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Cambridge University Press eBooks

Wavelets in atomic physics and in solid state physics

Jean-Pierre Antoine, Ph Antoine, B. Piraux

Published Aug 19, 1999• 4 citations
Publisher

Abstract

In the field of atomic and solid state physics, wavelet analysis has been applied so far in three different directions: (i) time-frequency analysis of harmonic generation in laser-atom interactions; (ii) ab initio electronic structure calculations in atoms and molecules; and (iii) construction of localized bases for the lowest Landau level of a 2-D electron gas submitted to a strong magnetic field. We survey these three types of applications, with more emphasis on methods than on precise results. Introduction There are two ways in which wavelets could play a role in atomic physics and possibly in solid state physics. First one may envisage them as physical objects , namely quantum states or wave functions. It is commonplace to remark that coherent states (CS) have a privileged role in atomic physics. Laser-atom interactions, revival phenomena, Rydberg wave packets and various semi-classical situations are all instances in which a coherent state description is clearly well-adapted. Of course, what is implied here are canonical CS, associated to the harmonic oscillator or the electromagnetic field [36]. But wavelets are also coherent states, namely those associated to the affine groups in various space dimensions, as we have seen in Chapter 2 (see [1] for a review on coherent states). Thus wavelets could well be thought of as convenient substitutes for canonical CS. However, this suggestion is still speculative at the present moment, very little has been achieved in this direction.

DOI
10.1017/cbo9780511613265.011

Related papers

Higher-dimensional wavelets

Jean-Pierre Antoine, Romain Murenzi, Pierre Vandergheynst +1 more

In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ 3 , wavelets in ℝ n ( n > 3), and wavelets on the 2-sphere S 2 . Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis. In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform. Three-dimensional wavelets Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].

Cambridge University Press eBooks•Sep 23, 2004•0 citations

COHERENT STATES AND THEIR GENERALIZATIONS: A MATHEMATICAL OVERVIEW

S. Twareque Ali, Jean-Pierre Antoine, Jean‐Pierre Gazeau +1 more

We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.

