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.

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.

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.

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.

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.

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.

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.

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.