SOLUTIONS FOR THE LANDAU PROBLEM USING SYMPLECTIC REPRESENTATIONS OF THE GALILEI GROUP

2013 ◽  
Vol 28 (05n06) ◽  
pp. 1350013 ◽  
Author(s):  
R. G. G. AMORIM ◽  
M. C. B. FERNANDES ◽  
F. C. KHANNA ◽  
A. E. SANTANA ◽  
J. D. M. VIANNA
Keyword(s):  
Quantum Mechanics ◽  
Group Theory ◽  
Phase Space ◽  
Wigner Function ◽  
Physical Theory ◽  
Star Product ◽  
Unitary Representations ◽  
Theory Approach ◽  
Galilei Group ◽  
Symplectic Representations

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

scholarly journals Analytical Solution for the Gross-Pitaevskii Equation in Phase Space and Wigner Function

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
A. X. Martins ◽  
R. A. S. Paiva ◽  
G. Petronilo ◽  
R. R. Luz ◽  
R. G. G. Amorim ◽  
...  
Keyword(s):  
Group Theory ◽  
Phase Space ◽  
Analytical Solution ◽  
Wigner Function ◽  
Star Product ◽  
Unitary Representations ◽  
Probability Amplitude ◽  
Pitaevskii Equation ◽  
Theory Approach ◽  
Consistent Theory

In this work, we study symplectic unitary representations for the Galilei group. As a consequence a nonlinear Schrödinger equation is derived in phase space. The formalism is based on the noncommutative structure of the star product, and using the group theory approach as a guide a physically consistent theory is constructed in phase space. The state is described by a quasi-probability amplitude that is in association with the Wigner function. With these results, we solve the Gross-Pitaevskii equation in phase space and obtained the Wigner function for the system considered.

scholarly journals Quantum Physics in Phase Space: an Analysis of Simple Pendulum

2018 ◽  
Vol 1 (2) ◽  
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Schrödinger Equation ◽  
Wigner Function ◽  
Schrodinger Equation ◽  
Quantum Physics ◽  
Symplectic Structure ◽  
Unitary Representations ◽  
Galilei Group ◽  
Simple Pendulum

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

Analytical solution for quantum quartic oscillator in phase space

2020 ◽  
Vol 35 (20) ◽  
pp. 2050100
Author(s):  
A. X. Martins ◽  
T. M. R. Filho ◽  
R. G. G. Amorim ◽  
R. A. S. Paiva ◽  
G. Petronilo ◽  
...  
Keyword(s):  
Phase Space ◽  
Analytical Solution ◽  
Schrödinger Equation ◽  
Wigner Function ◽  
Schrodinger Equation ◽  
Unitary Representations ◽  
Algebraic Methods ◽  
Quantum Oscillator ◽  
Galilei Group ◽  
Lie Methods

In this work, we address the quartic quantum oscillator in phase space using two approaches: computational and algebraic methods. In order to achieve such an aim, we built simplistic unitary representations for Galilei group, as a consequence the Schrödinger equation is derived in the phase space. In this context, the amplitudes of quasi-probability are associated with the Wigner function. In a computational way, we apply the techniques of Lie methods. As a result, we determine the solution of the quantum oscillator in the phase space and calculate the corresponding Wigner function. We also calculated the negativity parameter of the analyzed system.

Phase-Space Approach to Berry Phases

2006 ◽  
Vol 13 (01) ◽  
pp. 67-74 ◽  
Author(s):  
Dariusz Chruściński
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Wigner Function ◽  
Berry Phase ◽  
Space Formulation ◽  
Berry Phases ◽  
Classical Phase Space ◽  
New Formula ◽  
Space Approach

We propose a new formula for the adiabatic Berry phase which is based on phase-space formulation of quantum mechanics. This approach sheds a new light onto the correspondence between classical and quantum adiabatic phases — both phases are related with the averaging procedure: Hannay angle with averaging over the classical torus and Berry phase with averaging over the entire classical phase space with respect to the corresponding Wigner function.

