In physics, quantum dynamics is the quantum version of classical dynamics. Quantum dynamics deals with the motions, and energy and momentum exchanges of systems whose behavior is governed by the laws of quantum mechanics.[1][2] Quantum dynamics is relevant for burgeoning fields, such as quantum computing and atomic optics.
In mathematics, quantum dynamics is the study of the mathematics behind quantum mechanics.[3] Specifically, as a study of dynamics, this field investigates how quantum mechanical observables change over time. Most fundamentally, this involves the study of one-parameter automorphisms of the algebra of all bounded operators on the Hilbert space of observables (which are self-adjoint operators). These dynamics were understood as early as the 1930s, after Wigner, Stone, Hahn and Hellinger worked in the field. Recently, mathematicians in the field have studied irreversible quantum mechanical systems on von Neumann algebras.[4]
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The Origin of Quantum Mechanics (feat. Neil Turok)
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The Quantum Mechanics of Time Travel
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Google and NASA's Quantum Artificial Intelligence Lab
Transcription
Where did quantum theory come from? It started, not as a crazy idea, but with a light bulb. In the early 1890s, the German Bureau of Standards asked Max Planck how to make light bulbs more efficient so that they would give out the maximum light for the least electrical power. The first task Planck faced was to predict how much light a hot filament gives off. He knew that light consists of electromagnetic waves, with different colors of light carried by different frequency waves. The problem was to ensure that as much light as possible was given off by visible waves rather than ultraviolet or infrared. He tried to work out how much light of each color a hot object emits, but his predictions based on electromagnetic theory kept disagreeing with experiments. Instead, in what he later called an “act of despair,” he threw the existing theory out the window and worked backwards from experimental measurements. The data pointed him to a new rule of physics: light waves carry energy only in packets, with high frequency light consisting of large packets of energy and low frequency light consisting of small packets of energy. The idea that light comes in packets, or "quanta", may sound crazy, and it was at the time, but Einstein soon related it to a much more familiar problem: sharing. If you want to make a kid happy... give them a cookie! But if there are two kids, and you only have one cookie, you'll only be able to cheer them up half as much. And if there are four, or eight, or sixteen hundred thousand, you're not going to make them very happy at all if they have to share one cookie between them. In fact, if you have a room with infinitely many kids but not infinitely many cookies, if you share the cookies evenly each kid will only get an infinitesimally small crumb, and none of them will be cheered up. And they'll still eat all your cookies. The difference between light waves and kids is that you can't actually have infinitely many kids in a room. But because light waves come in all sizes, you can have arbitrarily small light waves, so you can fit infinitely many into a room. And then the light waves would consume all your cookies… I mean, energy. In fact, all these infinitesimal waves together would have an infinite capacity to absorb energy, and they'd suck all the heat out of anything you put into the room… instantly freezing the tea in your cup, or the sun, or even a supernova. Luckily, the universe doesn't work that way… because, as Planck guessed, the tiny, high frequency waves can only carry away energy in huge packets. They're like fussy kids who'll only accept exactly thirty-seven cookies, or a hundred and sixty-two thousand cookies, no more and no less. Because they're so picky, the fussy high frequency waves lose out and most of the energy is carried away in lower-frequency packets that are willing to take an equal share. This common, average energy that the packets carry, is in fact what we mean by "temperature." So a higher temperature just means higher average energy, and thus by Planck's rule, a higher frequency of light emitted. That's why as an object gets hotter it glows first infrared, then red, yellow, white; hotter and hotter towards blue, violet, ultraviolet… and so on. Specifically, Planck's quantum theory of fussy light tells us that light bulb filaments should be heated to a temperature of about 3200 Kelvin to ensure that most of the energy is emitted as visible waves - much hotter, and we'd start tanning from the ultraviolet light. Actually, quantum physics has been staring us in the face since long before lightbulbs and tanning beds: human beings have been making fires for millennia, with the color of the flames spelling out "quantum" all along.
Relation to classical dynamics
Equations to describe quantum systems can be seen as equivalent to that of classical dynamics on a macroscopic scale, except for the important detail that the variables don't follow the commutative laws of multiplication.[5] Hence, as a fundamental principle, these variables are instead described as "q-numbers", conventionally represented by operators or Hermitian matrices on a Hilbert space.[6] Indeed, the state of the system in the atomic and subatomic scale is described not by dynamic variables with specific numerical values, but by state functions that are dependent on the c-number time. In this realm of quantum systems, the equation of motion governing dynamics heavily relies on the Hamiltonian, also known as the total energy. Therefore, to anticipate the time evolution of the system, one only needs to determine the initial condition of the state function |Ψ(t) and its first derivative with respect to time.[7]
For example, quasi-free states and automorphisms are the Fermionic counterparts of classical Gaussian measures[8] (Fermions' descriptors are Grassmann operators).[6]
See also
- Quantum Field Theory
- Perturbation theory
- Semigroups
- Pseudodifferential operators
- Brownian motion
- Dilation theory
- Quantum probability
- Free probability
References
- ^ Joan Vaccaro (2008-06-26). "Centre for Quantum Dynamics, Griffith University". Quantiki. Archived from the original on 2009-10-25. Retrieved 2010-01-25.
- ^ Wyatt, Robert Eugene; Corey J. Trahan (2005). Quantum dynamics with trajectories. Springer. ISBN 9780387229645.
- ^ Teufel, Stefan (1821-01-01). Adiabatic perturbation theory in quantum dynamics. Springer. ISBN 9783540407232.
- ^ Price, Geoffrey (2003). Advances in quantum dynamics : proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on Advances in Quantum Dynamics, June 16-20, 2002, Mount Holyoke College, South Hadley, Massachusetts. Providence, R.I: American Mathematical Society. ISBN 0-8218-3215-8. OCLC 52901091.
- ^ Dirac, P. A. M. (1927). "The physical interpretation of the quantum dynamics". Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character. 113 (765): 621–641. Bibcode:1927RSPSA.113..621D. doi:10.1098/rspa.1927.0012. ISSN 0950-1207.
- ^ a b Kuypers, Samuel (2022). "The quantum theory of time: a calculus for q-numbers". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 478 (2263). arXiv:2108.02771. Bibcode:2022RSPSA.47810970K. doi:10.1098/rspa.2021.0970. ISSN 1364-5021. PMC 9326976. PMID 35909420.
- ^ Tang, Chung Liang (2005). Fundamentals of quantum mechanics: for solid state electronics and optics. Cambridge: Cambridge Univ. Press. ISBN 978-0-521-82952-6.
- ^ Alicki, Robert; Fannes, Mark (2001). Quantum dynamical systems (1. publ ed.). Oxford: Oxford University Press. pp. 103–121. ISBN 978-0-19-850400-9.
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