After the birth of quantum mechanics, the notion in physics that the frequency of light is the only factor that determines the energy of a single photon has played a fundamental role. However, under the assumption tha...After the birth of quantum mechanics, the notion in physics that the frequency of light is the only factor that determines the energy of a single photon has played a fundamental role. However, under the assumption that the theory of Lewis-Riesenfeld invariants is applicable in quantum optics, it is shown in the present work that this widely accepted notion is valid only for light described by a time-independent Hamiltonian, i.e., for light in media satisfying the conditions, ε(t) = ε(0), u(t) = u(0), and σ(t) = 0 simultaneously. The use of the Lewis Riesenfeld invariant operator method in quantum optics leads to a marvelous result: the energy of a single photon propagating through time-varying linear media exhibits nontrivial time dependence without a change of frequency.展开更多
We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU...We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU(1,1) algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld's invariant operator theory of treating timedependent harmonic oscillators.展开更多
Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic me...Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.展开更多
We derive for crystal optics in coordinate-invariant way the cone approximation of refraction vectors in the neighborhood of optic axes and determine its invariants and eigenvectors. It proved to describe an elliptic ...We derive for crystal optics in coordinate-invariant way the cone approximation of refraction vectors in the neighborhood of optic axes and determine its invariants and eigenvectors. It proved to describe an elliptic cone. The second invariant of the operator of the wave equation with respect to similarity transformations determines the special cases of degeneration including the optic axes where the polarization of the waves due to self-intersection of the dispersion surface is not uniquely determined. This second invariant is included in all investigations and it is taken into account in the illustrations. It is biquadratic in the refraction vectors and the corresponding forth-order surface in three-dimensional space splits in two separate shells and a non-rational product decomposition describing this is found. We give also a more general classification of all possible solutions of an equation with an arbitrary three-dimensional operator.展开更多
We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is timedependent ...We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is timedependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.展开更多
For two kinds of the Moebius invariant subspace A_l^(a,2)(D) and A_l^(-a,2)(D) of L^(a,2)(D), we define big and small Hankel operators H_b^(ll') and h_b^(ll') for the analytic symbol function b(z), and st...For two kinds of the Moebius invariant subspace A_l^(a,2)(D) and A_l^(-a,2)(D) of L^(a,2)(D), we define big and small Hankel operators H_b^(ll') and h_b^(ll') for the analytic symbol function b(z), and study their boundedness, compactness and Schatten-von Neumanu classes S_p-estimates, and hence develope Schatten -von Neumann properties of these op- erators.展开更多
The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a lin...The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.展开更多
Using the method of Clerc and Stein study the multipliers of spherical Fourier transform on symmetric space to proof the multipliers theory for the space SL(3,H)/SP(3), completely avoid the complex theory of Anker, an...Using the method of Clerc and Stein study the multipliers of spherical Fourier transform on symmetric space to proof the multipliers theory for the space SL(3,H)/SP(3), completely avoid the complex theory of Anker, and we have gain the same result. Key words Riemannian symmetric space SL(3,H)/SP(3) - multipliers - spherical Fourier transform - invariant differential operator CLC number O 152.5 - O 186.12 Biography: LIAN Bao-sheng (1973-), male, Master, research direction: Li group and Lie algebra.展开更多
An invariant function (IF) is defined as a multiplier of a symmetry that means a symmetry multiplied by an IF is still a symmetry. Primary branch solutions of arbitrary first order scalar systems can be obtained by ...An invariant function (IF) is defined as a multiplier of a symmetry that means a symmetry multiplied by an IF is still a symmetry. Primary branch solutions of arbitrary first order scalar systems can be obtained by means of the IF and its related symmetry approach. Especially, one recursion operator and some sets of infinitely many high order symmetries are also explicitly given for arbitrary (l q-1)-dimensional first order autonomous systems. Because of the intrusion of the arbitrary function, various implicit special exact solutions can be found by fixing the arbitrary functions and selecting different seed solutions.展开更多
In this article, we extend the well known Wendel's theorem to the context of vector-valued L1-spaces on hypergroups. In this regard, various cases have been studied.
A generalized scheme for the construction of coherent states in the context of position-dependent effective mass systems has been presented. This formalism is based on the ladder operators and associated algebra of th...A generalized scheme for the construction of coherent states in the context of position-dependent effective mass systems has been presented. This formalism is based on the ladder operators and associated algebra of the system which are obtained using the concepts of supersymmetric quantum mechanics and the property of shape invariance. In order to exemplify the general results and to analyze the properties of the coherent states, several examples have been considered.展开更多
基金supported by National Research Foundation of Korea Grant funded by the Korean Government (No. 2009-0077951)
文摘After the birth of quantum mechanics, the notion in physics that the frequency of light is the only factor that determines the energy of a single photon has played a fundamental role. However, under the assumption that the theory of Lewis-Riesenfeld invariants is applicable in quantum optics, it is shown in the present work that this widely accepted notion is valid only for light described by a time-independent Hamiltonian, i.e., for light in media satisfying the conditions, ε(t) = ε(0), u(t) = u(0), and σ(t) = 0 simultaneously. The use of the Lewis Riesenfeld invariant operator method in quantum optics leads to a marvelous result: the energy of a single photon propagating through time-varying linear media exhibits nontrivial time dependence without a change of frequency.
