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Journal articles on the topic 'Commutation relations (Quantum mechanics)'

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Published: 4 June 2021

Last updated: 1 August 2024

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1

WIDOM, A., and Y. N. SRIVASTAVA. "QUANTUM FLUID MECHANICS AND QUANTUM ELECTRODYNAMICS." Modern Physics Letters B 04, no. 01 (1990): 1–8. http://dx.doi.org/10.1142/s0217984990000027.

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The commutation relations of Landau quantum fluid mechanics are compared with those of quantum electrodynamics. In both cases, the operator representation of the commutators require a macroscopic phase, and a wavefunction periodic in that phase. A physical discussion is given for analogous effects in superfluids and superconductors, with regard to quantum coherence on a macroscopic scale. Other applications are then briefly described.

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2

SHABANOV, SERGEI V. "q-OSCILLATORS, NON-KÄHLER MANIFOLDS AND CONSTRAINED DYNAMICS." Modern Physics Letters A 10, no. 12 (1995): 941–48. http://dx.doi.org/10.1142/s0217732395001034.

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It is shown that q-deformed quantum mechanics (systems with q-deformed Heisenberg commutation relations) can be interpreted as an ordinary quantum mechanics on Kähler manifolds, or as a quantum theory with second- (or first-) class constraints.

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3

Man'ko, V. I., G. Marmo, F. Zaccaria, and E. C. G. Sudarshan. "Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics." International Journal of Modern Physics B 11, no. 10 (1997): 1281–96. http://dx.doi.org/10.1142/s0217979297000666.

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It is shown that for quantum systems the vector field associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrödinger and Heisenberg picture. We illustrate these ambiguities in terms of simple examples.

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4

IORIO, ALFREDO, and GIUSEPPE VITIELLO. "QUANTUM GROUPS AND VON NEUMANN THEOREM." Modern Physics Letters B 08, no. 04 (1994): 269–76. http://dx.doi.org/10.1142/s0217984994000285.

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We discuss the q-deformation of Weyl-Heisenberg algebra in connection with the von Neumann theorem in quantum mechanics. We show that the q-deformation parameter labels the Weyl systems in quantum mechanics and the unitarily inequivalent representations of the canonical commutation relations in quantum field theory.

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5

Shakhova, E. A., P. P. Rymkevich, A. S. Gorshkov, M. Y. Egorov, and A. S. Stepashkina. "Energy processes with natural quantization." E3S Web of Conferences 124 (2019): 01046. http://dx.doi.org/10.1051/e3sconf/201912401046.

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The paper shows that the quantum-mechanical approach is applicable to most macro processes occurring in nature include the power industry. The mathematical apparatus of the isomorphic Heisenberg algebra is proposed. A non-commutative ring is constructed within which the commutation relations are given. The transition from quantum to classical theory is shown.

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6

PEDRAM, POURIA. "A CLASS OF GUP SOLUTIONS IN DEFORMED QUANTUM MECHANICS." International Journal of Modern Physics D 19, no. 12 (2010): 2003–9. http://dx.doi.org/10.1142/s0218271810018153.

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Various candidates of quantum gravity such as string theory, loop quantum gravity and black hole physics all predict the existence of a minimum observable length which modifies the Heisenberg uncertainty principle to the so-called generalized uncertainty principle (GUP). This approach results from the modification of the commutation relations and changes all Hamiltonians in quantum mechanics. In this paper, we present a class of physically acceptable solutions for a general commutation relation without directly solving the corresponding generalized Schrödinger equations. These solutions satisfy the boundary conditions and exhibit the effect of the deformed algebra on the energy spectrum. We show that this procedure prevents us from doing equivalent but lengthy calculations.

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7

Kober, Martin. "Quaternionic quantization principle in general relativity and supergravity." International Journal of Modern Physics A 31, no. 04n05 (2016): 1650004. http://dx.doi.org/10.1142/s0217751x16500044.

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A generalized quantization principle is considered, which incorporates nontrivial commutation relations of the components of the variables of the quantized theory with the components of the corresponding canonical conjugated momenta referring to other space–time directions. The corresponding commutation relations are formulated by using quaternions. At the beginning, this extended quantization concept is applied to the variables of quantum mechanics. The resulting Dirac equation and the corresponding generalized expression for plane waves are formulated and some consequences for quantum field theory are considered. Later, the quaternionic quantization principle is transferred to canonical quantum gravity. Within quantum geometrodynamics as well as the Ashtekar formalism, the generalized algebraic properties of the operators describing the gravitational observables and the corresponding quantum constraints implied by the generalized representations of these operators are determined. The generalized algebra also induces commutation relations of the several components of the quantized variables with each other. Finally, the quaternionic quantization procedure is also transferred to [Formula: see text] supergravity. Accordingly, the quantization principle has to be generalized to be compatible with Dirac brackets, which appear in canonical quantum supergravity.