QUANTUM MECHANICAL GALILEI GROUP AND Q-PLANES

1992 ◽  
Vol 07 (34) ◽  
pp. 3169-3177 ◽  
Author(s):  
A.E.F. DJEMAI
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Quantum Mechanical ◽  
Galilei Group

In this work, we show that for a particular choice of translations in the phase space in the context of quantum mechanics, we get a Manin plane. In this framework, we construct the quantum mechanical Galilei group.

scholarly journals QUANTUM CANONICAL TRANSFORMATIONS IN WEYL–WIGNER–GROENEWOLD–MOYAL FORMALISM

2009 ◽  
Vol 24 (24) ◽  
pp. 4573-4587 ◽  
Author(s):  
TEKİN DERELİ ◽  
TUĞRUL HAKİOĞLU ◽  
ADNAN TEĞMEN
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Generating Function ◽  
Canonical Transformation ◽  
Star Product ◽  
Product Form ◽  
Canonical Transformations ◽  
Exponential Form ◽  
Alternative Approach ◽  
Point Transformations

A conjecture in quantum mechanics states that any quantum canonical transformation can decompose into a sequence of three basic canonical transformations; gauge, point and interchange of coordinates and momenta. It is shown that if one attempts to construct the three basic transformations in star-product form, while gauge and point transformations are immediate in star-exponential form, interchange has no correspondent, but it is possible in an ordinary exponential form. As an alternative approach, it is shown that all three basic transformations can be constructed in the ordinary exponential form and that in some cases this approach provides more useful tools than the star-exponential form in finding the generating function for given canonical transformation or vice versa. It is also shown that transforms of c-number phase space functions under linear–nonlinear canonical transformations and intertwining method can be treated within this argument.

Geometric Hamiltonian quantum mechanics and applications

2016 ◽  
Vol 13 (Supp. 1) ◽  
pp. 1630017 ◽  
Author(s):  
Davide Pastorello
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Projective Space ◽  
Star Product ◽  
Hamiltonian Formulation ◽  
Classical Mechanics ◽  
Point Of View ◽  
Quantum Theories ◽  
Geometric Point ◽  
Set Up

Adopting a geometric point of view on Quantum Mechanics is an intriguing idea since, we know that geometric methods are very powerful in Classical Mechanics then, we can try to use them to study quantum systems. In this paper, we summarize the construction of a general prescription to set up a well-defined and self-consistent geometric Hamiltonian formulation of finite-dimensional quantum theories, where phase space is given by the Hilbert projective space (as Kähler manifold), in the spirit of celebrated works of Kibble, Ashtekar and others. Within geometric Hamiltonian formulation quantum observables are represented by phase space functions, quantum states are described by Liouville densities (phase space probability densities), and Schrödinger dynamics is induced by a Hamiltonian flow on the projective space. We construct the star-product of this phase space formulation and some applications of geometric picture are discussed.

scholarly journals The group structure of dynamical transformations between quantum reference frames

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 470
Author(s):  
Angel Ballesteros ◽  
Flaminia Giacomini ◽  
Giulia Gubitosi
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Reference Frames ◽  
Group Structure ◽  
Building Blocks ◽  
Quantum Systems ◽  
Galilei Group ◽  
Dynamical Features ◽  
Group Symmetries ◽  
Galilei Algebra

Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations.

Star Product on the Euclidean Motion Group in the Plane

2021 ◽  
pp. 209-218
Author(s):  
Laarni B. Natividad ◽  
Job A. Nable
Keyword(s):  
Image Analysis ◽  
Quantum Mechanics ◽  
Phase Space ◽  
Star Product ◽  
Weyl Quantization ◽  
Wigner Transform ◽  
Motion Group ◽  
Euclidean Motion Group ◽  
Euclidean Motion ◽  
Motion Groups

In this work, we perform exact and concrete computations of star-product of functions on the Euclidean motion group in the plane, and list its $C$-star-algebra properties. The star-product of phase space functions is one of the main ingredients in phase space quantum mechanics, which includes Weyl quantization and the Wigner transform, and their generalizations. These methods have also found extensive use in signal and image analysis. Thus, the computations we provide here should prove very useful for phase space models where the Euclidean motion groups play the crucial role, for instance, in quantum optics.

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