Reviews in Mathematical Physics•Oct 1, 1995•149 citations

Coherent states: a contemporary panorama

S. Twareque Ali, Jean-Pierre Antoine, Fabio Bagarello +1 more

Coherent states (CS) of the harmonic oscillator (also called canonical CS) were introduced in 1926 by Schrödinger in answer to a remark by Lorentz on the classical interpretation of the wave function. They were rediscovered in the early 1960s, first (somewhat implicitly) by Klauder in the context of a novel representation of quantum states, then by Glauber and Sudarshan for the description of coherence in lasers. Since then, CS have grown into an extremely rich domain that pervades almost every corner of physics and have also led to the development of several flourishing topics in mathematics. Along the way, a number of review articles have appeared in the literature, devoted to CS, notably the 1985 reprint volume of Klauder and Skagerstam [1], the 1990 review paper by Zhang et al [2], the 1993 Oak Ridge Conference [3] and the 1995 review paper by Ali et al [4]. Textbooks also have been published, among which one might mention the ground breaking text of Perelomov [5] focusing on the group-theoretical aspects, that of Ali et al [6] 1 analyzing systematically the mathematical structure beyond the group-theoretical approach and also the relation to wavelet analysis, that of Dodonov and Man'ko [7] mostly devoted to quantum optics, that of Gazeau [8] more oriented towards the physical, probabilistic and quantization aspects, and finally the very recent one by Combescure and Robert [9]. In retrospect, one can see that the development of CS has gone through a two-phase transition. First, the (simultaneous) discovery in 1972 by Gilmore and Perelomov that CS were rooted in group theory, then the realization that CS can be defined in a purely algebraic way, as an eigenvalue problem or by a series expansion (Malkin and Man'ko 1969, Barut and Girardello 1971, Gazeau and Klauder 1999; references to the original articles may be found in the textbooks quoted above). Both facts resulted in an explosive expansion of the CS literature. We thought, therefore, that the time was ripe to devote a special issue of Journal of Physics A: Mathematical and Theoretical to CS. However, because of limitations of space and time, it would have been impossible to get a fully representative cross-section of papers, covering all the different facets of the subject. Consequently, we have selected 37 articles, including some by a few of the originators of the field. We thank all the authors for submitting their up-to-date thoughts on this fascinating subject. The contents of this special issue are subdivided into five categories: (1) review papers; (2) physics-oriented CS; (3) physics and quantum information; (4) mathematics, general topics; and (5) mathematics, particular problems. (1) Review papers We start with five review papers. The first paper, by Klauder, surveys the many possible applications of affine variables, both in classical and quantum physics. The second, by Sanders, proposes a grand tour of entangled CS, which are present in many fields, such as quantum optics, quantum information processing, etc. The next paper, by Rowe, surveys the field of vector CS and the attendant group representation problems (including induced representations). Then Oriti et al describe a particular class of CS relevant to (loop) quantum gravity. Finally, Combescure and Robert present a comprehensive review of fermionic CS, including all mathematical details. (2) Physics-oriented CS The six contributions in this section deal with specific physical problems: (i) Dajka–Luczka study Gazeau–Klauder cat states associated with a nonlinear Kerr oscillator, instead of the usual canonical CS leading to Schrödinger cat states; (ii) Angelova et al discuss squeezed CS associated with a 1D Morse potential, used in molecular physics; (iii) Bagrov et al study CS in a magnetic solenoid field and prove their completeness; (iv) Blasone–Jizba treat Nambu–Goldstone dynamics in spontaneously broken symmetries, using CS functional integrals; (v) Calixto et al describe accelerated relativistic particles in the context of spontaneous breakdown of conformal SU(2,2) symmetry, using SU(2,2) CS; and (vi) Mortazavi–Tavassoly study f -deformed charge CS and their physical properties (nonclassical features, sub-Poissonian statistical behavior, etc). (3) Physics and quantum information The second group of physically related CS contains four contributions with a distinct quantum information theoretic flavor. First, Thilagam describes the dynamical behavior of entanglement of a pair of qubits (excitons), using a CS basis. Next, Lavoie–de Guise study SU(3) intelligent states (i.e., minimal uncertainty states), of interest in the quantum information community. Then Muñoz constructs discrete CS for n qubits. Finally, Wagner–Kendon explore the continuous variable Deutsch–Jozsa algorithm known in quantum computing in a discrete formulation. (4) Mathematics, general topics In this subgroup, there are eight papers dealing with general properties of CS, independently of any particular system or application. A whole series discusses the interaction between CS and various mathematical objects: pseudodifferential operators and Weyl calculus (Unterberger); induced representations of the affine group and intertwining operators (Elmabrok–Hutnik); measure-free CS and reproducing kernels (Horzela–Szafraniec); extremal POV measures (Heinosaari–Pellonpää); Hilbert W*-modules (Bhattacharyya–Roy); Toeplitz operators (Hutníková–Hutník); and operator localization and homogeneous structure of nilpotent Lie groups (Kisil). In addition, Balazs et al consider multipliers for continuous frames, including CS or wavelet frames. (5) Mathematics, particular problems The second group of mathematically oriented papers contains 14 contributions, devoted to CS in particular systems. We start with a paper by Gilmore, which explores the (sometimes chaotic) evolution of atomic CS under a time-periodic driving field, using sphere maps S 2 → S 2 . Next, we include a paper on CS on the 2-sphere in a magnetic field (Hall–Mitchell); a paper on CS for a quantum particle on a Möbius strip (Cirilo–Lombardo); a discussion of quantization on the circle (Chadzitaskos et al ); SUSY CS for Pöschl–Teller potentials (Bergeron et al ); generalized Bargmann functions and von Neumann lattices (Vourdas et al ); partial reconstruction for a finite CS system, using the Fock–Bargmann representation (Calixto et al ); phase operators for SU(3) irreps, thus for finite quantum systems (de Guise); semiclassical CS in periodic potentials (Carles–Sparber); complexified CS with non-Hermitian Hamiltonians (Graefe–Schubert); minimal uncertainty states in the context of (semisimple) group representation theory (Oszmaniec); localization operators in the time-frequency domain, i.e., in Gabor analysis (Muzhikyan–Avanesyan); and, finally, two papers about fermionic CS (Daoud–Kibler and Trifonov). This brief description illustrates perfectly the extreme versatility of the CS concept. As already stressed, coherent states constitute nowadays a flourishing research topic, with applications to a wide spectrum of domains. Indeed, CS are everywhere in physics: condensed matter physics, atomic physics, nuclear and particle physics, quantum optics, dynamics—both quantum and classical potentials—quantum gravity, quantization and quantum information theory. On the other hand, CS have grown into a fully-fledged domain in mathematics, incorporating many tools such as group representations, POV measures, frames, holomorphic functions, orthogonal polynomials and so on. Interestingly enough, the majority of contributions to this special issue (22 out of 37) are mathematically minded, demonstrating the widespread interest CS have generated in various areas of mathematics. A third field related to CS (but almost not represented in the present collection) is signal processing. Indeed both Gabor analysis and wavelet analysis derive in the first place from CS theory, namely, CS associated to the Weyl–Heisenberg and the ax + b group, respectively. Here too, a tremendous development has taken place in recent years, another testimony to the richness of the notion of CS. We leave it to the jury of public opinion to judge whether the call for a special issue of the journal, devoted to coherent states, has been justified. References [1] Klauder J R and Skagerstam B S 1985 Coherent States—Applications in Physics and Mathematical Physics (Singapore: World Scientific) [2] Zhang W-M, Feng D H and Gilmore R 1990 Coherent states: theory and some applications Rev. Mod. Phys. 62 867–927 [3] Feng D H, Klauder J R and Strayer M (ed) 1994 Coherent States: Past, Present and Future (Singapore: World Scientific) [4] Ali S T, Antoine J-P, Gazeau J-P and Mueller U A 1995 Coherent states and their generalizations: a mathematical overview Rev. Math. Phys. 7 1013–104 [5] Perelomov A M 1986 Generalized Coherent States and Their Applications (New York: Springer) [6] Ali S T, Antoine J-P and Gazeau J-P 2000 Coherent States, Wavelets and Their Generalizations (New York: Springer) [7] Dodonov V V and Man'ko V I (ed) 2003 Theory of Nonclassical States of Light (London: Taylor & Francis) [8] Gazeau J-P 2009 Coherent States in Quantum Physics (Berlin: Wiley) [9] Combescure M and Robert D 2012 Coherent States and Applications in Mathematical Physics (New York: Springer) 1 A second edition of that volume is in preparation.

