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The Connection Between Density Matrix Method, Supersymmetric Quantum Mechanics and Lewis-Riesenfeld Invariant Theory  [PDF]
Jian Qi Shen
Physics , 2003,
Abstract: This paper is concerned with the connection between density matrix method, supersymmetric quantum mechanics and Lewis-Riesenfeld invariant theory. It is shown that these three formulations share the common mathematical structure: specifically, all of them have the invariant operators which satisfies the Liouville-Von Neumann equation and the solutions to the time-dependent Schr\"{o}dinger equation and/or Schr\"{o}dinger eigenvalue equation can be constructed in terms of the eigenstates of the invariants.
Lewis-Riesenfeld approach to the solutions of Schrodinger equation in the presence of the presence of a time-dependent linear potential  [PDF]
Pi-Gang Luan,Chi-Shung Tang
Physics , 2003, DOI: 10.1103/PhysRevA.71.014101
Abstract: We reexamine the general solution of a Schr\"{o}dinger equation in the presence of a time-dependent linear potential in configuration space based on the Lewis-Riesenfeld framework. For comparison, we also solve the problem in momentum space and then Fourier transform the solution to get the general wave function. Appropriately choosing the weight function in the latter method, we can obtain the same wave function as the former method. It is found that non-Hermitian time-dependent linear invariant can be used to obtain Gaussian-type wave-packet (GTWP) solutions of the time-dependent system. This operator is a specific linear combination of the initial momentum and initial position operators. This fact indicates that the constants of integration such as the initial position and initial momentum that determine the classical motion play important roles in the time-dependent quantum system. The eigenfunction of the linear invariant is interpreted as a wave packet with a "center of mass" moving along the classical trajectory, while the ratio between the coefficients of the initial position and initial momentum determines the width of the wave packet.
Invariant operator theory for the single-photon energy intime-varying media

Choi Jeong-Ryeol,

中国物理 B , 2010,
Abstract: 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, $\epsilon(t)=\epsilon(0)$, $\mu(t)=\mu(0)$, and $\sigma(t)=0$ 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.
LEWIS-RIESENFELD PHASES AND BERRY PHASES IN THEQUANTUM SYSTEM OF TIME-DEPENDENT HARMONICOSCILLATOR WITH A MOVING BOUNDARY
动边界量子含时谐振子系统的Lewis-Riesenfeld相位与Berry相位

LI LING,LI BO-ZANG,LIANG JIU-QING,
李玲
,李伯臧,梁九卿

物理学报 , 2001,
Abstract: 根据Lewis-Riesenfeld的量子不变量理论,计算了一维动壁无限深势阱内频率随时间变化的谐振子的Lewis-Riesenfeld相位,发现刘登云文中“非绝热Berry相位”与Lewis-Riesenfeld相位中的几何部分完全一致.也许更为重要的是,证明了至少对于做正弦振动的边界,在绝热近似下,该系统不存在非零的Berry相位.
Remarks on “Lewis- Riesenfeld phase” and quantum geometric phase
关于Lewis-Riesenfeld相位和量子几何相位

Li Hua-Zhong,
李华钟

物理学报 , 2004,
Abstract: 评注了《大学物理》21 23一文关于量子几何相位与Lewis相位论述与《物理学报》48 2018一文的结论有极大不同.指出前者对Lewis导出的相位与量子几何相位关系的错误陈述,而后者的结论是正确的.
NOTES ON THE QUANTUM INVARIANT AND ON THE-RELATION BETWEEN IT AND QUANTUM PHASE
关于量子不变量及其与量子相位关系的几点注记

LI BO-ZANG,ZHANG LING-YUN,ZHANG XIANG-DONG,
李伯臧
,张凌云,张向东

物理学报 , 1997,
Abstract: Based on the construction theorem of quantum invariant,the results of Lewis and Riesenfeld are rededuced in a much simpler way and extended significantly.Meanwhile,it is pointed out that the Lewis Riesenfeld phases are in general not of physical meaning unless the invariants take specific forms which have been worked out.As an example,the spin system in a magnetic field is discussed in detail.
Time-dependent massless Dirac fermions in graphene  [PDF]
Boubakeur Khantoul,Andreas Fring
Physics , 2015, DOI: 10.1016/j.physleta.2015.08.011
Abstract: Using the Lewis-Riesenfeld method of invariants we construct explicit analytical solutions for the massless Dirac equation in 2+1 dimensions describing quasi-particles in graphene. The Hamiltonian of the system considered contains some explicit time-dependence in addition to one resulting from being minimally coupled to a time-dependent magnetic field. The eigenvalue equations for the two spinor components of the Lewis-Riesenfeld invariant are found to decouple into a pair of supersymmetric invariants in a similar fashion as the known decoupling for the time-independent Dirac Hamiltonians.
Dalgarno-Lewis Method Revisited  [PDF]
A. B. Balantekin,A. Malkus
Physics , 2010,
Abstract: Proving the existence of an operator that connects non-perturbed states to perturbed states, an alternative derivation of the Dalgarno-Lewis method is given. To illustrate that the Dalgarno-Lewis method is an apt tool for algebraic Hamiltonians, the method is applied to one class of such systems, namely deep three-dimensional potentials with positive parity.
Exact expression for decoherence factor in the time-dependent generalized Cini model  [PDF]
Jian Qi Shen,San Shui Xiao,Qiang Wu
Physics , 2003,
Abstract: The present letter finds the complete set of exact solutions of the time-dependent generalized Cini model by making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation and, based on this, the general explicit expression for the decoherence factor is therefore obtained. This study provides us with a useful method to consider the geometric phase and topological properties in the time-dependent quantum decoherence process.
Exact solution for the thermo Jaynes--Cummings model

Yuan Hong-Chun,Fan Hong-Yi,

中国物理 B , 2009,
Abstract: Based on the construction of supersymmetric generators, we use the Lewis--Riesenfeld invariant method to deduce the exact and explicit eigen-energy spectrum with the time-dependent thermo Jaynes--Cummings model. One of the advantages of this approach is that it can transform the hidden form, related to the chronological product, of the time evolution operator into an explicit expression. Moreover, the dynamical and statistics properties of physical quantities are obtained for the given initial states in the thermo Jaynes--Cummings system.
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