Coherent states coupling

Also, rotational state resolution of cross-sections can be obtained by employing a coherent state analysis [51] for the situation of weak coupling between rotational and vibrational degrees of freedom. A suitable rotational coherent state can be expressed as... 

It is known that if the system described by the Hamiltonian in Eq. (146) has been in a coherent state a(0) = a and /3m(0) = pm at the initial moment of time, then its state will also be a coherent one at subsequent moments of time, and the dependence of the eigenvalues on time will be determined by the coupled equations... 

Semiclassics and propagation of coherent states for spin-orbit coupling problems... 

Abstract. The Dirac equation is discussed in a semiclassical context, with an emphasis on the separation of particles and anti-particles. Classical spin-orbit dynamics are obtained as the leading contribution to a semiclassical approximation of the quantum dynamics. In a second part the propagation of coherent states in general spin-orbit coupling problems is studied in two different semiclassical scenarios. 

Keywords Dirac equation, semiclassics, spin-orbit coupling, coherent states... 

In a second part we study the propagation of coherent states in general spin-orbit coupling problems with semiclassical means. This is done in two semiclassical scenarios h 0 with either spin quantum number s fixed (as above), or such that hs = S is fixed. In both cases, first approximate Hamiltonians are introduced that propagate coherent states exactly. The full Hamiltonians are then treated as perturbations of the approximate ones. The full quantum dynamics is seen to follow appropriate classical spin-orbit trajectories, with a semiclassical error of size yfh. As opposed to the first case,... 

Semiclassical studies of the propagation of coherent states have proven useful in many circumstances see, e.g., (Klauder and Skagerstam, 1982 Perelomov, 1986). Here we consider spin-orbit coupling problems that result from the Dirac equation either in a semiclassical or in a non-relativistic approximation (see, e.g., the Hamiltonians (30) and (31)). The Hamiltonians H that arise in such a context can be viewed as Weyl quantisations of symbols... 

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

Another possibility to introduce a semiclasscial initial value representation for the spin-coherent state propagator is to exploit the close relation between Schwinger s representation of a spin system and the spin-coherent state theory [100, 133-135]. To illustrate this approach, we consider an electronic two-level system coupled to Vvib nuclear DoF. Within the mapping approach the semiclassical propagator for this system is given by... 

The scheme we employ uses a Cartesian laboratory system of coordinates which avoids the spurious small kinetic and Coriolis energy terms that arise when center of mass coordinates are used. However, the overall translational and rotational degrees of freedom are still present. The unconstrained coupled dynamics of all participating electrons and atomic nuclei is considered explicitly. The particles move under the influence of the instantaneous forces derived from the Coulombic potentials of the system Hamiltonian and the time-dependent system wave function. The time-dependent variational principle is used to derive the dynamical equations for a given form of time-dependent system wave function. The choice of wave function ansatz and of sets of atomic basis functions are the limiting approximations of the method. Wave function parameters, such as molecular orbital coefficients, z,(f), average nuclear positions and momenta, and Pfe(0, etc., carry the time dependence and serve as the dynamical variables of the method. Therefore, the parameterization of the system wave function is important, and we have found that wave functions expressed as generalized coherent states are particularly useful. A minimal implementation of the method [16,17] employs a wave function of the form ... 

We have applied the above approach to a harmonic oscillator coupled to a spin by means of a photon number - nondemolition Hamiltonian. The spin is being measured periodically, whereas the measurement outcome is ignored. For a sufficiently high measurement frequency, the state of the harmonic oscillator evolves in a unitary manner which can be influenced by a choice of the meter basis. In practice however, the time interval At between two subsequent measurements always remains finite and, therefore, the system evolution is subject to decoherence. As an example of application, we have simulated the evolution of an initially coherent state of the harmonic oscillator into a Schrodinger cat-like superposition state. The state departs from the superposition as time increases. The simulations confirm that the decoherence rate increases dramatically with the amplitude of the initial coherent state, thus destroying very rapidly all macroscopic superposition states. 

The END theory was proposed in 1988 [11] as a general approach to deal with time-dependent non-adiabatic processes in quantum chemistry. We have applied the END method to the study of time-dependent processes in energy loss [12-16]. The END method takes advantage of a coherent state representation of the molecular wave function. A quantum mechanical Lagrangian formulation is employed to approximate the Schrodinger equation, via the time-dependent variational principle, by a set of coupled first-order differential equations in time to describe the END. 

We consider a system of two nonidentical and nonoverlapping atoms at positions ri and rj, coupled to the quantized three-dimensional electromagnetic held. The initial state of the held is the product of a single-mode coherent state of a driving laser held, and the vacuum state of the rest of the modes. Each atom is assumed to have only two levels the ground level g,) and the excited level e,)(i = 1,2), separated by an energy hot), =Ee. — Egi, and connected by an... 

Amirav and Jortner29 did make an attempt to redefine N, the number of states coupled, to JVeff, the dilution factor, which can be used when Lahmani et al. s5 conditions are not fulfilled. It should then also be equal to A+/A, but their treatment takes only partially into account the effect of the coherence width of the laser. This Neff arose from their consideration of the quantum yield and its dependence on J. In Section V we will discuss the quantum yield of the J = 0, K = 0 state, while in Section VI we consider its J dependence. 

In Ch. 21 Buntkowsky and Limbach review recent NMR work on the dynamics of dihydrogen and dideuterium in the coordination sphere of transition metals. In addition to inelastic neutron scattering and liquid state NMR, the effects of coherent (exchange couplings) and incoherent rotational tunneling of D2 pairs in transi-... 

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