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theories:quantum_mechanics:phase_space [2018/05/04 15:16]
jakobadmin [Why is it interesting?]
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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   ​   ​
 <tabbox Concrete> ​ <tabbox Concrete> ​
 +**Time Evolution**
 +
 +The time evolution in the phase space formulation of quantum mechanics is described by the vonNeumann equation
 +
 +$$ \frac{\partial f}{\partial t} = -\frac{1}{i \hbar} (f \star H - H \star f ). $$
 +
 +For the Wigner function, this equation reads
 +
 +$$ \frac{\partial W}{\partial t} = - \{\{ W,H \}\} = \frac{2}{ \hbar} W \left( \frac{\hbar}{2 ​ } ( \overleftarrow \partial_x \overrightarrow ​ \partial_p - \overleftarrow \partial_p \overrightarrow \partial_x)\right) H = - \{ W,H\} + \mathcal{O}(\hbar^2) , $$
 +
 +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 [[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.
 +
 +
 +
 +----
 +
 **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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   * [[https://​www.cambridge.org/​core/​services/​aop-cambridge-core/​content/​view/​9D0DC7453AD14DB641CF8D477B3C72A2/​S0305004100000487a.pdf/​quantum_mechanics_as_a_statistical_theory.pdf|Quantum Mechanics as a statistical theory]] by J. E. Moyal   * [[https://​www.cambridge.org/​core/​services/​aop-cambridge-core/​content/​view/​9D0DC7453AD14DB641CF8D477B3C72A2/​S0305004100000487a.pdf/​quantum_mechanics_as_a_statistical_theory.pdf|Quantum Mechanics as a statistical theory]] by J. E. Moyal
 <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.]] ​
  
-<note tip> +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.
-The motto in this section is: //the higher the level of abstraction, ​the better//. +
-</​note>​+
  
 <tabbox Why is it interesting?> ​ <tabbox Why is it interesting?> ​
theories/quantum_mechanics/phase_space.1525439780.txt.gz · Last modified: 2018/05/04 13:16 (external edit)