Noncommuting terms

Relationship between Noncommuting Terms in H and the Most Appropriate Hund s Case... 

Actually, for most of the well-known JT problems, the JT Hamiltonian includes noncommutative matrices. To my knowledge, there are just three lucky exceptions. One, the so-called E (g> big case, was considered above. Another one is the E (g) b2g case for a tetragonal E term linearly coupled to b2g mode of vibrations. The respective Hamiltonian is [cf. (1)] ... 

Care must be exercised to distinguish the concept of adiabatic Floquet dynamics introduced here, which refers to an adiabatic time-evolution, or to the slow variations of the Floquet basis with time, from the concept of adiabatic representation defined in the previous section, which refers to the slow variations of the electronic Hamiltonian (Floquet or not) with respect to nuclear motions (i.e., the noncommutativity of the electronic Hamiltonian Hei and the nuclear KE operator Tjv). Where confusion is possible and to be avoided, we shall refer to this concept of adiabaticity related to the BO approximation as the R-adiabaticity, while adiabaticity in actual time evolution will be termed t-adiabaticity. Non-adiabatic effects in time evolution are due to a fast variation of the (Floquet) Hamiltonian with time, causing Floquet states to change rapidly in time, to the extent that in going from one time slice to another, a resonance may be projected onto many new resonances as well as diffusion (continuum) states [40], and the Floquet analysis breaks down completely. We will see in Section 5 how one can take advantage of such effects to image nuclear motions by an ultrafast pump-probe process. 

A very remote analogy is conservation laws in classical and quantum physics. Description of quantum mechanics in terms of classical mechanics is not well defined, which happens because of commutativity of classical values and noncommutativity of their quantum analogs. We should regularize it and as a result part of classical symmetries may be realized in such a way that some conservation laws cannot be measured at all (e.g., conservation of the angular momentum as a vector). That example turns our attention to problem of observations. 

We apply the split-operator method Eq. (3.7) to separately handle the potential and kinetic energy terms of the Hamiltonian. We further apply the split-operator scheme to separate the kinetic energy term into the two exponentially noncommutative parts Tr + Tr and Tg to obtain a numerical short-time propagation method. 

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