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8

FLORATOS, EMMANUEL. "MATRIX QUANTIZATION OF TURBULENCE." International Journal of Bifurcation and Chaos 22, no. 09 (2012): 1250213. http://dx.doi.org/10.1142/s0218127412502136.

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Based on our recent work on Quantum Nambu Mechanics [Axenides & Floratos 2009], we provide an explicit quantization of the Lorenz chaotic attractor through the introduction of noncommutative phase space coordinates as Hermitian N × N matrices in R3. For the volume preserving part, they satisfy the commutation relations induced by one of the two Nambu Hamiltonians, the second one generating a unique time evolution. Dissipation is incorporated quantum mechanically in a self-consistent way having the correct classical limit without the introduction of external degrees of freedom. Due to its volume phase space contraction, it violates the quantum commutation relations. We demonstrate that the Heisenberg–Nambu evolution equations for the Matrix Lorenz system develop fast decoherence to N independent Lorenz attractors. On the other hand, there is a weak dissipation regime, where the quantum mechanical properties of the volume preserving nondissipative sector survive for long times.

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9

Brooke, J. A., and E. Prugovečki. "Relativistic canonical commutation relations and the geometrization of quantum mechanics." Il Nuovo Cimento A 89, no. 2 (1985): 126–48. http://dx.doi.org/10.1007/bf02804855.

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10

Chester, David, Xerxes D. Arsiwalla, Louis H. Kauffman, Michel Planat, and Klee Irwin. "Quantization of a New Canonical, Covariant, and Symplectic Hamiltonian Density." Symmetry 16, no. 3 (2024): 316. http://dx.doi.org/10.3390/sym16030316.

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We generalize Koopman–von Neumann classical mechanics to poly symplectic fields and recover De Donder–Weyl’s theory. Compared with Dirac’s Hamiltonian density, it inspires a new Hamiltonian formulation with a canonical momentum field that is Lorentz-covariant with symplectic geometry. We provide commutation relations for the classical and quantum fields that generalize the Koopman–von Neumann and Heisenberg algebras. The classical algebra requires four fields that generalize spacetime, energy–momentum, frequency–wavenumber, and the Fourier conjugate of energy–momentum. We clarify how first and second quantization can be found by simply mapping between operators in classical and quantum commutator algebras.

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11

Chevalier, Hadrien, Hyukjoon Kwon, Kiran E. Khosla, Igor Pikovski, and M. S. Kim. "Many-body probes for quantum features of spacetime." AVS Quantum Science 4, no. 2 (2022): 021402. http://dx.doi.org/10.1116/5.0079675.

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Many theories of quantum gravity can be understood as imposing a minimum length scale the signatures of which can potentially be seen in precise table top experiments. In this work, we inspect the capacity for correlated many-body systems to probe non-classicalities of spacetime through modifications of the commutation relations. We find an analytic derivation of the dynamics for a single mode light field interacting with a single mechanical oscillator and with coupled oscillators to first order corrections to the commutation relations. Our solution is valid for any coupling function as we work out the full Magnus expansion. We numerically show that it is possible to have superquadratic scaling of a nonstandard phase term, arising from the modification to the commutation relations, with coupled mechanical oscillators.

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12

Skála, Lubomír, and Vojtěch Kapsa. "Quantum Mechanics Needs No Interpretation." Collection of Czechoslovak Chemical Communications 70, no. 5 (2005): 621–37. http://dx.doi.org/10.1135/cccc20050621.

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Probabilistic description of results of measurements and its consequences for understanding quantum mechanics are discussed. It is shown that the basic mathematical structure of quantum mechanics like the probability amplitudes, Born rule, probability density current, commutation and uncertainty relations, momentum operator, rules for including scalar and vector potentials and antiparticles can be derived from the definition of the mean values of powers of space coordinates and time. Equations of motion of quantum mechanics, the Klein-Gordon equation, Schrödinger equation and Dirac equation are obtained from the requirement of the relativistic invariance of the theory. The limit case of localized probability densities leads to the Hamilton-Jacobi equation of classical mechanics. Many-particle systems are also discussed.

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13

Başkal, Sibel, Young Kim, and Marilyn Noz. "Poincaré Symmetry from Heisenberg’s Uncertainty Relations." Symmetry 11, no. 3 (2019): 409. http://dx.doi.org/10.3390/sym11030409.

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It is noted that the single-variable Heisenberg commutation relation contains the symmetry of the S p ( 2 ) group which is isomorphic to the Lorentz group applicable to one time-like dimension and two space-like dimensions, known as the S O ( 2 , 1 ) group. According to Paul A. M. Dirac, from the uncertainty commutation relations for two variables, it possible to construct the de Sitter group S O ( 3 , 2 ) , namely the Lorentz group applicable to three space-like variables and two time-like variables. By contracting one of the time-like variables in S O ( 3 , 2 ) , it is possible to construct the inhomogeneous Lorentz group I S O ( 3 , 1 ) which serves as the fundamental symmetry group for quantum mechanics and quantum field theory in the Lorentz-covariant world. This I S O ( 3 , 1 ) group is commonly known as the Poincaré group.

