Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Berry phase, Group Structure of theHydrogen Atom and the CoulombicMonopole in Non-commutativeQuantum Mechanics

Juhi Rajhans

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

This work introduces the reader to non-commutativity in quantum mechan- ics and explores the interaction picture of an electron and a monopole in the given scenario. It has been speculated that the unnatural vector po- tential in the case of a monopole magnetic �eld is due to the co-ordinate representation of the kinetic momentum operators. Thus if the co-ordinate representation is suitably constructed, one can generate a Hilbert space of self-adjoint operators which commute with the Hamiltonian. The �rst part of the work involves the study of the Berry phase and its manifestations in non-commutative quantum mechanics. The second part of the work involves the study of the group structure of the hydrogen atom and the preservation of the group structure in non-commutative quantum mechanics. The third part of the thesis deals with a charge-monopole interaction problem and the exact mapping of the group structure to the hydrogen atom problem. The concluding section gives a short description of the ongoing work of gener- alising the study of the group structure in the charge-monopole problem in non-commutative quantum mechanics.

Related papers

A charged particle interacting with a stationary magnetic monopole: quantum mechanics based on the kinetic momentum operators

Milun J. Raković

The standard quantum mechanical description of the motion of a charged particle in the field of a stationary magnetic monopole is notorious for the presence of unnatural singularities in the Hamiltonian operator originating in the vector potential A(r) used to describe the magnetic field of the monopole. In this paper, an elementary quantum mechanical formulation of the problem which involves only the physically observable field B(r) is presented. This is achieved by treating as a fundamental observable of the charged particle its kinetic momentum instead of the linear momentum p . An irreducible representation of the fundamental commutation relations involving the operators and is explicitly constructed. It is shown that the existence of an irreducible representation requires that Dirac’s charge quantization condition is satisfied. Also, it is demonstrated that, from the quantum mechanical perspective, the singularities (appearing when the vector potential is introduced) are in fact properties of coordinate representations of the fundamental commutation relations.

Journal of Physics A Mathematical and Theoretical•Nov 4, 2011•1 citation

Connection Constraints from Non-Abelian Supersymmetric Quantum Mechanics

Gianni Tallarita

We generalise the study of constraints imposed by supersymmetry on the Berry connection to transformations with component fields in representations of an internal symmetry group G. Since the fields act as co-ordinates of the underlying space one finds a non-trivial extension to its structure and, correspondingly, there are new non-abelian constraints on the Berry connection. The specific case of G=SU(2) is shown to constrain the connection to behave as a magnetic monopole over su(2), its Lie algebra.

arXiv:1005.3245•May 18, 2010•0 citations

Non-Relativistic Quantum Theory: Dynamics, Symmetry, and Geometry

Kai Shun Lam

Why Is Group Theory Useful in Quantum Mechanics? Irreducible Representations of SU(2) and SO(3), Rotation Matrices The Symmetric Groups The Lie Algebra of SO(4) and the Hydrogen Atom Geometric Phases: The Aharonov-Bohm Effect and the Magnetic Monopole The Berry Phase in Molecular Dynamics The Dynamic Phase: Riemann Surfaces in the Semiclassical Theory of Non-Adiabatic Collisions The Connection Is the Gauge Field and the Curvature Is the Force: Some Topology and Differential Geometry Topological Quantum (Chern) Numbers: The Integer Quantum Hall Effect Chern-Simons Theory: The Fractional Quantum Hall Effect and other chapters.

•Aug 24, 2009•3 citations

Resolution of monopole singularity in Berry's phase

Shinichi Deguchi, Kazuo Fujikawa

A monopole-like singularity generated by Berry's phase in momentum space and associated anomalous Poisson brackets have been discussed in various fields. In the applications, it is crucially important to understand precisely what is the monopole in momentum space implied by Berry's phase. Berry's phase is sensitive to how it is measured and thus very different from the genuine Dirac monopole; the monopole-like singularity in Berry's phase is a result of adiabatic approximation while the singularity disappears in non-adiabatic approximation. The resolution of monopole-like singularity in the non-adiabatic domain implies that no Dirac string appears close to the monopole position and the anomalous terms containing the monopole flux in quantum mechanical commutation relations are absent. The absence of modified commutators is also concluded if one understands Berry's phase in momentum space as an $O(\hbar)$ quantum mechanical effect with a quantized magnetic charge $e_{M}=2\pi\hbar$.

•Feb 27, 2019•1 citation

Quantum mechanics in phase space

Philip Watson

The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.

•Jan 1, 1991•0 citations

The quantum mechanics of non-holonomic systems

Richard J. Eden

Abstract Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q(0)r, then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q(0)r.

