Stochastic quantum mechanics

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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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Stochastic quantum mechanics is a framework for describing the dynamics of particles that are subjected to an intrinsic random processes as well as various external forces. The framework provides a derivation of the diffusion equations associated to these stochastic particles. It is best known for its derivation of the Schrödinger equation as the Kolmogorov equation for a certain type of conservative (or unitary) diffusion.

The derivation can be based on the extremization of an action in combination with a quantization prescription. This quantization prescription can be compared to canonical quantization and the path integral formulation, and is often referred to as Nelson’s stochastic quantization or stochasticization. As the theory allows for a derivation of the Schrödinger equation, it has given rise to the stochastic interpretation of quantum mechanics. This interpretation has served as the main motivation for developing the theory of stochastic mechanics.

In the 1930s both Erwin Schrodinger and Reinhold Furth recognised a similarity between the equations of classical diffusion and the formalism of quantum theory, but the first relatively coherent stochastic theory of quantum mechanics was put forward in 1946 by Hungarian physicist Imre Fényes. Louis de Broglie felt compelled to incorporate a stochastic process underlying quantum mechanics to make particles switch from one pilot wave to another. The theory of stochastic quantum mechanics is ascribed to Edward Nelson, who independently discovered a derivation of the Schrödinger equation within this framework. This theory was also developed by Davidson, Guerra, Ruggiero, Pavon and others.