In this review, a general algorithm for constructing coherent states of dynamical groups for a given quantum physical system is presented. The result is that, for a given dynamical group, the coherent states are isomorphic to a coset space of group geometrical space. Thus the topological and algebraic structure of the coherent states as well as the associated dynamical system can be extensively discussed. In addition, a quantum-mechanical phase-space representation is constructed via the coherent-state theory. Several useful methods for employing the coherent states to study the physical phenomena of quantum-dynamic systems, such as the path integral, variational principle, classical limit, and thermodynamic limit of quantum mechanics, are described.

Coherent states: Theory and some Applications

The development of wave optics for light brought many new insights into our understanding of physics, driven by fundamental experiments like the ones by Young, Fizeau, Michelson-Morley and others. Quantum mechanics, and especially the de Broglie’s postulate relating the momentum p of a particle to the wave vector k of an matter wave: k = 2 λ = p/ℏ, suggested that wave optical experiments should be also possible with massive particles (see table 1), and over the last 40 years electron and neutron interferometers have demonstrated many fundamental aspects of quantum mechanics [1].

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Optics and interferometry with atoms and molecules

Coupled reaction channels calculations in nuclear physics

The complete active space (CAS) SCF method is presented in detail with special emphasis on computational aspects. The CASSCF wave function is formed from a complete distribution of a number of active electrons in a set of active orbitals, which in general constitute a subset of the total occupied space. In contrast to other MCSCF schemes, a CASSCF calculation involves no selection of individual configurations, and the wave function therefore typically consists of a large number of terms. The largest case treated here includes 10 416 spin and space adapted configurations. To be able to treat such large CI expansions, a density‐matrix oriented formalism is used. The Newton–Raphson scheme is applied to calculate the orbital rotations, and the secular problem is solved with recent developments of CI techniques. The applicability of the method is demonstrated in calculations on the HNO molecule in ground and excited states, using a triple‐zeta basis and different sizes of the active space. With a reasonable choice of active space, the calculations converge in 6–10 iterations. This is true also for states which are not the lowest state of the symmetry in question. The equilibrium geometry for the ground state is RNO=1.215(1.212) A, RNH =1.079(1.063) A, ϑHNO=108.8(108.6) °, the experimental values given in parenthesis for comparison. The best estimates for the transition energies to the lowest 3A″ and 1A″ states are 0.67(0.85) eV and 1.52(1.63) eV, respectively. The results obtained indicate that the choice of active space may be crucial for the convergence properties of CASSCF calculations.

The complete active space SCF (CASSCF) method in a Newton–Raphson formulation with application to the HNO molecule

Seventy five years ago, three remarkable papers by Schr¨ odinger, Kennard and Darwin were published. They were devoted to the evolution of Gaussian wave packets for an oscillator, a free particle and a particle moving in uniform constant electric and magnetic fields. From the contemporary point of view, these packets can be considered as prototypes of the coherent and squeezed states, which are, in a sense, the cornerstones of modern quantum optics. Moreover, these states are frequently used in many other areas, from solid state physics to cosmology. This paper gives a review of studies performed in the field of so-called ‘nonclassical states’ (squeezed states are their simplest representatives) over the past seventy five years, both in quantum optics and in other branches of quantum physics. My starting point is to elucidate who introduced different concepts, notions and terms, when, and what were the initial motivations of the authors. Many new references have been found which enlarge the ‘standard citation package’ used by some authors, recovering many undeservedly forgotten (or unnoticed) papers and names. Since it is practically impossible to cite several thousand publications, I have tried to include mainly references to papers introducing new types of quantum states and studying their properties, omitting many publications devoted to applications and to the methods of generation and experimental schemes, which can be found in other well known reviews. I also mainly concentrate on the initial period, which terminated approximately at the border between the end of the 1980s and the beginning of the 1990s, when several fundamental experiments on the generation of squeezed states were performed and the first conferences devoted to squeezed and ‘nonclassical’ states commenced. The 1990s are described in a more ‘squeezed’ manner: I have confined myself to references to papers where some new concepts have been introduced, and to the most recent reviews or papers with extensive bibliographical lists.

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'Nonclassical' states in quantum optics: a 'squeezed' review of the first 75 years

We show that the group of linear canonical transformations in a 2N‐dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well‐known dynamical group of the N‐dimensional harmonic oscillator. Finally, we study the case of n particles in a q‐dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)× O (q). An application to the calculation of matrix elements is given in a following paper.

Linear Canonical Transformations and Their Unitary Representations

Transformation brackets for harmonic oscillator functions

The one-body problem the two-body problem the three-body problem the four-body problem the n-body problem in the hartree-fock approximation the harmonic oscillator in scattering and reaction theory group theory of harmonic oscillators four dimensional harmonic oscillator and coulomb problem the five dimensional oscillator and nuclear collective motions the six dimensional oscillatory and the interacting oscillator the one body relativistic oscillator the two body relativistic oscillator the n-body relativistic oscillator.

The harmonic oscillator in modern physics

In a previous note a dynamical description of resonance scattering was given, and transient terms appeared in the wave function describing the process. To understand the physical significance of these terms, the transient effects that appear when a shutter is opened are discussed in this paper. For a nonrelativistic beam of particles, the transient current has a close mathematical resemblance with the intensity of light in the Fresnel diffraction by a straight edge. This is the reason for calling the transient phenomena by the name of diffraction in time.The shutter problem is discussed for particles whose wave functions satisfy the Schr\"odinger equation, the ordinary wave equation, and the Klein-Gordon equation. Only for the Schr\"odinger time-dependent equation do the transient wave functions resemble the solutions that appear in Sommerfeld's theory of diffraction. The connection of transient phenomena with the time-energy uncertainty relation, and the interpretation of the transient current in a scattering process, are briefly discussed. The relativistic wave functions for the shutter problem may play an important role in the dynamical description of a relativistic scattering process.

Diffraction in time

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Marcos Moshinsky

Papers

Linear Canonical Transformations and Their Unitary Representations

Transformation brackets for harmonic oscillator functions

The harmonic oscillator in modern physics

Diffraction in time

The harmonic oscillator in modern physics : from atoms to quarks