Ladder operator

About: Ladder operator is a research topic. Over the lifetime, 3019 publications have been published within this topic receiving 63439 citations.

Spin‐s Spherical Harmonics and ð

Abstract: Recent work on the Bondi‐Metzner‐Sachs group introduced a class of functions sYlm (θ, φ) defined on the sphere and a related differential operator ð. In this paper the sYlm are related to the representation matrices of the rotation group R 3 and the properties of ð are derived from its relationship to an angular‐momentum raising operator. The relationship of the sTlm (θ, φ) to the spherical harmonics of R 4 is also indicated. Finally using the relationship of the Lorentz group to the conformal group of the sphere, the behavior of the sTlm under this latter group is shown to realize a representation of the Lorentz group.

Phase properties of the quantized single-mode electromagnetic field.

Abstract: The usual mathematical model of the single-mode electromagnetic field is the harmonic oscillator with an infinite-dimensional state space, which unfortunately cannot accommodate the existence of a Hermitian phase operator. Recently we indicated that this difficulty may be circumvented by using an alternative, and physically indistinguishable, mathematical model of the single-mode field involving a finite but arbitrarily large state space, the dimension of which is allowed to tend to infinity after physically measurable results, such as expectation values, are calculated. In this paper we investigate the properties of a Hermitian phase operator which follows directly and uniquely from the form of the phase states in this space and find them to be well behaved. The phase-number commutator is not subject to the difficulties inherent in Dirac's original commutator, but still preserves the commutator--Poisson-bracket correspondence for physical field states. In the quantum regime of small field strengths, the phase operator predicts phase properties substantially different from those obtained using the conventional Susskind-Glogower operators. In particular, our results are consistent with the vacuum being a state of random phase and the phases of two vacuum fields being uncorrelated. For higher-intensity fields, the quantum phase properties agree with those previously obtained by phenomenological and semiclassical approaches, where such approximations are valid. We illustrate the properties of the phase with a discussion of partial phase states. The Hermitian phase operator also allows us to construct a unitary number-shift operator and phase-moment generating functions. We conclude that the alternative mathematical description of the single-mode field presented here provides a valid, and potentially useful, quantum-mechanical approach for calculating the phase properties of the electromagnetic field.

A non-Hermitian Hamilton operator and the physics of open quantum systems

Abstract: The Hamiltonian Heff of an open quantum system consists formally of a first-order interaction term describing the closed (isolated) system with discrete states and a second-order term caused by the interaction of the discrete states via the common continuum of scattering states. Under certain conditions, the last term may be dominant. Due to this term, Heff is non-Hermitian. Using the Feshbach projection operator formalism, the solution ΨEc of the Schrodinger equation in the whole function space (with discrete as well as scattering states, and the Hermitian Hamilton operator H) can be represented in the interior of the localized part of the system in the set of eigenfunctions λ of Heff. Hence, the characteristics of the eigenvalues and eigenfunctions of the non-Hermitian operator Heff are contained in observable quantities. Controlling the characteristics by means of external parameters, quantum systems can be manipulated. This holds, in particular, for small quantum systems coupled to a small number of channels. The paper consists of three parts. In the first part, the eigenvalues and eigenfunctions of non-Hermitian operators are considered. Most important are the true and avoided crossings of the eigenvalue trajectories. In approaching them, the phases of the λ lose their rigidity and the values of observables may be enhanced. Here the second-order term of Heff determines decisively the dynamics of the system. The time evolution operator is related to the non-Hermiticity of Heff. In the second part of the paper, the solution ΨEc and the S matrix are derived by using the Feshbach projection operator formalism. The regime of overlapping resonances is characterized by non-rigid phases of the ΨEc (expressed quantitatively by the phase rigidity ρ). They determine the internal impurity of an open quantum system. Here, level repulsion passes into width bifurcation (resonance trapping): a dynamical phase transition takes place which is caused by the feedback between environment and system. In the third part, the internal impurity of open quantum systems is considered by means of concrete examples. Bound states in the continuum appearing at certain parameter values can be used in order to stabilize open quantum systems. Of special interest are the consequences of the non-rigidity of the phases of λ not only for the problem of dephasing, but also for the dynamical phase transitions and questions related to them such as phase lapses and enhancement of observables.

Quasi-Exactly Solvable Models in Quantum Mechanics

Abstract: QUASI-EXACT SOLVABILITY-WHAT DOES THAT MEAN? Introduction Completely algebraizable spectral problems The quartic oscillator The sextic oscillator Non-perturbative effects in an explicit form and convergent perturbation theory Partial algebraization of the spectral problem The two-dimensional harmonic oscillator Completely integrable quantum systems Deformation of completely integrable models Quasi-exact solvability and the Gaudin model The classical multi-particle Coulomb problem Classical formulation of quantal problems The Infeld-Hull factorization method and quasi-exact solvability The Gelfand-Levitan equation Summary Historical comments SIMPLEST ANALYTIC METHODS FOR CONSTRUCTING QUASI-EXACTLY SOLVABLE MODELS The Lanczos tridiagonalization procedure The sextic oscillator with a centrifugal barrier The electrostatic analogue-the quartic oscillator Higher oscillators with centrifugal barriers The electrostatic analogue-the general case The inverse method of separation of variables The Schrodinger equations with separable variables Multi-dimensional models The "field-theoretical" case Other quasi-exactly solvable models THE INVERSE METHOD OF SEPARATION OF VARIABLES Multi-parameter spectral equations The method-general formulation The case of differential equations Algebraically solvable multi-parameter spectral equations An analytic method Reduction to exactly solvable models The one-dimensional case-classification Elementary exactly solvable models The multi-dimensional case-classification CLASSIFICATION OF QUASI-EXACTLY SOLVABLE MODELS WITH SEPARATE VARIABLE Preliminary comments The one-dimensional non-degenerate case The non-degenerate case-the first type The non-degenerate case-the second type The non-degenerate case-the third type The one-dimensional simplest degenerate case The simplest degenerate case-the first type The simplest degenerate case-the second type The simplest degenerate case-the third type The one-dimensional twice-degenerate case The twice-degenerate-the first type The twice-degenerate case-the second type The one-dimensional most degenerate case The multi-dimensional case COMPLETELY INTEGRABLE GAUDIN MODELS AND QUASI-EXACT SOLVABILITY Hidden symmetries Partial separation of variables Some properties of simple Lie algebras Special decomposition in simple Lie algebras The generalized Gaudin model and its solutions Quasi-exactly solvable equations Reduction to the Schrodinger form Conclusions Appendices A: The Inverse Schrodinger Problem and Its Solution for Several Given States Appendices B: The Generalized Quantum Tops and Exact Solvability Appendices C: The Method of Raising and Lowering Operators Appendices D: Lie Algebraic Hamiltonians and Quasi-Exact Solvability References Index