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14

Benzair, H., M. Merad, and T. Boudjedaa. "Path integral for quantum dynamics with position-dependent mass within the displacement operator approach." Modern Physics Letters A 35, no. 30 (2020): 2050246. http://dx.doi.org/10.1142/s0217732320502466.

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In the context of quantum mechanics reformulated in a modified Hilbert space, we can formulate the Feynman’s path integral approach for the quantum systems with position-dependent mass particle using the formulation of position-dependent infinitesimal translation operator. Which is similar a deformed quantum mechanics based on modified commutation relations. An illustration of calculation is given in the case of the harmonic oscillator, the infinite square well and the inverse square plus Coulomb potentials.

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15

Moretti, P., and L. Cianchi. "Feynman's approach to quantum mechanics: Trajectories, commutation relations and uncertainty principle." Il Nuovo Cimento D 11, no. 1-2 (1989): 229–40. http://dx.doi.org/10.1007/bf02450241.

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16

Wu, Kunlong. "Embarking on the path to quantum field theory." Theoretical and Natural Science 26, no. 1 (2023): 221–26. http://dx.doi.org/10.54254/2753-8818/26/20241083.

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The harmonic oscillator represents as a unique tool in modern physics since it is not only exactly solvable but also relates many phenomena. This paper introduces the relation between Quantum Field Theory and other physical theories and discusses how harmonic oscillator behave in Quantum Mechanics and Quantum Field Theory by the quantization of Klein-Gordon Field. To this end, this paper applies energy-momentum equation given by special relativity to Schrdingers Equation to get the Klein-Gordon Field, and then uses canonical method to quantize the field, which is called the second quantization of the field. The comparison is made between the result quantized field, the quantum harmonic oscillator and classical harmonic oscillator. The similarity of two physical systems is demonstrate by the comparison between commutation relations of operators. The similar commutation relations then help people to interpret quantized Klein-Gordon Field as a system that contain a quantized harmonic oscillator in each space coordinate.

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17

Mathieu, J., L. Marchildon, and D. Rochon. "The bicomplex quantum Coulomb potential problem." Canadian Journal of Physics 91, no. 12 (2013): 1093–100. http://dx.doi.org/10.1139/cjp-2013-0261.

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Generalizations of the complex number system underlying the mathematical formulation of quantum mechanics have been known for some time, but the use of the commutative ring of bicomplex numbers for that purpose is relatively new. This paper provides an analytical solution of the quantum Coulomb potential problem formulated in terms of bicomplex numbers. We define the problem by introducing a bicomplex hamiltonian operator and extending the canonical commutation relations to the form [Formula: see text], where ξ is a bicomplex number. Following Pauli’s algebraic method, we find the eigenvalues of the bicomplex hamiltonian. These eigenvalues are also obtained, along with appropriate eigenfunctions, by solving the extension of Schrödinger’s time-independent differential equation. Examples of solutions are displayed. There is an orthonormal system of solutions that belongs to a bicomplex Hilbert space.

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18

Meljanac, Stjepan, and Salvatore Mignemi. "Quantum Mechanics of the Extended Snyder Model." Symmetry 15, no. 7 (2023): 1373. http://dx.doi.org/10.3390/sym15071373.

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We investigate a quantum mechanical harmonic oscillator based on the extended Snyder model. This realization of the Snyder model is constructed as a quantum phase space generated by D spatial coordinates and D(D−1)/2 tensorial degrees of freedom, together with their conjugated momenta. The coordinates obey nontrivial commutation relations and generate a noncommutative geometry, which admits nicer properties than the usual realization of the model, in particular giving rise to an associative star product. The spectrum of the harmonic oscillator is studied through the introduction of creation and annihilation operators. Some physical consequences of the introduction of the additional degrees of freedom are discussed.

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19

Wheeler, James T. "Quanta Without Quantization." Modern Physics Letters A 12, no. 29 (1997): 2175–81. http://dx.doi.org/10.1142/s0217732397002223.

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The dimensional properties of fields in classical general relativity lead to a tangent tower structure which gives rise directly to quantum mechanical and quantum field theory structures without quantization. We derive all the fundamental elements of quantum mechanics from the tangent tower structure, including fundamental commutation relations, a Hilbert space of pure and mixed states, measurable expectation values, Schrödinger time evolution, "collapse" of a state and the probability interpretation. The most central elements of string theory also follow, including an operator valued mode expansion like that in string theory as well as the Virasoro algebra with central charges.

