Weyl transformation in Fermi systems

In the path integral representation, the Hamiltonian in a quantum system is associated with the Hamiltonian in a classical system through the Weyl transformation. From this, it is possible to describe the time evolution in a quantum system by the Hamiltonian in a classical system. In a Bose system, the Weyl transformation is defined by the eigenstates of the canonical operators, since the Hamiltonian is given by a function of the canonical operators. On the other hand, in a Fermi system, the Hamiltonian is usually described by a function of the creation and annihilation operators, and hence the Weyl transformation is defined by the coherent states which are the eigenstate of an annihilation operator. Here, we formulate the Weyl transformation in Fermi systems in terms of the eigenstates of the canonical operators so as to clarify the correspondence between both systems. Using this, we can derive the path integral representation in Fermi systems.     

Quantum Systems Connected by a Time-Dependent Canonical Transformation

We study both classical and quantum relation between two Hamiltoniansystems which are mutually connected by time-dependent canonical transformation. One is ordinary conservative system and the other istime-dependent Hamiltonian system. The quantum unitary operatorrelevant 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.     

Exact Invariants for a Time-Dependent Hamiltonian System

We have a classical look for a quantum system which is exactly solvable. We construct the invariant manifolds analytically, and then apply the semiclassical quantization rules in a final step to compute the quasienergies. The invariant is obtained by performing a canonical transformation of the initially time-dependent Hamiltonian to a time-independent one. The correspondence between classical and quantum mechanics is elucidated.     

LO-phonon overheating in quantum dots

Longitudinal optical phonons have been used to interpret the electronic energy relaxation in quantum dots and at the same time they served as a reservoir, with which the electronic subsystem is in contact. Such a phonon subsystem is expected to be passive, namely, in a long-time limit the whole system should be able to achieve such a stationary state, in which statistical distributions of both subsystems do not change in time. We pay attention to this property of the LO phonon bath. We show the passivity property of the so far used approximations to electronic transport in quantum dots. Also we show a way how to improve the passivity of LO phonon bath using canonical Lang-Firsov transformation. Presented at the X-th Symposium on Suface Physics, Prague, Czech Republic, July 11–15, 2005.     

A physical perspective on classical cloning

The celebrated quantum no-cloning theorem states that an arbitrary quantum state cannot be cloned perfectly. This raises questions about cloning of classical states, which have also attracted attention. Here, we present a physical approach to the classical cloning process showing how cloning can be realised using Hamiltonians. After writing down a canonical transformation that clones classical states, we show how this can be implemented by Hamiltonian evolution. We then propose an experiment using the tools of nonlinear optics to realise the ideas presented here. Finally, to understand the cloning process in a more realistic context, we introduce statistical mechanical noise to the system and study how this affects the cloning process. While most of our work deals with linear systems and harmonic oscillators, we give some examples of cloning maps on manifolds and show that any system whose configuration space is a group manifold admits a cloning canonical transformation.     

Orbits of hybrid systems as qualitative indicators of quantum dynamics

Hamiltonian theory of hybrid quantum–classical systems is used to study dynamics of the classical subsystem coupled to different types of quantum systems. It is shown that the qualitative properties of orbits of the classical subsystem clearly indicate if the quantum subsystem does or does not have additional conserved observables.     

Measure synchronization in hybrid quantum-classical systems

Measure synchronization in hybrid quantum-classical systems is investigated in this paper. The dynamics of the classical subsystem is described by the Hamiltonian equations, while the dynamics of the quantum subsystem is governed by the Schrödinger equation. By increasing the coupling strength in between the quantum and classical subsystems, we reveal the existence of measure synchronization in coupled quantum-classical dynamics under energy conservation for the hybrid systems.     

Effects of barrier fluctuation on the tunneling dynamics in the presence of classical chaos in a mixed quantum-classical system

We present a numerical investigation of the tunneling dynamics of a particle moving in a bistable potential with fluctuating barrier which is coupled to a non-integrable classical system and study the interplay between classical chaos and barrier fluctuation in the tunneling dynamics. We found that the coupling of the quantum system with the classical subsystem decreases the tunneling rate irrespective of whether the classical subsystem is regular or chaotic and also irrespective of the fact that whether the barrier fluctuates or not. Presence of classical chaos always enhances the tunneling rate constant. The effect of barrier fluctuation on the tunneling rate in a mixed quantum-classical system is to suppress the tunneling rate. In contrast to the case of regular subsystem, the suppression arising due to barrier fluctuation is more visible when the subsystem is chaotic.      

Squeezing and Quantum Canonical Transformations

In this paper we present an approach to quantum mechanical canonical transformations. Our main result is that time-dependent quantum canonical transformations can always be cast in the form of squeezing operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the time-dependent quantum harmonic oscillator. The issue of the systematic construction of quantum canonical transformations is also discussed along the lines of Dirac, Wigner, and Schwinger ideas and to the more recent work by Lee. The main conclusion is that the classical phase space transformation can be maintained in the operator formalism but the construction of the quantum canonical transformation is not clearly related to the classical generating function of a classical canonical transformation. We hereby propose the much more efficient method given by the squeezing operators. This method has also been proved to be very useful, by one of the authors, in the framework of the dynamical symmetries (Cerveró, J. M. (1999). International Journal of Theoretical Physics 38, 2095–2109).     

