Quantum variable

TABLE Illb Evaluation of Higher Moments Property Exact Quantum Variable Quadratic... 

It has also been shown that the technique referred to as multiple-quantum variable angle spinning (MQVAS) can be used for the purpose of separating the CSA and quadrupolar lineshapes [155]. In this experiment, a judicious choice of the rotor orientation is fjRL = 70.12°, which leads to cancelation of the fourth-rank term of the second-order interaction (Fig. 2). Thus, the refocusing ratio k for the quadrupolar interaction depends on the ratio of the second-rank terms... 

It is less obvious how to assign quantum variables. While, for instance, the total classical angular momentum of a collision fragment is clearly defined, in the case of small fragments product rotational quantum numbers are typically sought. The standard procedure for linear molecules is to assume rigid rotor conditions and to solve the usual correspondence principle relations to obtain the quasiclassical product quantum numbers (restricted to integer values). 

An equation for y(P, R, t) follows from the trace over quantum variables,... 

The resulting equation of motion for pw also leads to a new equation for the phase space density y, after taking the trace over quantum variables. 

By definition, a mixed quantum-classical method treats the various degrees of freedom (DoF) of a system on a different djmamical footing, e.g. quantum mechanics for the electronic DoF and classical mechanics for the nuclear DoF. As was discussed above, some of the problems with these methods are related to inconsistencies inherent in this mixed quantum-classical ansatz. To avoid these problems, recently a conceptually different way to incorporate quantum mechanical DoF into a semiclassical or quasiclassical theory has been proposed, the so-called mapping approach. " In this formulation, the problem of a classical treatment of discrete DoF such as electronic states is bypassed by transforming the discrete quantum variables to continuous variables. In this section we briefly introduce the general concept of the mapping approach and discuss the quasiclassical implementation of this method as well as applications to the three models introduced above. The semiclassical version of the mapping approach is discussed in Sec. 7. 

P. van Loock, S. L. Braunstein. Telecloning of Continuous Quantum Variables. Physical Review... 

Quantum variables mapped to classical ones Meyer-Miller method... 

The squared amplitudes are either quantum state probabilities or they may represent quantum operators if the hamiltonian is expanded in quantum operators rather than quantum states. Aside from a set of equations of the type (1) we have some classical equations of motion for the classical part of the system. In order to couple the classical and quantum subsystems one introduces an average hamiltonian, i.e. an average over the total hamiltonian of the system such that the quantum variables only appear with some average or expectation value in the hamiltonian governing the classical motion. Thus we also obtain a set of equations of motion of the type ... 

ShalashUin DV, Irene Burghardt (2008) Gaussian-based techniques for quantum propagation from the time-dependent variational principle formulation in terms of trajectories of coupled classical and quantum variables. J Chem Phys 129 084104... 

In [6] such variables are called quantum variables , by resemblance with the hidden variables theories of quantum mechanics. 

The appearance of non-commuting quantum variables can now also be traced back to the non-commutative algebra of 4D hypercomplex functions. On projection into 3D by the separation of space and time variables, the quaternion variables are reduced to complex functions that characterize orbital angular momentum, but the commutation properties remain. Not appreciating the essence of complex wave... 

The classical particle motion should be governed by an average potential Veff = ( IV/l I ), where is the total wavefunction of the quantum system, i.e., the phonons in the present case. The averaging is carried out over the quantum variables Qk-... 

---