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20

Chashchina, Olga I., Abhijit Sen, and Zurab K. Silagadze. "On deformations of classical mechanics due to Planck-scale physics." International Journal of Modern Physics D 29, no. 10 (2020): 2050070. http://dx.doi.org/10.1142/s0218271820500704.

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Several quantum gravity and string theory thought experiments indicate that the Heisenberg uncertainty relations get modified at the Planck scale so that a minimal length do arises. This modification may imply a modification of the canonical commutation relations and hence quantum mechanics at the Planck scale. The corresponding modification of classical mechanics is usually considered by replacing modified quantum commutators by Poisson brackets suitably modified in such a way that they retain their main properties (antisymmetry, linearity, Leibniz rule and Jacobi identity). We indicate that there exists an alternative interesting possibility. Koopman–von Neumann’s Hilbert space formulation of classical mechanics allows, as Sudarshan remarked, to consider the classical mechanics as a hidden variable quantum system. Then, the Planck scale modification of this quantum system naturally induces the corresponding modification of dynamics in the classical substrate. Interestingly, it seems this induced modification in fact destroys the classicality: classical position and momentum operators cease to be commuting and hidden variables do appear in their evolution equations.

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21

Contreras González, Mauricio, Marcelo Villena, and Roberto Ortiz Herrera. "An Optimal Control Perspective on Classical and Quantum Physical Systems." Symmetry 15, no. 11 (2023): 2033. http://dx.doi.org/10.3390/sym15112033.

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This paper analyzes classical and quantum physical systems from an optimal control perspective. Specifically, we explore whether their associated dynamics can correspond to an open- or closed-loop feedback evolution of a control problem. Firstly, for the classical regime, when it is viewed in terms of the theory of canonical transformations, we find that a closed-loop feedback problem can describe it. Secondly, for a quantum physical system, if one realizes that the Heisenberg commutation relations themselves can be considered constraints in a non-commutative space, then the momentum must depend on the position of any generic wave function. That implies the existence of a closed-loop strategy for the quantum case. Thus, closed-loop feedback is a natural phenomenon in the physical world. By way of completeness, we briefly review control theory and the classical mechanics of constrained systems and analyze some examples at the classical and quantum levels.

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22

ANDERSON, LARA B., and JAMES T. WHEELER. "QUANTUM MECHANICS AS A MEASUREMENT THEORY ON BICONFORMAL SPACE." International Journal of Geometric Methods in Modern Physics 03, no. 02 (2006): 315–40. http://dx.doi.org/10.1142/s0219887806001168.

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Biconformal spaces contain the essential elements of quantum mechanics, making the independent imposition of quantization unnecessary. Based on three postulates characterizing motion and measurement in biconformal geometry, we derive standard quantum mechanics, and show how the need for probability amplitudes arises from the use of a standard of measurement. Additionally, we show that a postulate for unique, classical motion yields Hamiltonian dynamics with no measurable size changes, while a postulate for probabilistic evolution leads to physical dilatations manifested as measurable phase changes. Our results lead to the Feynman path integral formulation, from which follows the Schrödinger equation. We discuss the Heisenberg uncertainty relation and fundamental canonical commutation relations.

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23

Eckhardt, W. "The quantum-mechanical harmonic oscillator: Markovian limit and commutation relations." Physics Letters A 114, no. 2 (1986): 75–76. http://dx.doi.org/10.1016/0375-9601(86)90482-2.

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24

Giesel, Kristina, and Michael Kobler. "An Open Scattering Model in Polymerized Quantum Mechanics." Mathematics 10, no. 22 (2022): 4248. http://dx.doi.org/10.3390/math10224248.

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We derive a quantum master equation in the context of a polymerized open quantum mechanical system for the scattering of a Brownian particle in an ideal gas environment. The model is formulated in a top-down approach by choosing a Hamiltonian with a coupling between the system and environment that is generally associated with spatial decoherence. We extend the existing work on such models by using a non-standard representation of the canonical commutation relations, inspired by the quantization procedure applied in loop quantum gravity, which yields a model in which position operators are replaced by holonomies. The derivation of the master equation in a top-down approach opens up the possibility to investigate in detail whether the assumptions, usually used in such models in order to obtain a tractable form of the dissipator, hold also in the polymerized case or whether they need to be dropped or modified. Furthermore, we discuss some physical properties of the master equation associated to effective equations for the expectation values of the fundamental operators and compare our results to the already existing models of collisional decoherence.

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25

Özcan, Özgür. "Investigating students’ conceptual difficulties on commutation relations and expectation value problems in quantum mechanics." SHS Web of Conferences 26 (2016): 01123. http://dx.doi.org/10.1051/shsconf/20162601123.

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26

Bhattacharya, Ranjan, and Siddhartha Bhowmick. "Do trilinear commutation relations in quantum mechanics admit coordinate space realization in three dimensions?" Journal of Mathematical Physics 28, no. 6 (1987): 1290–92. http://dx.doi.org/10.1063/1.527532.