Quantum Dynamics of Systems Connected by a Canonical Transformation

Quantum Hamiltonian systems corresponding to classical systems related by a general canonical transformation are considered. The differential equation to find the unitary operator, which corresponds to the canonical transformation and connects quantum states of the original and transformed systems, is obtained. The propagator associated with their wave functions is found by the unitary operator. Quantum systems related by a linear canonical point transformation are analyzed. The results are tested by finding the wave functions of the under-, critical-, and over-damped harmonic oscillator from the wave functions of the harmonic oscillator, free-particle system, and negative harmonic potential system, using the unitary operator to connect them, respectively.     

Analytical Bethe Ansatz,Canonical Bäcklund Transformation and Q-Operator For A New Discrete Integrable Heirarchy

A new discrete heiarchy of integrable equations is generated from a new Lax Operator and a canonical Bäcklund transformation of the system is derived using Sklyanin’s formalism, based on the classical r-matrix. By quantising the system a quantum analogue of the corresponding canonical Bäcklund transformation is obtained and certain properties of the associated Q-operator are examined. Finally the analytical Bethe Ansatz is used to solve for the spectrum.     

Quantum Noether identities for non-local transformationsin higher-order derivatives theories

Based on the phase-space generating functional of the Green function for a system with a regular/singular higher-order Lagrangian, the quantum canonical Noether identities (NIs) under a local and non-local transformation in phase space have been deduced, respectively. For a singular higher-order Lagrangian, one must use an effective canonical action I_eff^P in quantum canonical NIs instead of the classical I^P in classical canonical NIs. The quantum NIs under a local and non-local transformation in configuration space for a gauge-invariant system with a higher-order Lagrangian have also been derived. The above results hold true whether or not the Jacobian of the transformation is equal to unity or not. It has been pointed out that in certain cases the quantum NIs may be converted to conservation laws at the quantum level. This algorithm to derive the quantum conservation laws is significantly different from the quantum first Noether theorem. The applications of our formulation to the Yang-Mills fields and non-Abelian Chern-Simons (CS) theories with higher-order derivatives are given, and the conserved quantities at the quantum level for local and non-local transformations are found, respectively.Received: 12 February 2002, Revised: 16 June 2003, Published online: 25 August 2003Z.-P. Li: Corresponding authorAddress for correspondence: Department of Applied Physics, Beijing Polytechnic University, Beijing 100022, P.R. China     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

Canonical transformations in quantum mechanics

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.     

Removability of the topological term in theO(3) nonlinear δ-model

It has been shown that the topological term in theO(3) nonlinear δ-model can be removed by a suitable canonical transformation using classical theory. In this paper, the quantum unitary transformation corresponding to the classical canonical transformation is presented. The meaning of the unitary transformation and the removability of the topological term are then discussed.     

A One-Dimensional Model of Hydrogen Molecular Ion

We present a study on a one-dimensional hydrogen molecular ion under the Born-Oppenheimer approximation. A canonical transformation produces the classical system directlyto be a pendulum. The quantum Schrodinger equation is solved analytically and theelectronic energy curves show that the bound states of this 1D model differ from the 2D and 3DH₂⁺. The vibration spectroscopy is also obtained by employing the Morse's eigen wavefunctionsas basis vectors to diagonalize the Hamiltonian for R. The semiclassical quantization yieldselectronic energies in agreement with the quantum ones reasonably.     

Nonlinear supersymmetry,quantum anomaly and quasi-exactly solvable systems

The nonlinear supersymmetry of one-dimensional systems is investigated in the context of the quantum anomaly problem. Any classical supersymmetric system characterized by the nonlinear in the Hamiltonian superalgebra is symplectomorphic to a supersymmetric canonical system with the holomorphic form of the supercharges. Depending on the behaviour of the superpotential, the canonical supersymmetric systems are separated into the three classes. In one of them the parameter specifying the supersymmetry order is subject to some sort of classical quantization, whereas the supersymmetry of another extreme class has a rather fictive nature since its fermion degrees of freedom are decoupled completely by a canonical transformation. The nonlinear supersymmetry with polynomial in momentum supercharges is analysed, and the most general one-parametric Calogero-like solution with the second order supercharges is found. Quantization of the systems of the canonical form reveals the two anomaly-free classes, one of which gives rise naturally to the quasi-exactly solvable systems. The quantum anomaly problem for the Calogero-like models is “cured” by the specific superpotential-dependent term of order ℏ². The nonlinear supersymmetry admits the generalization to the case of two-dimensional systems.     

介观耗散电容耦合电路量子化中的正则变换

针对介观耗散电容耦合RLC电路提出一种一般的正则变换, 并证明了其正确性和合理性. 用这种正则变换研究了双回路介观耗散电容耦合电路的量子化问题, 得出的对角化哈密顿量比文献中多出一非线性项. 这种具有普遍性的一般正则变换可能对研究介观多回路耗散系统的量子涨落、量子噪声等性质具有重要的意义. 关键词： RLC电路')" href="#">RLC电路 量子化 正则变换