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theories:quantum_mechanics:phase_space [2018/05/04 15:29]
jakobadmin [Concrete]
theories:quantum_mechanics:phase_space [2018/10/11 15:02] (current)
jakobadmin [Concrete]
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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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 where $\{\{ ,\}\}$ denotes the Moyal bracket and $\{ ,\}$ the [[advanced_notions:​poisson_bracket|Poisson bracke]]t. ​ where $\{\{ ,\}\}$ denotes the Moyal bracket and $\{ ,\}$ the [[advanced_notions:​poisson_bracket|Poisson bracke]]t. ​
  
-In the limit, $\hbar \to 0$ the von Neumann equation reduces to the [[basic_tools:​phase_space:​liouvilles_theorem|Liouville equation]]. ​+In the limit, $\hbar \to 0$ the von Neumann equation reduces to the [[theorems:​liouvilles_theorem|Liouville equation]]. ​
  
 The difference between the von Neumann equation and the Liouville equation is that in the former the density of points in phase space is not conserved. Formulated differently,​ the probability fluid is diffusive and compressible. The difference between the von Neumann equation and the Liouville equation is that in the former the density of points in phase space is not conserved. Formulated differently,​ the probability fluid is diffusive and compressible.
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 **Reading Recommendations** **Reading Recommendations**
  
 +  * A great introduction is [[https://​arxiv.org/​abs/​1602.06071|The algebraic way]] by Hiley
   * See also the corresponding chapter in Ballentine'​s Quantum Mechanics book, and also    * See also the corresponding chapter in Ballentine'​s Quantum Mechanics book, and also 
   * [[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?> ​
theories/quantum_mechanics/phase_space.1525440559.txt.gz · Last modified: 2018/05/04 13:29 (external edit)