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Comptes Rendus Physique

Moving bases as an alternative to ‘interaction representations’

J. Jeener

Published Apr 1, 2004• 3 citations
Publisher

Abstract

The use of multiple bases moving with respect to each other in quantum dynamics is discussed. This procedure is formally equivalent to the use of interaction pictures, but it leads to an intuitively simpler interpretation of the calculations. A close analogy is shown between the rotating frames of classical NMR theory and the use of moving (rotating) bases in the corresponding quantum presentation, for the traditional basic NMR experiment and for two problems involving Berry's phase.

DOI
10.1016/j.crhy.2004.03.002

Related papers

Interchange of Pictures in Quantum Mechanics and its Applications

Xue Zheng

Picture in Quantum Mechanics corresponds to reference frame in classical mechanics.This paper discusses the three usual pictures in Quantum Mechanics.Rotating reference frame is a kind of frame which is more general than interaction picture.With proper choice of involved parameters,one can simplify the Hamiltonian as well as its derivation procedures,and be clearer of the physical nature.

Journal of Hefei Normal University•Jan 1, 2010•0 citations

The Dirac picture

Dževad Belkić

In this chapter, the authors show that various pictures of quantum mechanics are interrelated by the unitary transformations. Calling upon the correspondence principle, the quantum-mechanical scattering problem would be quite analogous to the mentioned classical model. The two analysed pictures of quantum mechanics are not the only possible ones. Quite the opposite, all the unitary transformations carried out on the state vectors and the observables of the Schrödinger picture, in fact, define a new formalism of quantum mechanics. There are infinitely many such formalisms but they are all rigorously equivalent. Among all of the possible, alternative pictures of quantum mechanics, of particular importance for scattering theory is the so-called interaction picture, which was first introduced by Dirac. Hence, the alternative name: the Dirac picture.

CRC Press eBooks•Jan 15, 2020•0 citations

Time dependence in quantum mechanics : Floquet theory and the Berry phase

D J Moore

We give a short review of some aspects of the dynamics of time-dependent quantum systems. In particular we discuss how to recast time-periodic quantum problems into an equivalent time-independent form, contrasting the results with those gained for a similar reformulation in the case of general time-dependence. This approach, usually known under the name of Floquet theory, can be used to clarify the structure of, for example, invariants of the given quantum system. We also apply it to the calculation of the Berry phase and show how it enables us to understand Berry phases for time-independent systems. This involves rewriting earlier work of the author in a coordinate-free fashion

•Feb 15, 1993•1 citation

Quantum Description Of Nmr

M. Goldman

Abstract We are now equipped with the necessary mathematical background to analyse NMR in quantum-mechanical terms. In this chapter, the analysis is restricted to standard NMR, essentially the free-induction decay following a single r.f. pulse. The first section deals with the case when the only interaction experienced by the spins is the Zeeman interaction with an external field. It illustrates the way in which quantum mechanics is used in practice and its main result is a quantum justification of the equation of motion of the magnetization derived in Chapter 1 by a purely classical approach. The second section is a simplified description of chemical shifts and indirect interactions and of their influence on the NMR spectra. For such a simple problem, it will already become apparent that the formal machinery developed in the previous chapters is far superior to the classical or naive quantum methods, as regards the ease and the simplicity with which quantitative results are obtained.

•Mar 28, 2001•0 citations

Berry phases and unitary transformations

Thomas F. Jordan

Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.

Journal of Mathematical Physics•Sep 1, 1988•56 citations

On magnetic monopoles, the anomalous g-factor of the electron and the spin-orbit coupling in the Dirac theory

G. Coddens

We discuss the algebra and the interpretation of the anomalous Zeeman effect and the spin-orbit coupling within the Dirac theory. Whereas the algebra for the anomalous Zeeman effect is impeccable and therefore in excellent agreement with experiment, the physical interpretation of that algebra uses images that are based on macroscopic intuition but do not correspond to the meaning of this algebra. The interpretation violates the Lorentz symmetry. We give an alternative intuitive description of the meaning of this effect, which respects the symmetry and is exact. It can be summarized by stating that a magnetic field makes any charged particle spin. We show also that the traditional discussion about magnetic monopoles confuses two issues, viz. the symmetry of the Maxwell equations and the quantization of charge. These two issues define each a different concept of magnetic monopole. They cannot be merged together into a unique all-encompassing issue. We also generalize the minimal substitution for a charged particle, and provide some intuition for the magnetic vector potential. We finally explore the algebra of the spin-orbit coupling, which turns out to be badly wrong. The traditional theory that is claimed to reproduce the Thomas half is based on a number of errors. An error-free application of the Dirac theory cannot account for the Thomas precession, because it only accounts for the instantaneous local boosts, not for the rotational component of the Lorentz transformation. This runs contrary to established beliefs, but Thomas precession can be understood in terms of the Berry phase on a path through the velocity space of the Lorentz group manifold. These results clearly reveal the limitations of the prevailing working philosophy to "shut up and calculate".