Journal of Physics A Mathematical and Theoretical•May 30, 2012•9 citations

Atomic physics

Michel Le Bellac

This chapter is devoted to an introduction to atomic physics which will be mainly concerned with one-electron atoms. After a brief discussion of the perturbation and variational methods in Section 14.1, in Section 14.2 we study the fine and hyperfine structure of the energy levels as well as the effect of a magnetic field on these levels. In Section 14.3 we examine the coupling of an atom to an electromagnetic field and important applications of this coupling such as the photoelectric effect and the rate of spontaneous emission. In Section 14.4 we give a brief introduction to a subject which has been expanding enormously in the last twenty years, the laser manipulation of atoms, and we discuss Doppler cooling and magneto-optical traps. Finally, Section 14.5 is devoted to a short discussion of two-electron atoms. Approximation methods Generalities In classical physics it is only in exceptional cases that it is possible to solve the Newton or Maxwell equations analytically given the initial conditions at time t = t 0 and, in the first case, the forces, or, in the second, the sources of electromagnetic field. In general, it is necessary to resort to an approximation method such as numerical integration of the equations, the perturbation method, or something else. The situation is no different in quantum physics: only in exceptional cases do we know how to “solve the Schrödinger equation” exactly, that is, how to obtain the time evolution of the state vector |ϕ( t 〉 as a function of its value |ϕ( t 0 〉 at initial time t = t 0 .

