theories:classical_mechanics:koopman_von_neumann_mechanics
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theories:classical_mechanics:koopman_von_neumann_mechanics [2018/04/12 15:46] bogumilvidovic created |
theories:classical_mechanics:koopman_von_neumann_mechanics [2021/11/15 18:07] (current) 109.147.252.211 [Why is it interesting?] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | <note tip> | + | Koopman-von-Neumann Mechanics is a reformulation of [[theories:classical_mechanics:newtonian|classical mechanics]] using the same language that we usually only use in [[theories:quantum_mechanics:canonical|quantum mechanics]]. |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | Usually in classical mechanics, our main focus is the trajectories of objects. |
| - | + | ||
| + | In contrast, in quantum mechanics, our main focus are observables, states and measurements. This means, instead of describing the trajectories of objects, we now ask instead for example: If we measure the momentum of this ball, what's the probability that the result will be $10$ kg m/s? | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
| + | The equation of motion in Koopmann-von-Neumann mechanics is a complexified reformulation of the classical Liouville equation (which handles probability distributions over the phase space). | ||
| - | * See the notes by Frank Wilczek on [[http://frankwilczek.com/2015/koopmanVonNeumann02.pdf|"Koopman von Neumann Mechanics, and a Step Beyond"]] | + | * See the notes by Frank Wilczek on [[https://web.archive.org/web/20170206103038/http://frankwilczek.com/2015/koopmanVonNeumann02.pdf|"Koopman von Neumann Mechanics, and a Step Beyond"]] |
| * See also this [[https://physics.stackexchange.com/a/80822|nice answer at StackExchange]] | * See also this [[https://physics.stackexchange.com/a/80822|nice answer at StackExchange]] | ||
| + | * [[https://arxiv.org/pdf/quant-ph/0301172.pdf|Topics in Koopman-von Neumann Theory]] -a Ph.D. Thesis by Danilo Mauro | ||
| <tabbox Abstract> | <tabbox Abstract> | ||
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| </note> | </note> | ||
| - | <tabbox Why is it interesting?> | + | <tabbox Why is it interesting?> |
| + | The Koopman-von-Neumann reformulation of classical mechanics is extremely helpful to understand the differences between classical and quantum mechanics. Usually these theories are described in completely different languages which makes it hard to compare them. Using the Koopman-von-Neumann reformulation we describe classical mechanics in exactly the same language that we use in quantum mechanics and thus can perfectly analyze what is different. | ||
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| + | ---- | ||
| + | <blockquote>Ordinary mechanics must also be statistically formulated: the determinism | ||
| + | of classical physics turns out to be an illusion, it is an idol, not an ideal in | ||
| + | scientific research. | ||
| + | <cite>Max Born, 1954 Nobel Prize Lecture.</cite></blockquote> | ||
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