arXiv:1604.05160•Apr 1, 2016•2 citations
The use of multiple bases moving with respect to each other in quantum dynamics is discussed. This procedure is formally equivalent to the use of interaction pictures, but it leads to an intuitively simpler interpretation of the calculations. A close analogy is shown between the rotating frames of classical NMR theory and the use of moving (rotating) bases in the corresponding quantum presentation, for the traditional basic NMR experiment and for two problems involving Berry's phase.
In this chapter, the authors show that various pictures of quantum mechanics are interrelated by the unitary transformations. Calling upon the correspondence principle, the quantum-mechanical scattering problem would be quite analogous to the mentioned classical model. The two analysed pictures of quantum mechanics are not the only possible ones. Quite the opposite, all the unitary transformations carried out on the state vectors and the observables of the Schrödinger picture, in fact, define a new formalism of quantum mechanics. There are infinitely many such formalisms but they are all rigorously equivalent. Among all of the possible, alternative pictures of quantum mechanics, of particular importance for scattering theory is the so-called interaction picture, which was first introduced by Dirac. Hence, the alternative name: the Dirac picture.
We give a short review of some aspects of the dynamics of time-dependent quantum systems. In particular we discuss how to recast time-periodic quantum problems into an equivalent time-independent form, contrasting the results with those gained for a similar reformulation in the case of general time-dependence. This approach, usually known under the name of Floquet theory, can be used to clarify the structure of, for example, invariants of the given quantum system. We also apply it to the calculation of the Berry phase and show how it enables us to understand Berry phases for time-independent systems. This involves rewriting earlier work of the author in a coordinate-free fashion
Abstract We are now equipped with the necessary mathematical background to analyse NMR in quantum-mechanical terms. In this chapter, the analysis is restricted to standard NMR, essentially the free-induction decay following a single r.f. pulse. The first section deals with the case when the only interaction experienced by the spins is the Zeeman interaction with an external field. It illustrates the way in which quantum mechanics is used in practice and its main result is a quantum justification of the equation of motion of the magnetization derived in Chapter 1 by a purely classical approach. The second section is a simplified description of chemical shifts and indirect interactions and of their influence on the NMR spectra. For such a simple problem, it will already become apparent that the formal machinery developed in the previous chapters is far superior to the classical or naive quantum methods, as regards the ease and the simplicity with which quantitative results are obtained.
Berry phases for spin are defined for any closed loop made by a vector changing direction in three-dimensional space. A sequence of rotations moves the vector along the loop. Each rotation is around the axis perpendicular to the moving vector. The Berry phases come from the eigenvalues of the unitary operator for the product of these rotations. The angle of the product rotation is shown to be the solid angle enclosed by the loop. The proof uses the ordinary language of quantum mechanics. The product is calculated from the commutation relations for spin. A general framework is set up to define Berry phases for other transformations and states like those for rotations and spin. The integral formula is derived. Alternatives for dynamics are shown to provide different applications and interpretations of the same mathematics. An example is used to show how one Hamiltonian may be simpler than others. Adiabatic evolution is obtained in the limit as a coupling constant goes to zero, so the adiabatic changes are made by a weak perturbation that acts over a long time. The Berry phase is the same whether the coupling constant is large or small. A rationale for the definition of Berry phases is obtained from any of the alternatives for dynamics. It is particularly clear in both the limit of adiabatic evolution, where the Hamiltonian does not contain the operator that generates Berry phases, and the opposite extreme where the Hamiltonian is the operator that generates Berry phases. The latter is illustrated with an example for spin. The general definition of Berry phases and the method of calculating them are illustrated by obtaining Berry phases from Lorentz transformations. They are similar to those obtained from rotations. A vector traces a loop on a unit hyperboloid instead of the unit sphere. In place of the solid angle, the Berry phases contain the analogous measure of the area enclosed by the loop on the hyperboloid. The sign is opposite what it is for rotations. The general definition is shown to fit any unitary representation of a semisimple Lie group. The complete set of commuting operators is chosen to contain a basis for a Cartan subalgebra of the Lie algebra of generators. If a sequence of transformations by unitary operators in the group representation takes each operator in the Cartan subalgebra around a loop in the Lie algebra back to the same operator, it can be made by unitary operators in the group representation that have all the properties required for the definition of Berry phases. The loops are made by vectors moving in a real space, the Lie algebra. The motion of the vectors must maintain the lengths and angles defined by the Cartan metric of the Lie algebra. The Berry phases are determined by one loop made by one suitably chosen operator from the Cartan subalgebra.
We discuss the algebra and the interpretation of the anomalous Zeeman effect and the spin-orbit coupling within the Dirac theory. Whereas the algebra for the anomalous Zeeman effect is impeccable and therefore in excellent agreement with experiment, the physical interpretation of that algebra uses images that are based on macroscopic intuition but do not correspond to the meaning of this algebra. The interpretation violates the Lorentz symmetry. We give an alternative intuitive description of the meaning of this effect, which respects the symmetry and is exact. It can be summarized by stating that a magnetic field makes any charged particle spin. We show also that the traditional discussion about magnetic monopoles confuses two issues, viz. the symmetry of the Maxwell equations and the quantization of charge. These two issues define each a different concept of magnetic monopole. They cannot be merged together into a unique all-encompassing issue. We also generalize the minimal substitution for a charged particle, and provide some intuition for the magnetic vector potential. We finally explore the algebra of the spin-orbit coupling, which turns out to be badly wrong. The traditional theory that is claimed to reproduce the Thomas half is based on a number of errors. An error-free application of the Dirac theory cannot account for the Thomas precession, because it only accounts for the instantaneous local boosts, not for the rotational component of the Lorentz transformation. This runs contrary to established beliefs, but Thomas precession can be understood in terms of the Berry phase on a path through the velocity space of the Lorentz group manifold. These results clearly reveal the limitations of the prevailing working philosophy to "shut up and calculate".