基金The project supported by National Natural Science Foundation of China under Grant No. 10475056.
文摘We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU(1,1) algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld's invariant operator theory of treating timedependent harmonic oscillators.
文摘Two concepts of phenomenological optics of homogeneous, anisotropic and dispersive media are compared, the younger and more general concept of media with spatial dispersion and the older concept of (bi)-anisotropic media with material tensors for electric and magnetic induction which only depend on the frequency. The general algebraic form of the polarization vectors for the electric field and their one-dimensional projection operators is discussed without the degenerate cases of optic axis for which they become two-dimensional projection operators. Group velocity and diffraction coefficients in an approximate equation for the slowly varying amplitudes of beam solutions are calculated. As special case a polariton permittivity for isotropic media with frequency dispersion but without losses is discussed for the usual passive case and for the active case (occupation inversion of two energy levels that goes in direction of laser theory) and the group velocity is calculated. For this active case, regions of frequency and wave vector with group velocities greater than that of light in vacuum were found. This is not fully understood and due to large diffraction is likely only to realize in guided resonator form. The notion of “negative refraction” is shortly discussed but we did not find agreement with its assessment in the original paper.
文摘We derive for crystal optics in coordinate-invariant way the cone approximation of refraction vectors in the neighborhood of optic axes and determine its invariants and eigenvectors. It proved to describe an elliptic cone. The second invariant of the operator of the wave equation with respect to similarity transformations determines the special cases of degeneration including the optic axes where the polarization of the waves due to self-intersection of the dispersion surface is not uniquely determined. This second invariant is included in all investigations and it is taken into account in the illustrations. It is biquadratic in the refraction vectors and the corresponding forth-order surface in three-dimensional space splits in two separate shells and a non-rational product decomposition describing this is found. We give also a more general classification of all possible solutions of an equation with an arbitrary three-dimensional operator.
基金Supported by the Korea Science and Engineering Foundation (KOSEF) Grant Funded by the Korea Government (MOST) under Grant No.F01-2007-000-10075-0
文摘We study both classical and quantum relation between two Hamiltonian systems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other is timedependent Hamiltonian system. The quantum unitary operator relevant to classical canonical transformation between the two systems are obtained through rigorous evaluation. With the aid of the unitary operator, we have derived quantum states of the time-dependent Hamiltonian system through transforming the quantum states of the conservative system. The invariant operators of the two systems are presented and the relation between them are addressed. We showed that there exist numerous Hamiltonians, which gives the same classical equation of motion. Though it is impossible to distinguish the systems described by these Hamiltonians within the realm of classical mechanics, they can be distinguishable quantum mechanically.
文摘For two kinds of the Moebius invariant subspace A_l^(a,2)(D) and A_l^(-a,2)(D) of L^(a,2)(D), we define big and small Hankel operators H_b^(ll') and h_b^(ll') for the analytic symbol function b(z), and study their boundedness, compactness and Schatten-von Neumanu classes S_p-estimates, and hence develope Schatten -von Neumann properties of these op- erators.
文摘The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.
文摘Using the method of Clerc and Stein study the multipliers of spherical Fourier transform on symmetric space to proof the multipliers theory for the space SL(3,H)/SP(3), completely avoid the complex theory of Anker, and we have gain the same result. Key words Riemannian symmetric space SL(3,H)/SP(3) - multipliers - spherical Fourier transform - invariant differential operator CLC number O 152.5 - O 186.12 Biography: LIAN Bao-sheng (1973-), male, Master, research direction: Li group and Lie algebra.
基金Supported by the National Natural Science Foundations of China under Grant Nos.11435005,11471004,11175092,and 11205092Shanghai Knowledge Service Platform for Trustworthy Internet of Things No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘An invariant function (IF) is defined as a multiplier of a symmetry that means a symmetry multiplied by an IF is still a symmetry. Primary branch solutions of arbitrary first order scalar systems can be obtained by means of the IF and its related symmetry approach. Especially, one recursion operator and some sets of infinitely many high order symmetries are also explicitly given for arbitrary (l q-1)-dimensional first order autonomous systems. Because of the intrusion of the arbitrary function, various implicit special exact solutions can be found by fixing the arbitrary functions and selecting different seed solutions.
基金supported by senior research fellowship of CSIR,India
文摘In this article, we extend the well known Wendel's theorem to the context of vector-valued L1-spaces on hypergroups. In this regard, various cases have been studied.
文摘A generalized scheme for the construction of coherent states in the context of position-dependent effective mass systems has been presented. This formalism is based on the ladder operators and associated algebra of the system which are obtained using the concepts of supersymmetric quantum mechanics and the property of shape invariance. In order to exemplify the general results and to analyze the properties of the coherent states, several examples have been considered.