Quantum Hamiltonian

To begin with, we compare the stepsizes used in the simulations (Fig. 3). As pointed out before, it seems to be unreasonable to equip the Pickaback scheme with a stepsize control, because, as we indeed observe in Fig. 3, the stepsize never increases above a given level. This level depends solely on the eigenvalues of the quantum Hamiltonian. When analyzing the other integrators, we observe that the stepsize control just adapts to the dynamical behavior of the classical subsystem. The internal (quantal) dynamics of the Hydrogen-Chlorine subsystem does not lead to stepsize reductions. 

Eor ease of presentation only, we here consider the case of two particles having spatial coordinates x and y, and masses m and M, with m interaction potential V x, y), the quantum Hamiltonian H is given by... 

In cases where the elassieal energy, and henee the quantum Hamiltonian, do not eontain terms that are explieitly time dependent (e.g., interaetions with time varying external eleetrie or magnetie fields would add to the above elassieal energy expression time dependent terms diseussed later in this text), the separations of variables teehniques ean be used to reduee the Sehrodinger equation to a time-independent equation. 

The location of the quantum/classical boundary across a covalent bond also has implications for the energy terms evaluated in the Emm term. Classical energy terms that involve only quantum atoms are not evaluated. These are accounted for by the quantum Hamiltonian. Classical energy terms that include at least one classical atom are evaluated. Referring to Figure 2, the Ca—Cp bond term the N — Ca—Cp, C — Ca—Cp, Ha— Ca—Cp, Ca—Cp — Hpi, and Ca—Cp — Hp2 angle tenns and the proper dihedral terms involving a classical atom are all included. 

The critical points of the equivalent classical Hamiltonian occur at stationary state energies of the quantum Hamiltonian H and correspond to stationary states in both the quantum and generalized classical pictures. These points are characterized by the constrained generalized eigenvalue equation obtained by setting the time variation to zero in Eq. (4.17)... 

Alternatively, proton double quantum (DQ) NMR, based on a combined DQ excitation and a reconversion block of the pulse sequence, has been utilized to gain direct access to residual DCCs for cross-linked systems.69,83-89 For this purpose, double-quantum buildup curves are obtained with use of a well-defined double-quantum Hamiltonian along with a specific normalization approach. Residual interactions are directly proportional to a dynamic order parameter Sb of the polymer backbone,87... 

Keywords Intramolecular kirchhoff laws Molecular electronics Molecular logic gates Single molecule electronic circuits Quantum hamiltonian computing... 

Our approach is based on a systematic semiclassical study of the Dirac equation. After separating particles and anti-particles to arbitrary powers in h, a semiclassical expansion of the quantum dynamics in the Heisenberg picture is developed. To leading order this method produces classical spin-orbit dynamics for particles and anti-particles, respectively, that coincide with the findings of Rubinow and Keller Hamiltonian relativistic (anti-) particles drive a spin precession along their trajectories. A modification of that method leads to a semiclassical equivalent of the Foldy-Wouthuysen transformation resulting in relativistic quantum Hamiltonians with spin-orbit coupling. 

Another ambiguity in defining the classical mapping Hamiltonian is related to the fact that different bosonic quantum Hamiltonians may correspond to the same original quantum Hamiltonian H. This problem was already discussed in Section VI.A.2 for A-level systems. In the context of nonadiabatic dynamics, a different version of the mapping Hamiltonian is given by... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

In the last few years we have witnessed the successful development of several methods for the numerical solution of multi-dimensional quantum Hamiltonians Monte Carlo methods centroid methods,mixed quantum-classical methods, and recently a revival of semiclassical methods. We have developed another approach to this problem, the exponential resummation of the evolution operator. - The rest of this Section will explain briefly this method. 

G. Casati, B. Chirkov, J. Ford, and F.M. Izrailev, in G. Casati and J. Ford, (Eds.), Stochastic Behavior in Classical and Quantum Hamiltonian Systems of Lecture Notes in Physics, Vol. 93, Springer, Berlin, 1979. 

Fig. 6. A closed loop apparatus for optimally identifying quantum Hamiltonian information. The closed loop operations aim to reveal one or more control experiments that identify the best quality Hamiltonian information. Hamiltonian quality is used as the feedback signal for the learning algorithm guiding the laboratory experiments.

A pulse scheme recovering the zero-quantum Hamiltonian was proposed by Baldus and Meier.142 It is weakly dependent on spectral parameters and a faithful measure of internuclear distances. This sequence is based on the former rotor-synchronized R/L-driven polarization transfer experiments.143,144 It uses the LG or FS-LG, which is used to decouple the high-7 spins, and combined MAS and RF irradiation of low-7 spins to decouple the hetero-nuclear dipolar interactions. With phase-inversion and amplitude attenuation in the rotating frame and refocusing pulses in the laboratory frame part of the pulse sequence, a zero-quantum average Hamiltonian can be obtained with optimum chemical-shift/offset independence. 

The quantum Hamiltonian of the classical kicked rotor, defined by the classical Hamiltonian function (5.1.2), is easily obtained by canonical quantization. On replacing the classical angular momentum L by the quantum angular momentum operator L according to... 

We turn now to a discussion of the quantum mechanics of surface state electrons. Written in the units (6.1.7) the quantum Hamiltonian is given by... 

Casati, G., Chirikov, B.V., Izraelev, F.M. and Ford, J. (1979). Stochastic behavior of a quantum pendulum under a periodic perturbation, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, eds. G. Casati and J. Ford (Springer, New York). 

In quantum mechanics, when using variation methods, one encounters the same finite-size problem in studying the critical behavior of a quantum Hamiltonian. .., A ) as a function of its set of parameters X,. In this... 

The third method is a direct finite-size scaling approach to study the critical behavior of the quantum Hamiltonian without the need to make any explicit analogy to classical statistical mechanics [54,88]. The truncated wave function that approximate the eigenfunction Eq. (52) is given by... 

The transitions between the 0) ) and ) AT) manifold of vibrational states can be obtained by diagonalization of the two-quantum Hamiltonian in the site basis ... 

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