Moyal's Characteristic Function, the Density Matrix and von Neumann's Idempotent
| Autor: | Hiley, Basil J. |
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| Rok vydání: | 2014 |
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| Druh dokumentu: | Working Paper |
| Popis: | In the Wigner-Moyal approach to quantum mechanics, we show that Moyal's starting point, the characteristic function $M(\tau,\theta)=\int \psi^{*}(x)e^{i(\tau {\hat p}+\theta{\hat x})}\psi(x)dx$, is essentially the primitive idempotent used by von Neumann in his classic paper "Die Eindeutigkeit der Schr\"odingerschen Operatoren". This paper provides the original proof of the Stone-von Neumann equation. Thus the mathematical structure Moyal develops is simply a re-expression of what is at the heart of quantum mechanics and reproduces exactly the results of the quantum formalism. The "distribution function" $F(X,P,t)$ is simply the quantum mechanical density matrix expressed in an $( X,P)$-representation, where $X$ and $P$ are the mean co-ordinates of a cell structure in phase space. The whole approach therefore clearly has little to do with classical statistical theories but is a consequence of a non-commutative nature of the theory. Comment: 12 |
| Databáze: | arXiv |
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