Proceedings of the Royal Society of London A Mathematical and Physical Sciences•Mar 7, 1951•30 citations
This work introduces the reader to non-commutativity in quantum mechan-
ics and explores the interaction picture of an electron and a monopole in
the given scenario. It has been speculated that the unnatural vector po-
tential in the case of a monopole magnetic �eld is due to the co-ordinate
representation of the kinetic momentum operators. Thus if the co-ordinate
representation is suitably constructed, one can generate a Hilbert space of
self-adjoint operators which commute with the Hamiltonian. The �rst part
of the work involves the study of the Berry phase and its manifestations in
non-commutative quantum mechanics. The second part of the work involves
the study of the group structure of the hydrogen atom and the preservation
of the group structure in non-commutative quantum mechanics. The third
part of the thesis deals with a charge-monopole interaction problem and the
exact mapping of the group structure to the hydrogen atom problem. The
concluding section gives a short description of the ongoing work of gener-
alising the study of the group structure in the charge-monopole problem in
non-commutative quantum mechanics.
The standard quantum mechanical description of the motion of a charged particle in the field of a stationary magnetic monopole is notorious for the presence of unnatural singularities in the Hamiltonian operator originating in the vector potential A(r) used to describe the magnetic field of the monopole. In this paper, an elementary quantum mechanical formulation of the problem which involves only the physically observable field B(r) is presented. This is achieved by treating as a fundamental observable of the charged particle its kinetic momentum instead of the linear momentum p . An irreducible representation of the fundamental commutation relations involving the operators and is explicitly constructed. It is shown that the existence of an irreducible representation requires that Dirac’s charge quantization condition is satisfied. Also, it is demonstrated that, from the quantum mechanical perspective, the singularities (appearing when the vector potential is introduced) are in fact properties of coordinate representations of the fundamental commutation relations.
We generalise the study of constraints imposed by supersymmetry on the Berry connection to transformations with component fields in representations of an internal symmetry group G. Since the fields act as co-ordinates of the underlying space one finds a non-trivial extension to its structure and, correspondingly, there are new non-abelian constraints on the Berry connection. The specific case of G=SU(2) is shown to constrain the connection to behave as a magnetic monopole over su(2), its Lie algebra.
Why Is Group Theory Useful in Quantum Mechanics? Irreducible Representations of SU(2) and SO(3), Rotation Matrices The Symmetric Groups The Lie Algebra of SO(4) and the Hydrogen Atom Geometric Phases: The Aharonov-Bohm Effect and the Magnetic Monopole The Berry Phase in Molecular Dynamics The Dynamic Phase: Riemann Surfaces in the Semiclassical Theory of Non-Adiabatic Collisions The Connection Is the Gauge Field and the Curvature Is the Force: Some Topology and Differential Geometry Topological Quantum (Chern) Numbers: The Integer Quantum Hall Effect Chern-Simons Theory: The Fractional Quantum Hall Effect and other chapters.
A monopole-like singularity generated by Berry's phase in momentum space and associated anomalous Poisson brackets have been discussed in various fields. In the applications, it is crucially important to understand precisely what is the monopole in momentum space implied by Berry's phase. Berry's phase is sensitive to how it is measured and thus very different from the genuine Dirac monopole; the monopole-like singularity in Berry's phase is a result of adiabatic approximation while the singularity disappears in non-adiabatic approximation. The resolution of monopole-like singularity in the non-adiabatic domain implies that no Dirac string appears close to the monopole position and the anomalous terms containing the monopole flux in quantum mechanical commutation relations are absent. The absence of modified commutators is also concluded if one understands Berry's phase in momentum space as an O(ℏ)O(\hbar)O(ℏ) quantum mechanical effect with a quantized magnetic charge eM=2πℏe_{M}=2\pi\hbareM​=2πℏ.
The objective of this thesis is to describe the fundamental concepts relating to the reformulation of quantum mechanics in phase space. It is assumed that the interested reader is familiar with the principles and techniques of ordinary quantum mechanics. In classical statistical mechanics the expectation values of physical quantities are calculated as averages over phase space distribution functions. It is possible to obtain a similar procedure in quantum mechanics. However, phase space is a classical concept and thus has no quantum mechanical equivalent, owing to the non-commutability of the operators p and q. A sensible basic requirement for the formulation of quantum mechanics in phase space is a linear one-one mapping between quantum operators and classical functions. This was achieved by the pioneering work of Weyl, Wigner and Moyal. The phase space that they devised we shall call "pseudo phase space". It is completely quantum. The pseudo phase space variables are represented by commuting c-numbers of momentum, p, and position, q; they are the result of applying the Weyl correspondence rule to the operators p and q, respectively. In some ways the pseudo phase space formulation of quantum mechanics appears to return us to an "almost" classical arena, since we dispense with the calculus of non-commuting operators for purely algebraic methods. The pseudo phase space formulation of quantum mechanics is self-contained. That is, there is no need, in principle, to switch to the Schroedinger or Heisenberg pictures when solving physical problems, although it may often be convenient to do so. The classical appearance of the pseudo phase space scheme is especially useful when considering the semi-classical limit of quantum mechanics. In fact, for potentials up to and including the harmonic oscillator the equations represent classical motion. The structure of the thesis is such that it takes the reader steadily through the major concepts of pseudo phase space theory. The mathematical techniques that shall be needed are developed early in Chapter 1. These are used throughout the thesis. This presents the advantage that the reader is not faced with learning new mathematical methods as the thesis proceeds. The pseudo phase space formulation of quantum mechanics as found applications in all areas of mathematical physics. Hence, there is a considerable amount of specialist literature available on the subject. Itis hoped that this thesis serves has an introduction.
Abstract Interactions of a non-holonomic type are fundamentally different from interactions which can be treated as part of the Hamiltonian of a system. They usually lead to constraints which do not commute with the Hamiltonian, and cause important alterations in the development of a state vector. This paper deals with the Heisenberg equations of motion by analogy with classical dynamics using the Poisson bracket formalism of a previous paper (Eden 1951). The Schrödinger equation is investigated in co-ordinate representation, and it is shown that the wave function will have a non-integrabie phase factor or quasi phase. The quasi phase leads to an indefiniteness in the wave function, but does not violate the fundamental laws of quantum mechanics nor lead to any ambiguity in the physical interpretation of the theory. The relation between the Schrödinger and the Heisenberg equations shows that the Schrödinger treatment is also consistent with the classical analogue. If there is a given initial probability that the non-holonomic system has co-ordinates q(0)r, then there will be the same probability that the wave function in the subsequent motion will be zero except in a certain region of co-ordinate space. This region is the part of co-ordinate space which is accessible in the classical theory from the point q(0)r.