Cambridge University Press eBooks•Mar 23, 2006•0 citations

Classical Structures in Quantum Mechanics and Applications

Augusto César Lobo, Clyffe de Assis Ribeiro

The theory of Non-Relativistic Quantum Mechanics was created (or discovered) back in the 1920's mainly by Schrödinger and Heisenberg, but it is fair enough to say that a more modern and unified approach to the subject was introduced by Dirac and Jordan with their (intrinsic) Transformation Theory. In his famous text book on quantum mechanics [1], Dirac introduced his well-known bra and ket notation and a view that even Einstein (who was, as well known, very critical towards the general quantum physical world-view) considered the most elegant presentation of the theory at that time[2]. One characteristic of this formulation is that the observables of position and momentum are truly treated equally so that an intrinsic phase-space approach seems a natural course to be taken. In fact, we may distinguish at least two different quantum mechanical approaches to the structure of the quantum phase space: The Weyl-Wigner (WW) formalism and the advent of the theory of Coherent States (CS). The Weyl-Wigner formalism has had many applications ranging from the discussion of the Classical/Quantum Mechanical transition and quantum chaos to signal analysis[3,4]. The Coherent State formalism had a profound impact on Quantum Optics and during the course of time has found applications in diverse areas such as geometric quantization, wavelet and harmonic analysis [5]. In this chapter we present a compact review of these formalisms (with also a more intrinsic and coordinate independent notation) towards some non-standard and up-to-date applications such as modular variables and weak values.