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27

ZUMINO, BRUNO. "DEFORMATION OF THE QUANTUM MECHANICAL PHASE SPACE WITH BOSONIC OR FERMIONIC COORDINATES." Modern Physics Letters A 06, no. 13 (1991): 1225–35. http://dx.doi.org/10.1142/s0217732391001305.

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We describe the one-parameter deformation of the phase space of a quantum mechanical system and show that this twisted phase space is covariant under the action of the symplectic quantum group. The analogous case of a system with fermionic coordinates is also considered and the phase space is shown to be covariant under the action of the orthogonal quantum group. Twisted commutation relations occur in the description of deformed spaces or superspaces as well as in the formulation of field theories with generalized statistics. The many-parameter case is briefly discussed.

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28

SOW, C. L., and T. T. TRUONG. "QUANTUM GROUP APPROACH TO A SOLUBLE VERTEX MODEL WITH GENERALIZED ICE RULE." International Journal of Modern Physics A 11, no. 10 (1996): 1747–61. http://dx.doi.org/10.1142/s0217751x96000936.

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Using the representation of the quantum group SL q(2) by the Weyl operators of the canonical commutation relations in quantum mechanics, we construct and solve a new vertex model on a square lattice. Random variables on horizontal bonds are Ising variables, and those on the vertical bonds take half positive integer values. The vertex is subjected to a generalized form of the so-called “ice rule,” its property is studied in detail and its free energy calculated with the method of quantum inverse scattering. Remarkably, in analogy with the usual six-vertex model, there exists a “free-fermion” limit with a novel rich operator structure. The existing algebraic structure suggests a possible connection with a lattice neutral plasma of charges, via the fermion-boson correspondence.

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29

Kryvobok, Artem, and Alan D. Kathman. "Quantum mechanical four-dimensional non-polarizing beamsplitter." Quantum Studies: Mathematics and Foundations 9, no. 1 (2021): 55–70. http://dx.doi.org/10.1007/s40509-021-00256-8.

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AbstractSome quantum optics researchers might not realize that classical electromagnetism predicts a $$\mathbf {\pi }$$ π phase shift between S- and P-polarized reflection and might think the reflection coefficients of the transverse modes are independent, or that such a phase shift has no measurable consequences. In this paper, we discuss theoretical grounds to define elements of a 4x4 matrix to represent the beamsplitter, accurately accounting for transverse polarization modes and phase relations between them. We also provide experimental evidence confirming this matrix representation. From a scientific point of view, the paper addresses a non-trivial equivalence between the classical fields Fresnel formalism and the canonical commutation relations of the quantized photonic fields. That the formalism can be readily verified with a simple experiment provides further benefit. The beamsplitter expression derived can be applied in the field of quantum computing.

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30

Bracci, Luciano, and Luigi E. Picasso. "Nonequivalent representations of canonical commutation relations in quantum mechanics: The case of the Aharonov-Bohm effect." American Journal of Physics 75, no. 3 (2007): 268–71. http://dx.doi.org/10.1119/1.2360994.

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DULAT, SAYIPJAMAL, and KANG LI. "COMMUTATOR ANOMALY IN NONCOMMUTATIVE QUANTUM MECHANICS." Modern Physics Letters A 21, no. 39 (2006): 2971–76. http://dx.doi.org/10.1142/s0217732306020585.

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In this paper, the Schrödinger equation on noncommutative phase space is given by using a generalized Bopp's shift. Then the anomaly term of commutator of arbitrary physical observable operators on noncommutative phase space is obtained. Finally, the basic uncertainty relations for space–space and space–momentum as well as momentum–momentum operators in noncommutative quantum mechanics (NCQM), and uncertainty relation for arbitrary physical observable operators in NCQM are discussed.

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Vladimirov, Igor G., and Ian R. Petersen. "Decoherence quantification through commutation relations decay for open quantum harmonic oscillators." Systems & Control Letters 178 (August 2023): 105585. http://dx.doi.org/10.1016/j.sysconle.2023.105585.

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Szasz A., Vincze Gy. "Rosen-Chambers Variation Theory of Linearly-Damped Classic and Quantum Oscillator." JOURNAL OF ADVANCES IN PHYSICS 4, no. 1 (2014): 404–26. http://dx.doi.org/10.24297/jap.v4i1.6966.

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Phenomena of damped harmonic oscillator is important in the description of the elementary dissipative processes of linear responses in our physical world. Its classical description is clear and understood, however it is not so in the quantum physics, where it also has a basic role. Starting from the Rosen-Chambers restricted variation principle a Hamilton like variation approach to the damped harmonic oscillator will be given. The usual formalisms of classical mechanics, as Lagrangian, Hamiltonian, Poisson brackets, will be covered too. We shall introduce two Poisson brackets. The first one has only mathematical meaning and for the second, the so-called constitutive Poisson brackets, a physical interpretation will be presented. We shall show that only the fundamental constitutive Poisson brackets are not invariant throughout the motion of the damped oscillator, but these show a kind of universal time dependence in the universal time scale of the damped oscillator. The quantum mechanical Poisson brackets and commutation relations belonging to these fundamental time dependent classical brackets will be described. Our objective in this work is giving clearer view to the challenge of the dissipative quantum oscillator.

