Answer by NinjaDarth for What is the connection between Poisson brackets and...
Algebraically, the relation $[a,b] = iħ\{a,b\}$ isn't just a "correspondence" or something that holds "only in the limit", but is true as is. More precisely, when taken with the definition $a·b ≡½(ab +...
View ArticleAnswer by Qmechanic for What is the connection between Poisson brackets and...
According to the topic of deformation quantization, the first few entries in the dictionary between $$ \text{Quantum Mechanics}\quad\longleftrightarrow\quad\text{Classical Mechanics}\tag{0}$$read $$...
View ArticleAnswer by Nikolaj-K for What is the connection between Poisson brackets and...
Regarding the significance of the observables momentum and position there are many similarities between Classical and Quantum mechanics. Some of the algebraic relations have been pointed out. In the...
View ArticleAnswer by joseph f. johnson for What is the connection between Poisson...
Both the commutator (of matrices) and the Poisson bracket satisfy the Jacobi identity,$[A,[B,C]]+[B,[C,A]]+[C,[A,B]]=0$.This is why Dirac was inspired by Heisenberg's use of commutators to develop a...
View ArticleAnswer by David Z for What is the connection between Poisson brackets and...
Poisson brackets play more or less the same role in classical mechanics that commutators do in quantum mechanics. For example, Hamilton's equation in classical mechanics is analogous to the Heisenberg...
View ArticleAnswer by Siyuan Ren for What is the connection between Poisson brackets and...
I don't know any link between Poisson bracket and anti-commutator, but I do know the link between Poisson bracket and commutator.$$[\hat a,\hat b]=i\hbar\{a,b\}_\text{Poisson}$$SubtletiesAs the...
View ArticleWhat is the connection between Poisson brackets and commutators?
The Poisson bracket is defined as:$$\{f,g\} ~:=~ \sum_{i=1}^{N} \left[ \frac{\partial f}{\partial q_{i}} \frac{\partial g}{\partial p_{i}} -\frac{\partial f}{\partial p_{i}} \frac{\partial g}{\partial...
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