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The fractional Schrödinger equation, a fundamental equation of fractional quantum mechanics, has the following form: [73] [74] (,) = (,) + (,) (,). where the solution of the equation is the wavefunction ψ ( r , t ) – the quantum mechanical probability amplitude for the particle to have a given position vector r at any given time t , and ħ ...
The fractional quantum Hall effect (FQHE) is a collective behavior in a 2D system of electrons. In particular magnetic fields, the electron gas condenses into a remarkable liquid state, which is very delicate, requiring high quality material with a low carrier concentration, and extremely low temperatures.
In quantum mechanics, fractionalization is the phenomenon whereby the quasiparticles of a system cannot be constructed as combinations of its elementary constituents. One of the earliest and most prominent examples is the fractional quantum Hall effect, where the constituent particles are electrons but the quasiparticles carry fractions of the electron charge.
The h-calculus is the calculus of finite differences, which was studied by George Boole and others, and has proven useful in combinatorics and fluid mechanics. In a sense, q -calculus dates back to Leonhard Euler and Carl Gustav Jacobi , but has only recently begun to find usefulness in quantum mechanics , given its intimate connection with ...
The fractional quantum Hall effect is more complicated and still considered an open research problem. [2] Its existence relies fundamentally on electron–electron interactions. In 1988, it was proposed that there was a quantum Hall effect without Landau levels. [3] This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect.
In 1982, Frank Wilczek published two papers exploring the fractional statistics of quasiparticles in two dimensions, giving them the name "anyons" to indicate that the phase shift upon permutation can take any value. [10] Daniel Tsui and Horst Störmer discovered the fractional quantum Hall effect in
To solve this problem, several promising approaches are proposed in condensed matter physics, including Josephson junction qubits, spintronic qubits using the spin orientation of magnetic materials, and the topological non-Abelian anyons from fractional quantum Hall effect states. [78] Condensed matter physics also has important uses for ...
In quantum mechanics, the quantum revival [1] is a periodic recurrence of the quantum wave function from its original form during the time evolution either many times in space as the multiple scaled fractions in the form of the initial wave function (fractional revival) or approximately or exactly to its original form from the beginning (full ...