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theories:quantum_mechanics:phase_space [2018/05/04 15:29] jakobadmin [Concrete] |
theories:quantum_mechanics:phase_space [2018/05/05 12:23] jakobadmin ↷ Page name changed from theories:quantum_mechanics:phase_space_quantum_mechanics to theories:quantum_mechanics:phase_space |
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<tabbox Intuitive> | <tabbox Intuitive> | ||
+ | <blockquote>Ever since Werner Heisenberg’s 1927 paper on uncertainty, there has | ||
+ | been considerable hesitancy in simultaneously considering positions and | ||
+ | momenta in quantum contexts, since these are incompatible observables. [...] However, they too are wrong. Quantum mechanics (QM) can be consistently | ||
+ | and autonomously formulated in phase space, with c-number position | ||
+ | and momentum variables simultaneously placed on an equal footing, in a way | ||
+ | that fully respects Heisenberg’s principle. [...] The net result is that quantum mechanics works smoothly and consistently | ||
+ | in phase space, where position coordinates and momenta blend together closely | ||
+ | and symmetrically. Thus, sharing a common arena and language with classical | ||
+ | mechanics [14], QMPS connects to its classical limit more naturally and intuitively | ||
+ | than in the other two familiar alternate pictures, namely, the standard | ||
+ | formulation through operators in Hilbert space, or the path integral formulation. [...] | ||
+ | |||
+ | Still, as every physics undergraduate learns early on, classical phase space is | ||
+ | built out of “c-number” position coordinates and momenta, x and p, ordinary | ||
+ | commuting variables characterizing physical particles; whereas such observables | ||
+ | are usually represented in quantum theory by operators that do not commute. | ||
+ | How then can the two be reconciled? The ingenious technical solution to this | ||
+ | problem was provided by Groenewold in 1946, and consists of a special binary | ||
+ | operation, the ⋆-product (see Star Product), which enables x and p to maintain | ||
+ | their conventional classical interpretation, but which also permits x and p to | ||
+ | combine more subtly than conventional classical variables; in fact to combine | ||
+ | in a way that is equivalent to the familiar operator algebra of Hilbert space | ||
+ | quantum theory. Nonetheless, expectation values of quantities measured in the lab (observables) | ||
+ | are computed in this picture of quantum mechanics by simply taking | ||
+ | integrals of conventional functions of x and p with a quasi-probability density | ||
+ | in phase space, the Wigner function — essentially the density matrix in this | ||
+ | picture. But, unlike a Liouville probability density of classical statistical mechanics, | ||
+ | this density can take provocative negative values and, indeed, these can | ||
+ | be reconstructed from lab measurements [11]. | ||
+ | <cite>[[https://arxiv.org/abs/1104.5269|Quantum Mechanics in Phase space]] by Curtright and Zachos</cite> | ||
+ | |||
+ | </blockquote> | ||
+ | |||
<blockquote>Classical statistical mechanics is a ' crypto-deterministic' | <blockquote>Classical statistical mechanics is a ' crypto-deterministic' | ||
theory, where each element of the probability distribution of the dynamical | theory, where each element of the probability distribution of the dynamical | ||
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* [[http://aapt.scitation.org/doi/pdf/10.1119/1.16475|Canonical transformation in quantum mechanics]] Y. S. Kim and E. Wigner | * [[http://aapt.scitation.org/doi/pdf/10.1119/1.16475|Canonical transformation in quantum mechanics]] Y. S. Kim and E. Wigner | ||
* [[https://arxiv.org/abs/1104.5269|A Concise Treatise on Quantum Mechanics in Phase Space]] by Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, World Scientific, 2014. | * [[https://arxiv.org/abs/1104.5269|A Concise Treatise on Quantum Mechanics in Phase Space]] by Thomas L. Curtright, David B. Fairlie, & Cosmas K. Zachos, World Scientific, 2014. | ||
+ | * [[https://en.wikipedia.org/wiki/Phase_space_formulation]] | ||
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<tabbox Abstract> | <tabbox Abstract> | ||
In the phase space formulation, the transition from classical mechanics to quantum mechanics is known as [[advanced_tools:quantization|deformation quantization.]] | In the phase space formulation, the transition from classical mechanics to quantum mechanics is known as [[advanced_tools:quantization|deformation quantization.]] | ||
+ | |||
+ | The map between function in the phase space formulation of quantum mechanics and the operators in the corresponding Hilbert space is known as Wigner–Weyl transform. | ||
<tabbox Why is it interesting?> | <tabbox Why is it interesting?> |