arXiv:1212.5340•Dec 21, 2012•0 citations

Wavelets from coherent states: II

John R. Klauder

The area of applicability of coherent states is generally quantum theory, while that of wavelets is generally signal analysis. Nevertheless, separated from their respective applications, coherent states and (continuous) wavelets have a great deal in common. To illustrate this connection we shall outline three general types of coherent states and a few of their applications. We address our presentation to interested wavelet practitioners.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Mar 22, 1999•3 citations
In the field of atomic and solid state physics, wavelet analysis has been applied so far in three different directions: (i) time-frequency analysis of harmonic generation in laser-atom interactions; (ii) ab initio electronic structure calculations in atoms and molecules; and (iii) construction of localized bases for the lowest Landau level of a 2-D electron gas submitted to a strong magnetic field. We survey these three types of applications, with more emphasis on methods than on precise results. Introduction There are two ways in which wavelets could play a role in atomic physics and possibly in solid state physics. First one may envisage them as physical objects , namely quantum states or wave functions. It is commonplace to remark that coherent states (CS) have a privileged role in atomic physics. Laser-atom interactions, revival phenomena, Rydberg wave packets and various semi-classical situations are all instances in which a coherent state description is clearly well-adapted. Of course, what is implied here are canonical CS, associated to the harmonic oscillator or the electromagnetic field [36]. But wavelets are also coherent states, namely those associated to the affine groups in various space dimensions, as we have seen in Chapter 2 (see [1] for a review on coherent states). Thus wavelets could well be thought of as convenient substitutes for canonical CS. However, this suggestion is still speculative at the present moment, very little has been achieved in this direction.
In the previous chapters, we have thoroughly discussed the 2-D CWT and some of its applications. Then we have made the connection with the group theoretical origins of the method, thus establishing a general framework, based on the coherent state formalism. In the present chapter, we will apply the same technique to a number of different situations involving higher dimensions: wavelets in 3-D space ℝ 3 , wavelets in ℝ n ( n > 3), and wavelets on the 2-sphere S 2 . Then, in the next chapter, we will treat time-dependent wavelets, that is, wavelets on space–time, designed for motion analysis. In all cases, the technique is the same. First one identifies the manifold on which the signals are defined and the appropriate group of transformations acting on the latter. Next one chooses a square integrable representation of that group, possibly modulo some subgroup. Then one constructs wavelets as admissible vectors and derives the corresponding wavelet transform. Three-dimensional wavelets Some physical phenomena are intrinsically multiscale and three-dimensional. Typical examples may be found in fluid dynamics, for instance the appearance of coherent structures in turbulent flows, or the disentangling of a wave train in (mostly underwater) acoustics, as discussed above. In such cases, a 3-D wavelet analysis is clearly more adequate and likely to yield a deeper understanding [56].
We present a survey of the theory of coherent states (CS) and some of their generalizations, with emphasis on the mathematical structure, rather than on physical applications. Starting from the standard theory of CS over Lie groups, we develop a general formalism, in which CS are associated to group representations which are square integrable over a homogeneous space. A further step allows us to dispense with the group context altogether, and thus obtain the so-called reproducing triples and continuous frames introduced in some earlier work. We discuss in detail a number of concrete examples, namely semisimple Lie groups, the relativity groups and various types of wavelets. Finally we turn to some physical applications, centering on quantum measurement and the quantization/dequantization problem, that is, the transition from the classical to the quantum level and vice versa.
Coherent states (CS) of the harmonic oscillator (also called canonical CS) were introduced in 1926 by Schrödinger in answer to a remark by Lorentz on the classical interpretation of the wave function. They were rediscovered in the early 1960s, first (somewhat implicitly) by Klauder in the context of a novel representation of quantum states, then by Glauber and Sudarshan for the description of coherence in lasers. Since then, CS have grown into an extremely rich domain that pervades almost every corner of physics and have also led to the development of several flourishing topics in mathematics. Along the way, a number of review articles have appeared in the literature, devoted to CS, notably the 1985 reprint volume of Klauder and Skagerstam [1], the 1990 review paper by Zhang et al [2], the 1993 Oak Ridge Conference [3] and the 1995 review paper by Ali et al [4]. Textbooks also have been published, among which one might mention the ground breaking text of Perelomov [5] focusing on the group-theoretical aspects, that of Ali et al [6] 1 analyzing systematically the mathematical structure beyond the group-theoretical approach and also the relation to wavelet analysis, that of Dodonov and Man'ko [7] mostly devoted to quantum