Momentum variables

We then say that the particle has spin and the three components Sl constitute the (pseudovector) spin operator. Note that by virtue of Eq. (9-55) the spin variables are not expressible in terms of the variables q and p. Since the angular momentum variables J are also the infinitesimal generators of rotations we deduce that... 

The algebraic description of three-dimensional problems is in terms of U(4). The coset space for this case is a complex three-dimensional vector, We denote its complex conjugate by q. One can then introduce the canonical position and momentum variables q and p by the transformation... 

In the system of quantum dipoles, dipole and momentum variables have to be replaced by the quantum operators, and quantum statistical mechanics has to be applied. Now, the kinetic energy given by Eq. 9 does affect the thermal average of quantity that depends on dipole variables, due to non-commutivity of dipole and momentiun operators. According to the Pl-QMC method, a quantum system of N dipoles can be approximated by P coupled classical subsystems of N dipoles, where P is the Trotter munber and this approximation becomes exact in the limit P oo. Each quantiun dipole vector is replaced by a cychc chain of P classical dipole vectors, or beads , i.e., - fii -I-. .., iii p, = Hi,I. This classical system of N coupled chains... 

The r-space and p-space representations of the ( th-order density matrices, whether spin-traced or not, are related [127] by a fif -dimensional Fourier transform because the parent wavefunctions are related by a 3A -dimensional Fourier transform. Substitution of Eq. (5.1) in Eq. (5.8), and integration over the momentum variables, leads to the following explicit spin-traced relationship ... 

An alternative coupling scheme for temperature and pressure, the Nose-Hoover scheme, adds new, independent variables that control these quantities to the simulation (Nose 1984 Hoover 1985). These variables are then propagated along with the position and momentum variables. 

In the analysis of the bulk periodic orbits, a simplification occurs for the bending oscillations. Because the Hamiltonian of a linear molecule depends quadratically on the angular momentum variable La, the time derivative of the conjugated angle given by = l2 c vanishes with La, in contrast to the time derivatives of the other angle variables, which are essentially equal to 0j - os j. Therefore, the subsystem La = 0 always contains bulk periodic orbits that are labeled by n, tr2,n-i). 

Other examples can be discussed in the same way. We note that all these results follow from simple dimensional analysis in terms of coil radius Rg, chain concentration cp, and (if present) momentum variables. The method is based on the assumptions that... 

The quantum-classical Liouville equation was expressed in the subsystem basis in Sec. 3.1. Based on this representation, it is possible to recast the equations of motion in a form where the discrete quantum degrees of freedom are described by continuous position and momentum variables [44-49]. In the mapping basis the eigenfunctions of the n-state subsystem can be replaced with eigenfunctions of n fictitious harmonic oscillators with occupation numbers limited to 0 or 1 A) —> toa) = 0i, , 1a, -0 ). This mapping basis representation then makes use of the fact that the matrix element of an operator Bw(X) in the subsystem basis, B y (X), can be written in mapping form as B(( (X) = (AIBy X A ) = m Bm(X) mx>), where... 

Like the momentum variable, which in coordinate space is represented by... 

In classical statistical mechanics, the system is defined with a set of position and momentum variables, qt and pt. The momentum conjugate to q, is defined by the equation... 

The angular momentum variables all enter as quantum-mechanical operators, whose eigenvalues may be measurable. 

In addition to the angular momentum variables (4.1.58) we introduce the position and momentum variables Xn and according to... 

It is helpful to introduce the scaled position and momentum variables X and P, respectively, defined according to... 

Integration of Equation [17] over the momentum variable leads to... 

Our calculations of cross sections were based on sets of coupled Integral equations In momentum variables. Although this is quite different from wave-mechanical approaches, it leads to results by simply solving large sets of coupled linear equations. In our case for as many as 1,500 variables. How this was done is explained In detail In Reference It can be seen that one of the biggest chal-... 

Here we would like to add some comment of a general character. Dirac (1958) argued that the transformation from classical to quantum mechanics should be made, first by constructing the classical Hamiltonian in the Cartesian coordinate system and then by replacing the positions and momenta by their quantum-mechanical operator equivalents, which are determined by the particular representation chosen. The important point is that this transformation should be performed in the Cartesian coordinate system, for it is only in this system that the Heisenberg uncertainty principle for the positions and momenta is usually enunciated. In this connection, notice that some momentum wave functions such as those obtained by Podolsky and Pauling (1929) are correct wave functions that are useful in calculations of the expectation value of any observable, but at the same time they have a drawback in that the momentum variables used there are not conjugate to any relevant position variables (see also, Lombardi, 1980). 

So in this case as in the previous one the operator acts not only on the "spin" variables but also on the momentum variables and. multiplies the function by o(/AK AyJ A V/ The method of the further solution of (65) and derivation of the static conductivity is in analogy with the previous section. Finally we get... 

That the time and frequency variables are conjugate in the same sense that the position and momentum variables are, was in principle clear in both classical and quantum mechanics. It is even possible to take the usual notion of the phase space, whose dimension equals twice the number of mechanical degrees of freedom and to add two more, those of time and energy, and this is possible also in quantum mechanics (29,30). The advantage of doing so in practice was first realized in the so called, linear response regime (31,32), where the change in the state of the system is linear in the perturbation. Spectroscopy with weak (i.e., ordinary) fields is a clear example and for reasons which we intend to discuss, the early applications were to simple molecules (typically diatomic)... 

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