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Mar 2013

Volume 54, Issue 3, Articles (03xxxx)

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J. Math. Phys. 54, 033512 (2013); http://dx.doi.org/10.1063/1.4794514 (20 pages)

Andrew Neate and Aubrey Truman
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back to top Quantum Mechanics

The symmetry groups of noncommutative quantum mechanics and coherent state quantization

S. Hasibul Hassan Chowdhury and S. Twareque Ali

J. Math. Phys. 54, 032101 (2013); http://dx.doi.org/10.1063/1.4793992 (21 pages)

Online Publication Date: 8 March 2013

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We explore the group theoretical underpinning of noncommutative quantum mechanics for a system moving on the two-dimensional plane. We show that the pertinent groups for the system are the two-fold central extension of the Galilei group in (2+1)-space-time dimensions and the two-fold extension of the group of translations of math4. This latter group is just the standard Weyl-Heisenberg group of standard quantum mechanics with an additional central extension. We also look at a further extension of this group and discuss its significance to noncommutative quantum mechanics. We build unitary irreducible representations of these various groups and construct the associated families of coherent states. A coherent state quantization of the underlying phase space is then carried out, which is shown to lead to exactly the same commutation relations as usually postulated for this model of noncommutative quantum mechanics.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.20.-a Group theory
03.65.Fd Algebraic methods

On Pauli pairs

Stanislav Shkarin

J. Math. Phys. 54, 032102 (2013); http://dx.doi.org/10.1063/1.4794087 (10 pages)

Online Publication Date: 11 March 2013

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The state of a system in classical mechanics can be uniquely reconstructed if we know the positions and the momenta of all its parts. In 1958 Pauli has conjectured that the same holds for quantum mechanical systems. The conjecture turned out to be wrong. In this paper we provide a new set of examples of Pauli pairs, being the pairs of quantum states indistinguishable by measuring the spatial location and momentum. In particular, we construct a new set of spatially localized Pauli pairs.
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03.65.Ta Foundations of quantum mechanics; measurement theory

About a new family of coherent states for some SU(1,1) central field potentials

Dusan Popov, Vjekoslav Sajfert, Nicolina Pop, and Viorel Chiritoiu

J. Math. Phys. 54, 032103 (2013); http://dx.doi.org/10.1063/1.4795137 (21 pages)

Online Publication Date: 19 March 2013

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In this paper, we shall define a new family of coherent states which we shall call the “mother coherent states,” bearing in mind the fact that these states are independent from any parameter (the Bargmann index, the rotational quantum number J, and so on). So, these coherent states are defined on the whole Hilbert space of the Fock basis vectors. The defined coherent states are of the Barut-Girardello kind, i.e., they are the eigenstates of the lowering operator. For these coherent states we shall calculate the expectation values of different quantum observables, the corresponding Mandel parameter, the Husimi's distribution function and also the P- function. Finally, we shall particularize the obtained results for the three-dimensional harmonic and pseudoharmonic oscillators.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra
03.65.Fd Algebraic methods

Quantum graph as a quantum spectral filter

Ondřej Turek and Taksu Cheon

J. Math. Phys. 54, 032104 (2013); http://dx.doi.org/10.1063/1.4795404 (17 pages)

Online Publication Date: 20 March 2013

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We study the transmission of a quantum particle along a straight input–output line to which a graph Γ is attached at a point. In the point of contact we impose a singularity represented by a certain properly chosen scale-invariant coupling with a coupling parameter α. We show that the probability of transmission along the line as a function of the particle energy tends to the indicator function of the energy spectrum of Γ as α → ∞. This effect can be used for a spectral analysis of the given graph Γ. Its applications include a control of a transmission along the line and spectral filtering. The result is illustrated with an example where Γ is a loop exposed to a magnetic field. Two more quantum devices are designed using other special scale-invariant vertex couplings. They can serve as a band-stop filter and as a spectral separator, respectively.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.10.Ox Combinatorics; graph theory
02.30.-f Function theory, analysis
02.50.Cw Probability theory
03.65.Fd Algebraic methods

Optimal volume Wegner estimate for random magnetic Laplacians on math2

David Hasler and Daniel Luckett

J. Math. Phys. 54, 032105 (2013); http://dx.doi.org/10.1063/1.4794082 (7 pages)

Online Publication Date: 21 March 2013

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We consider a two dimensional magnetic Schrödinger operator on a square lattice with a spatially stationary random magnetic field. We prove a Wegner estimate with optimal volume dependence. The Wegner estimate holds around the spectral edges, and it implies Hölder continuity of the integrated density of states in this region. The proof is based on the Wegner estimate obtained in Erdős and Hasler [“Wegner estimate for random magnetic Laplacians on 2Z2,” Ann. Henri Poincaré 12, 1719–1731 (2012)]10.1007/s00023-012-0177-9.
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03.65.Ge Solutions of wave equations: bound states
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
05.50.+q Lattice theory and statistics (Ising, Potts, etc.)
02.50.-r Probability theory, stochastic processes, and statistics

Group action in topos quantum physics

C. Flori

J. Math. Phys. 54, 032106 (2013); http://dx.doi.org/10.1063/1.4795803 (52 pages)

Online Publication Date: 29 March 2013

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Topos theory has been suggested first by Isham and Butterfield, and then by Isham and Döring, as an alternative mathematical structure within which to formulate physical theories. In particular, it has been used to reformulate standard quantum mechanics in such a way that a novel type of logic is used to represent propositions. In this paper, we extend this formulation to include the notion of a group and group transformation in such a way that we overcome the problem of twisted presheaves. In order to implement this we need to change the type of topos involved, so as to render the notion of continuity of the group action meaningful.
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03.65.Fd Algebraic methods
02.20.Uw Quantum groups
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