Momentum maps for mixed states in quantum and classical mechanics

Tronci, Cesare

doi:10.3934/jgm.2019032

Cited by 16 publications

(16 citation statements)

References 62 publications

(167 reference statements)

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“…where k w k = 1 and each Ψ k satisfies a separate (uncoupled) Schrödinger equation with Hamiltonian (1.1). We remark that (5.2) is the equivariant momentum map for unitary transformations of the {Ψ k } recently studied in [77]. Correspondingly, the overall dynamics of {Ψ k } is produced by the variational principle…”

Section: Factorization Of the Molecular Density Operatormentioning

confidence: 95%

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Geometry of Nonadiabatic Quantum Hydrodynamics

2019

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The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether's conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called 'collective'. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the = 0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called 'Bohmions', which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.

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Section: Factorization Of the Molecular Density Operatormentioning

confidence: 95%

“…with n w n = 1. For the momentum map aspects of quantum mixed states, see [65,77]. Equation and one can verify that the following Diff(R 3 )−action is Hamiltonian:…”

Section: Density Operators and Classical Closuresmentioning

confidence: 99%

Geometry of Nonadiabatic Quantum Hydrodynamics

2019

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“…We note in passing that the operators Z ± satisfy the commutation relations density [59]. More specifically, this quantity is a momentum map [45,46,60,61] for the group of strict contact transformations generated by the operator ih −1 L H [49], where…”

Section: ) See Appendix a For Further Explanations Therefore The Heisenberg Equation For L Hmentioning

confidence: 99%

Koopman wavefunctions and classical–quantum correlation dynamics

2019

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Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman–von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical–quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect—the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical–quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.

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“…Moreover, even the "ordinary value" of the dynamic phase is uniquely fixed according to Eq. (64). For the harmonic oscillator, the signs of the "ordinary values" are determined by tan( ωτ 2 ) according to Eq.…”

Section: Continuous Energy Spectrum Without Band Gapmentioning

confidence: 99%

“…One may be interested in the counterpart of the dynamic phase θ D for classical mixed states. However, the concept of the mixed state in classical mechanics is only sparsely explored in the literature [62][63][64], and it seems there is no broadly accepted definition. For example, the dynamic phase of a classical object following a closed curve in the parameter space was introduced in Ref.…”

Section: Implications For Classical Systemsmentioning

confidence: 99%

Dynamic process and Uhlmann process: Incompatibility and dynamic phase of mixed quantum states

Guo

Hou

et al. 2020

Phys. Rev. B

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While a pure quantum state may accumulate both the Berry phase and dynamic phase as it undergoes a cyclic path in the parameter space, the situation is more complicated when mixed quantum states are considered. From the Ulhmann bundle, a mixed quantum state can accumulate the Ulhmann phase if the parallel-transport condition is satisfied. However, we show that the Ulhmann process is in general not compatible with the evolution equation of the density matrix governed by the Hamiltonian. Thus, a mixed quantum state usually accumulates a dynamic phase during its time evolution. We present the expression of the dynamic phase for mixed quantum states. In examples of one-dimensional two-band models and simple harmonic oscillator, the dynamic phase can take multiple discrete values in quasi-static processes at infinitely high temperature due to the resonant points. However, the behavior differs if the energy spectrum is continuous without a band gap. Moreover, there is no natural analog of the dynamic phase in classical systems.

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Momentum maps for mixed states in quantum and classical mechanics

Cited by 16 publications

References 62 publications

Geometry of Nonadiabatic Quantum Hydrodynamics

Geometry of Nonadiabatic Quantum Hydrodynamics

Koopman wavefunctions and classical–quantum correlation dynamics

Dynamic process and Uhlmann process: Incompatibility and dynamic phase of mixed quantum states

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