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Gáliková, Veronika, and Peter Prešnajder. "COULOMB SCATTERING IN NON-COMMUTATIVE QUANTUM MECHANICS." Acta Polytechnica 53, no. 5 (2013): 427–32. http://dx.doi.org/10.14311/ap.2013.53.0427.

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Recently we formulated the Coulomb problem in a rotationally invariant NC configuration space specified by NC coordinates <em>x<sub>i</sub>, i</em> = 1, 2, 3, satisfying commutation relations<em> [x<sub>i</sub>, x<sub>j</sub> ] = 2iλε<sub>ijk</sub>x<sub>k</sub></em> (<em>λ</em> being our NC parameter). We found that the problem is exactly solvable: first we gave an exact simple formula for the energies of the negative bound states <em>E<sup>λ</sup><sub>n</sub></em> &lt; 0 (n being the principal quantum number), and later we found the full solution of the NC Coulomb problem. In this paper we present an exact calculation of the NC Coulomb scattering matrix <em>S<sup>λ</sup><sub>j</sub> (E)</em> in the <em>j</em>-th partial wave. As the calculations are exact, we can recognize remarkable non-perturbative aspects of the model: 1) energy cut-off — the scattering is restricted to the energy interval 0 &lt; <em>E</em> &lt; <em>E</em><sub>crit</sub> = 2/<em>λ</em><sup>2</sup>; 2) the presence of two sets of poles of the S-matrix in the complex energy plane — as expected, the poles at negative energy <em>E</em><sup>I</sup><sub><em>λ</em>n</sub> = <em>E</em><sup><em>λ</em></sup><sub>n</sub> for the Coulomb attractive potential, and the poles at ultra-high energies <em>E</em><sup>II</sup><sub><em>λ</em>n</sub> = <em>E</em><sub>crit</sub> − <em>E<sup>λ</sup></em><sub>n</sub> for the Coulomb <em>repulsive</em> potential. The poles at ultra-high energies disappear in the commutative limit <em>λ</em>→0.

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Beggs, Edwin J., and Shahn Majid. "Quantum Riemannian geometry of phase space and nonassociativity." Demonstratio Mathematica 50, no. 1 (2017): 83–93. http://dx.doi.org/10.1515/dema-2017-0009.

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Abstract Noncommutative or ‘quantum’ differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket while the data for quantum differential forms is a Poisson-compatible connection. We give an introduction to our recent result whereby further classical data such as classical bundles, metrics etc. all become quantised in a canonical ‘functorial’ way at least to 1st order in deformation theory. The theory imposes compatibility conditions between the classical Riemannian and Poisson structures as well as new physics such as typical nonassociativity of the differential structure at 2nd order. We develop in detail the case of ℂℙn where the commutation relations have the canonical form [wi, w̄j] = iλδij similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in λ.

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36

Palenik, Mark C. "Quantum mechanics from Newton's second law and the canonical commutation relation [ X , P ] = i." European Journal of Physics 35, no. 4 (2014): 045014. http://dx.doi.org/10.1088/0143-0807/35/4/045014.

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37

DORSCH, GLÁUBER CARVALHO, and JOSÉ ALEXANDRE NOGUEIRA. "MAXIMALLY LOCALIZED STATES IN QUANTUM MECHANICS WITH A MODIFIED COMMUTATION RELATION TO ALL ORDERS." International Journal of Modern Physics A 27, no. 21 (2012): 1250113. http://dx.doi.org/10.1142/s0217751x12501138.

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We construct the states of maximal localization taking into account a modification of the commutation relation between position and momentum operators to all orders of the minimum length parameter. To first-order, the algebra we use reproduces the one proposed by Kempf, Mangano and Mann. It is emphasized that a minimal length acts as a natural regulator for the theory, thus eliminating the otherwise ever appearing infinities. So, we use our results to calculate the first correction to the Casimir effect due to the minimal length. We also discuss some of the physical consequences of the existence of a minimal length, culminating in a proposal to reformulate the very concept of "position measurement."

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38

Cui, Dianzhen, T. Li, Jianning Li, and Xuexi Yi. "Detecting deformed commutators with exceptional points in optomechanical sensors." New Journal of Physics 23, no. 12 (2021): 123037. http://dx.doi.org/10.1088/1367-2630/ac3ff7.

