"The order of the rotation switches from first p then q, to first q then p, and Planck’s constant h is a measure of the minimum element of noncommutivity in this ordered process."
"Pauli, in addition to visualizing a kernel under rotation, also might have associated the above equation with the Pythagorean tetractys, where p, q, h, and i are the four elements, and the kernel is some visualizable whole uniting all four."
"Max Jammer focuses on Heisenberg’s substitution of quantum frequencies for
classical Fourier components, which resulted in the noncommutative matrix
multiplication that was unfamiliar to Heisenberg."
"Further, by emphasising the constancy of h-bar in the relation [change of]X times [change of]P approximates h-bar one tends to be led to the notion that the 'disturbance' is dependent only on the size of the cell in phase space. In this way the overall experimental conditions were tacitly dismissed as irrelevant."
It was in view of this elementary derivation of the radiation formula that Planck made for the first time the explicit statement that the energy, and not only the average energy, of an oscillator of frequency v is an integral multiple of the energy element hv. On the basis of these considerations Planck interpreted h as the finite extension of the elementary area in phase space.
It occurred to Rydberg and was later explicitly stated as a fundamental principle by Ritz that the frequency of every spectral line of an element could be expressed as the difference between two terms or “spectral terms,” each of which contained an integer. Ritz’s principle, or, as it was subsequently called, “the combination principle,” could not be accounted for by classical physics. In order to understand the reason for this incompatibility, it must be noted that the above-mentioned hypothesis according to which the spectrum as a whole was thought to be produced by the free vibrations of one single atom had meanwhile been refuted.
Conway showed convincingly, in 1907, that each atom could give rise to only one spectral line at a time. Conway’s contention was further elaborated by Bevan’s analysis of the anomalous dispersion by potassium vapor. Bevan showed that any theoretical explanation of this phenomenon, if worked out on the basis of the previous hypothesis, would necessarily imply an excessively great number of electrons per molecule. Each individual line in a spectrum had therefore to be associated with the periodic motion of an electron and the different lines of the spectrum with motions of excited electrons in different atoms.
Distortion of the Poisson Bracket by the Noncommutative Planck Constants
In this paper we introduce a kind of "noncommutative neighbourhood" of a semiclassical parameter corresponding to the Planck constant.
Hence by a simple rescaling of the noncommutativity parameters one can ensure the constancy of the Planck constant. On the other hand, in [21] it was shown that by assuming that , giving the fundamental length scale in noncommutative geometry, is smaller than the average neutron size, having an order of magnitude of around 1 fm, it follows that . Hence, in practical calculations one can ignore the deviations between the numerical values of the effective Planck constant and the usual Planck constant.
https://ujp.bitp.kiev.ua/index.php/ujp/article/view/2019529
The conceptual incompatibility of spacetime in gravity and quantum physics implies the existence of noncommutative spacetime and geometry on the Planck scale. We present the formulation of a noncommutative quantum mechanics based on the Seiberg–Witten map, and we study the Aharonov–Bohm effect induced by the noncommutative phase space. We investigate the existence of the persistent current in a nanoscale ring with an external magnetic field along the ring axis, and we introduce two observables to probe the signal coming from the noncommutative phase space. Based on this formulation, we give a value-independent criterion to demonstrate the existence of the noncommutative phase space.
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