convenient to describe this space using a vector basis in which every basis vector has a defined result of the measurement – e.g., a vector basis of defined momentum in case momentum is measured. The measurement operator is ...