optics, that of Gazeau [8] more oriented towards the physical, probabilistic and quantization aspects, and finally the very recent one by Combescure and Robert [9]. In retrospect, one can see that the development of CS has gone through a two-phase transition. First, the (simultaneous) discovery in 1972 by Gilmore and Perelomov that CS were rooted in group theory, then the realization that CS can be defined in a purely algebraic way, as an eigenvalue problem or by a series expansion (Malkin and Man'ko 1969, Barut and Girardello 1971, Gazeau and Klauder 1999; references to the original articles may be found in the textbooks quoted above). Both facts resulted in an explosive expansion of the CS literature. We thought, therefore, that the time was ripe to devote a special issue of Journal of Physics A: Mathematical and Theoretical to CS. However, because of limitations of space and time, it would have been impossible to get a fully representative cross-section of papers, covering all the different facets of the subject. Consequently, we have selected 37 articles, including some by a few of the originators of the field. We thank all the authors for submitting their up-to-date thoughts on this fascinating subject. The contents of this special issue are subdivided into five categories: (1) review papers; (2) physics-oriented CS; (3) physics and quantum information; (4) mathematics, general topics; and (5) mathematics, particular problems. (1) Review papers We start with five review papers. The first paper, by Klauder, surveys the many possible applications of affine variables, both in classical and quantum physics. The second, by Sanders, proposes a grand tour of entangled CS, which are present in many fields, such as quantum optics, quantum information processing, etc. The next paper, by Rowe, surveys the field of vector CS and the attendant group representation problems (including induced representations). Then Oriti et al describe a particular class of CS relevant to (loop) quantum gravity. Finally, Combescure and Robert present a comprehensive review of fermionic CS, including all mathematical details. (2) Physics-oriented CS The six contributions in this section deal with specific physical problems: (i) Dajka–Luczka study Gazeau–Klauder cat states associated with a nonlinear Kerr oscillator, instead of the usual canonical CS leading to Schrödinger cat states; (ii) Angelova et al discuss squeezed CS associated with a 1D Morse potential, used in molecular physics; (iii) Bagrov et al study CS in a magnetic solenoid field and prove their completeness; (iv) Blasone–Jizba treat Nambu–Goldstone dynamics in spontaneously broken symmetries, using CS functional integrals; (v) Calixto et al describe accelerated relativistic particles in the context of spontaneous breakdown of conformal SU(2,2) symmetry, using SU(2,2) CS; and (vi) Mortazavi–Tavassoly study f -deformed charge CS and their physical properties (nonclassical features, sub-Poissonian statistical behavior, etc). (3) Physics and quantum information The second group of physically related CS contains four contributions with a distinct quantum information theoretic flavor. First, Thilagam describes the dynamical behavior of entanglement of a pair of qubits (excitons), using a CS basis. Next, Lavoie–de Guise study SU(3) intelligent states (i.e., minimal uncertainty states), of interest in the quantum information community. Then Muñoz constructs discrete CS for n qubits. Finally, Wagner–Kendon explore the continuous variable Deutsch–Jozsa algorithm known in quantum computing in a discrete formulation. (4) Mathematics, general topics In this subgroup, there are eight papers dealing with general properties of CS, independently of any particular system or application. A whole series discusses the interaction between CS and various mathematical objects: pseudodifferential operators and Weyl calculus (Unterberger); induced representations of the affine group and intertwining operators (Elmabrok–Hutnik); measure-free CS and reproducing kernels (Horzela–Szafraniec); extremal POV measures (Heinosaari–Pellonpää); Hilbert W*-modules (Bhattacharyya–Roy); Toeplitz operators (Hutníková–Hutník); and operator localization and homogeneous structure of nilpotent Lie groups (Kisil). In addition, Balazs et al consider multipliers for continuous frames, including CS or wavelet frames. (5) Mathematics, particular problems The second group of mathematically oriented papers contains 14 contributions, devoted to CS in particular systems. We start with a paper by Gilmore, which explores the (sometimes chaotic) evolution of atomic CS under a time-periodic driving field, using sphere maps S 2 → S 2 . Next, we include a paper on CS on the 2-sphere in a magnetic field (Hall–Mitchell); a paper on CS for a quantum particle on a Möbius strip (Cirilo–Lombardo); a discussion of quantization on the circle (Chadzitaskos et al ); SUSY CS for Pöschl–Teller potentials (Bergeron et al ); generalized Bargmann functions and von Neumann lattices (Vourdas et al ); partial reconstruction for a finite CS system, using the Fock–Bargmann representation (Calixto et al ); phase operators for SU(3) irreps, thus for finite quantum systems (de Guise); semiclassical CS in periodic potentials (Carles–Sparber); complexified CS with non-Hermitian Hamiltonians (Graefe–Schubert); minimal uncertainty states in the context of (semisimple) group representation theory (Oszmaniec); localization operators in the time-frequency domain, i.e., in Gabor analysis (Muzhikyan–Avanesyan); and, finally, two papers about fermionic CS (Daoud–Kibler and Trifonov). This brief description