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Abstract Models of quantum gravity imply a modification of the canonical position-momentum commutation relations. In this paper, working with a binary mechanical system, we examine the effect of quantum gravity on the exceptional points of the system. On the one side, we find that the exceedingly weak effect of quantum gravity can be sensed via pushing the system towards a second-order exceptional point, where the spectra of the non-Hermitian system exhibits non-analytic and even discontinuous behavior. On the other side, the gravity perturbation will affect the sensitivity of the system to deposition mass. In order to further enhance the sensitivity of the system to quantum gravity, we extend the system to the other one which has a third-order exceptional point. Our work provides a feasible way to use exceptional points as a new tool to explore the effect of quantum gravity.

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39

Pedram, Pouria. "The Minimal Length and the Shannon Entropic Uncertainty Relation." Advances in High Energy Physics 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/5101389.

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In the framework of the generalized uncertainty principle, the position and momentum operators obey the modified commutation relationX,P=iħ1+βP2, whereβis the deformation parameter. Since the validity of the uncertainty relation for the Shannon entropies proposed by Beckner, Bialynicki-Birula, and Mycielski (BBM) depends on both the algebra and the used representation, we show that using the formally self-adjoint representation, that is,X=xandP=tan⁡βp/β, where[x,p]=iħ, the BBM inequality is still valid in the formSx+Sp≥1+ln⁡πas well as in ordinary quantum mechanics. We explicitly indicate this result for the harmonic oscillator in the presence of the minimal length.

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Kullie, Ossama. "Time Operator, Real Tunneling Time in Strong Field Interaction and the Attoclock." Quantum Reports 2, no. 2 (2020): 233–52. http://dx.doi.org/10.3390/quantum2020015.

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Attosecond science, beyond its importance from application point of view, is of a fundamental interest in physics. The measurement of tunneling time in attosecond experiments offers a fruitful opportunity to understand the role of time in quantum mechanics. In the present work, we show that our real T-time relation derived in earlier works can be derived from an observable or a time operator, which obeys an ordinary commutation relation. Moreover, we show that our real T-time can also be constructed, inter alia, from the well-known Aharonov–Bohm time operator. This shows that the specific form of the time operator is not decisive, and dynamical time operators relate identically to the intrinsic time of the system. It contrasts the famous Pauli theorem, and confirms the fact that time is an observable, i.e., the existence of time operator and that the time is not a parameter in quantum mechanics. Furthermore, we discuss the relations with different types of tunneling times, such as Eisenbud–Wigner time, dwell time, and the statistically or probabilistic defined tunneling time. We conclude with the hotly debated interpretation of the attoclock measurement and the advantage of the real T-time picture versus the imaginary one.

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Herrebrugh, Albert V. ‎. "Determinism In Quantum Slit-Experiments." Hyperscience International Journals 2, no. 3 (2022): 115–21. http://dx.doi.org/10.55672/hij2022pp115-121.

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A mathematical model for the slit experiments in the heart of quantum mechanics is developed to gain insight into quantum ‎theory. The proposed system-theoretical model is entirely based on commutative mathematics, i.e. convolution, and integral ‎transformations, and starts with spacetime functions with inherent energy-based cause and effect relations of the state-‎function Ѱ in the complex Hilbert space. The benefits of his approach are as 1-Invariance in time reversal. 2-Deterministic ‎result functions in the model in line with the outcome of slit experiments. 3- Separation of causality and cross-correlations of ‎attained states. 4- Disappearance of a posteriori probability of quantum states. 5- Quantum a priori fixed states after ‎causality interactions have ended, (even) when quanta are (light-years) separated. The model predicts the patterns in the ‎experiments with mathematical functions of the energy distributions. The quantum mechanical counterpart description of the ‎physical reality of slit experiments thus may be considered complete in A. Einstein’s definition. The patterns in double slit ‎experiments are found to be an effect of energy (amplitude-) modulation. An equivalent double-slit pattern can be retrieved ‎from an input modulated 1-slit experiment excluding interference interpretations. The system-theoretical model uses generic ‎properties of quanta and evolves into determinism in ‎quantum mechanics slit experiments. The mathematics in the model ‎handles beables by treatment ‎of momentum p in system theoretical I/O relations of the transformed functions and allows the ‎proposed description by the avoidance of a direct addressing of the individual quanta through variables. The following ‎method yields exact, non-probabilistic results.‎

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Ching, C. L., C. X. Yeo, and W. K. Ng. "Nonrelativistic anti-Snyder model and some applications." International Journal of Modern Physics A 32, no. 02n03 (2017): 1750009. http://dx.doi.org/10.1142/s0217751x17500099.

