scholarly journals A Geometric Approach to Time Evolution Operators of Lie Quantum Systems

2008 ◽  
Vol 48 (5) ◽  
pp. 1379-1404 ◽  
Author(s):  
José F. Cariñena ◽  
Javier de Lucas ◽  
Arturo Ramos
Keyword(s):  
Time Evolution ◽  
Geometric Approach ◽  
Quantum Systems ◽  
Evolution Operators

scholarly journals Geometry of variational methods: dynamics of closed quantum systems

2020 ◽  
Vol 9 (4) ◽  
Author(s):  
Lucas Hackl ◽  
Tommaso Guaita ◽  
Tao Shi ◽  
Jutho Haegeman ◽  
Eugene Demler ◽  
...  
Keyword(s):  
Variational Methods ◽  
Time Evolution ◽  
Coherent States ◽  
Geometric Approach ◽  
Quantum Systems ◽  
Imaginary Time ◽  
Excitation Spectra ◽  
Spectral Functions ◽  
Geometric Framework ◽  
Gaussian States

We present a systematic geometric framework to study closed quantum systems based on suitably chosen variational families. For the purpose of (A) real time evolution, (B) excitation spectra, (C) spectral functions and (D) imaginary time evolution, we show how the geometric approach highlights the necessity to distinguish between two classes of manifolds: Kähler and non-Kähler. Traditional variational methods typically require the variational family to be a Kähler manifold, where multiplication by the imaginary unit preserves the tangent spaces. This covers the vast majority of cases studied in the literature. However, recently proposed classes of generalized Gaussian states make it necessary to also include the non-Kähler case, which has already been encountered occasionally. We illustrate our approach in detail with a range of concrete examples where the geometric structures of the considered manifolds are particularly relevant. These go from Gaussian states and group theoretic coherent states to generalized Gaussian states.

Interacting Quantum Systems

2014 ◽  
pp. 1-37
Keyword(s):  
Density Matrix ◽  
Integral Equations ◽  
Time Evolution ◽  
Path Integral ◽  
Quantum Systems ◽  
Transition Rates ◽  
Feynman Path Integral ◽  
Matrix Representations ◽  
Evolution Operators ◽  
Feynman Path

In this chapter, basic quantum tools such as time-evolution operators, transition rates and amplitudes, statistical and projector operators, and interaction and density matrix representations are employed to characterize the open and interacting quantum systems with the aid of Schrödinger, quantum master, Fokker-Planck, and Feynman path integral equations and formulations.

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