illustrates perfectly the extreme versatility of the CS concept. As already stressed, coherent states constitute nowadays a flourishing research topic, with applications to a wide spectrum of domains. Indeed, CS are everywhere in physics: condensed matter physics, atomic physics, nuclear and particle physics, quantum optics, dynamics—both quantum and classical potentials—quantum gravity, quantization and quantum information theory. On the other hand, CS have grown into a fully-fledged domain in mathematics, incorporating many tools such as group representations, POV measures, frames, holomorphic functions, orthogonal polynomials and so on. Interestingly enough, the majority of contributions to this special issue (22 out of 37) are mathematically minded, demonstrating the widespread interest CS have generated in various areas of mathematics. A third field related to CS (but almost not represented in the present collection) is signal processing. Indeed both Gabor analysis and wavelet analysis derive in the first place from CS theory, namely, CS associated to the Weyl–Heisenberg and the ax + b group, respectively. Here too, a tremendous development has taken place in recent years, another testimony to the richness of the notion of CS. We leave it to the jury of public opinion to judge whether the call for a special issue of the journal, devoted to coherent states, has been justified. References [1] Klauder J R and Skagerstam B S 1985 Coherent States—Applications in Physics and Mathematical Physics (Singapore: World Scientific) [2] Zhang W-M, Feng D H and Gilmore R 1990 Coherent states: theory and some applications Rev. Mod. Phys. 62 867–927 [3] Feng D H, Klauder J R and Strayer M (ed) 1994 Coherent States: Past, Present and Future (Singapore: World Scientific) [4] Ali S T, Antoine J-P, Gazeau J-P and Mueller U A 1995 Coherent states and their generalizations: a mathematical overview Rev. Math. Phys. 7 1013–104 [5] Perelomov A M 1986 Generalized Coherent States and Their Applications (New York: Springer) [6] Ali S T, Antoine J-P and Gazeau J-P 2000 Coherent States, Wavelets and Their Generalizations (New York: Springer) [7] Dodonov V V and Man'ko V I (ed) 2003 Theory of Nonclassical States of Light (London: Taylor & Francis) [8] Gazeau J-P 2009 Coherent States in Quantum Physics (Berlin: Wiley) [9] Combescure M and Robert D 2012 Coherent States and Applications in Mathematical Physics (New York: Springer) 1 A second edition of that volume is in preparation.
This chapter is devoted to an introduction to atomic physics which will be mainly concerned with one-electron atoms. After a brief discussion of the perturbation and variational methods in Section 14.1, in Section 14.2 we study the fine and hyperfine structure of the energy levels as well as the effect of a magnetic field on these levels. In Section 14.3 we examine the coupling of an atom to an electromagnetic field and important applications of this coupling such as the photoelectric effect and the rate of spontaneous emission. In Section 14.4 we give a brief introduction to a subject which has been expanding enormously in the last twenty years, the laser manipulation of atoms, and we discuss Doppler cooling and magneto-optical traps. Finally, Section 14.5 is devoted to a short discussion of two-electron atoms. Approximation methods Generalities In classical physics it is only in exceptional cases that it is possible to solve the Newton or Maxwell equations analytically given the initial conditions at time t = t 0 and, in the first case, the forces, or, in the second, the sources of electromagnetic field. In general, it is necessary to resort to an approximation method such as numerical integration of the equations, the perturbation method, or something else. The situation is no different in quantum physics: only in exceptional cases do we know how to “solve the Schrödinger equation” exactly, that is, how to obtain the time evolution of the state vector |ϕ( t 〉 as a function of its value |ϕ( t 0 〉 at initial time t = t 0 .
The theory of Non-Relativistic Quantum Mechanics was created (or discovered) back in the 1920's mainly by Schrödinger and Heisenberg, but it is fair enough to say that a more modern and unified approach to the subject was introduced by Dirac and Jordan with their (intrinsic) Transformation Theory. In his famous text book on quantum mechanics [1], Dirac introduced his well-known bra and ket notation and a view that even Einstein (who was, as well known, very critical towards the general quantum physical world-view) considered the most elegant presentation of the theory at that time[2]. One characteristic of this formulation is that the observables of position and momentum are truly treated equally so that an intrinsic phase-space approach seems a natural course to be taken. In fact, we may distinguish at least two different quantum mechanical approaches to the structure of the quantum phase space: The Weyl-Wigner (WW) formalism and the advent of the theory of Coherent States (CS). The Weyl-Wigner formalism has had many applications ranging from the discussion of the Classical/Quantum Mechanical transition and quantum chaos to signal analysis[3,4]. The Coherent State formalism had a profound impact on Quantum Optics and during the course of time has found applications in diverse areas such as geometric quantization, wavelet and harmonic analysis [5]. In this chapter we present a compact review of these formalisms (with also a more intrinsic and coordinate independent notation) towards some non-standard and up-to-date applications such as modular variables and weak values.