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In this paper, we examine the (2[Formula: see text]+[Formula: see text]1)-dimensional Dirac equation in a homogeneous magnetic field under the nonrelativistic anti-Snyder model which is relevant to doubly/deformed special relativity (DSR) since it exhibits an intrinsic upper bound of the momentum of free particles. After setting up the formalism, exact eigensolutions are derived in momentum space representation and they are expressed in terms of finite orthogonal Romanovski polynomials. There is a finite maximum number of allowable bound states [Formula: see text] due to the orthogonality of the polynomials and the maximum energy is truncated at [Formula: see text]. Similar to the minimal length case, the degeneracy of the Dirac–Landau levels in anti-Snyder model are modified and there are states that do not exist in the ordinary quantum mechanics limit [Formula: see text]. By taking [Formula: see text], we explore the motion of effective massless charged fermions in graphene-like material and obtained a maximum bound of deformed parameter [Formula: see text]. Furthermore, we consider the modified energy dispersion relations and its application in describing the behavior of neutrinos oscillation under modified commutation relations.

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Chung, Won Sang. "Two Phases of the Non-Commutative Quantum Mechanics with the Generalized Uncertainty Relations." International Journal of Theoretical Physics 55, no. 4 (2015): 2174–81. http://dx.doi.org/10.1007/s10773-015-2856-z.

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ACCARDI, LUIGI. "NOISE AND DISSIPATION IN QUANTUM THEORY." Reviews in Mathematical Physics 02, no. 02 (1990): 127–76. http://dx.doi.org/10.1142/s0129055x90000065.

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A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.

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Plankensteiner, David, Christoph Hotter, and Helmut Ritsch. "QuantumCumulants.jl: A Julia framework for generalized mean-field equations in open quantum systems." Quantum 6 (January 4, 2022): 617. http://dx.doi.org/10.22331/q-2022-01-04-617.

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A full quantum mechanical treatment of open quantum systems via a Master equation is often limited by the size of the underlying Hilbert space. As an alternative, the dynamics can also be formulated in terms of systems of coupled differential equations for operators in the Heisenberg picture. This typically leads to an infinite hierarchy of equations for products of operators. A well-established approach to truncate this infinite set at the level of expectation values is to neglect quantum correlations of high order. This is systematically realized with a so-called cumulant expansion, which decomposes expectation values of operator products into products of a given lower order, leading to a closed set of equations. Here we present an open-source framework that fully automizes this approach: first, the equations of motion of operators up to a desired order are derived symbolically using predefined canonical commutation relations. Next, the resulting equations for the expectation values are expanded employing the cumulant expansion approach, where moments up to a chosen order specified by the user are included. Finally, a numerical solution can be directly obtained from the symbolic equations. After reviewing the theory we present the framework and showcase its usefulness in a few example problems.

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SHALABY, ABOUZEID M. "VACUUM STABILITY OF THE ${\mathcal{PT}}$-SYMMETRIC (-ϕ4) SCALAR FIELD THEORY". International Journal of Modern Physics A 28, № 08 (2013): 1350023. http://dx.doi.org/10.1142/s0217751x13500231.

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In this paper, we study the vacuum stability of the classical unstable (-ϕ4) scalar field potential. Regarding this, we obtained the effective potential, up to second-order in the coupling, for the theory in 1+1 and 2+1 space–time dimensions. We found that the obtained effective potential is bounded-from-below, which proves the vacuum stability of the theory in space–time dimensions higher than the previously studied 0+1 case. In our calculations, we used the canonical quantization regime in which one deals with operators rather than classical functions used in the path integral formulation. Therefore, the non-Hermiticity of the effective field theory is obvious. Moreover, the method we employ implements the canonical equal-time commutation relations and the Heisenberg picture for the operators. Thus, the metric operator is implemented in the calculations of the transition amplitudes. Accordingly, the method avoids the very complicated calculations needed in other methods for the metric operator. To test the accuracy of our results, we obtained the exponential behavior of the vacuum condensate for small coupling values, which has been obtained in the literature using other methods. We assert that this work is interesting, as all the studies in the literature advocate the stability of the (-ϕ4) theory at the quantum mechanical level while our work extends the argument to the level of field quantization.

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Speicher, Roland, and Moritz Weber. "Quantum groups with partial commutation relations." Indiana University Mathematics Journal 68, no. 6 (2019): 1849–83. http://dx.doi.org/10.1512/iumj.2019.68.7791.

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48

Goodearl, Kenneth R. "Commutation relations for arbitrary quantum minors." Pacific Journal of Mathematics 228, no. 1 (2006): 63–102. http://dx.doi.org/10.2140/pjm.2006.228.63.

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Chung, Won Sang. "Generalized Uncertainty Relation in the Non-commutative Quantum Mechanics." International Journal of Theoretical Physics 55, no. 6 (2016): 2989–3000. http://dx.doi.org/10.1007/s10773-016-2931-0.

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Wang, Simeng. "Quantum symmetries on noncommutative complex spheres with partial commutation relations." Infinite Dimensional Analysis, Quantum Probability and Related Topics 21, no. 04 (2018): 1850028. http://dx.doi.org/10.1142/s0219025718500285.

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We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

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