Statistics of Local Value in Quantum Mechanics |
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Authors: | Shunlong Luo |
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Institution: | (1) Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080, People's Republic of China;(2) Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong |
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Abstract: | Given a quantum mechanical observable and a state, one can construct a classical observable, that is, a real function on the configuration space, such that it is the optimal estimate of the quantum observable, in the sense of minimum variance. This optimal estimate turns out to be the quantum mechanical local value, which arises from several contexts such as de Broglie–Bohm's casual approach to quantum mechanics, instantaneous frequency in time–frequency analysis, Nelson's quantum fluctuations formalism, and phase-space approach to quantum mechanics. Accordingly, any observable can be decomposed into a local value part and a quantum fluctuation part, which are independent, both geometrically and statistically. Furthermore, the current density in quantum mechanics, the osmotic velocity in stochastic mechanics, and the Fisher information in classical statistical inference, arise naturally in connection with local value. In particular, Heisenberg uncertainty principle can be quantified more precisely by virtue of local value. |
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Keywords: | local value classical observable Fisher information conditionalexpectation Heisenberg uncertainty principle |
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