Conjugate momenta

With each random choice of y and its conjugate momentum Py, one can have a separate trajectory with a different final wave function. After a series of calculations, the energy and state resolved cross-sections are obtained. 

Using the fact that the quantum mechanical coordinate operators q = x, y, z as well as the conjugate momentum operators (pj = px, Py, Pz are Hermitian, it is possible to show that Lx, Ly, and L are also Hermitian, as they must be if they are to correspond to experimentally measurable quantities. 

To construct Nose-Hoover constant-temperature molecular dynamics, an additional coordinate, s, and its conjugate momentum p, are introduced. The Hamiltonian of the extended system of the N particles plus extended degrees of freedom can be expressed... 

The total space of system coordinates consists of a tagged coordinate Q (conjugate momentum P) and a set of mass-scaled bath coordinates q (conjugate momenta p). The Hamiltonian reads... 

In continuum field theory, the field dynamical variable and the coordinates X and t are only parameters. With the conjugate momentum f (T, t), and Hamiltonian operator... 

To achieve the desired separation of the reactive degree of freedom from the bath modes, we use time-dependent normal form theory [40,99]. As a first step, the phase space is extended through the addition of two auxiliary variables a canonical coordinate x, which takes the same value as time t, and its conjugate momentum PT. The dynamics on the extended phase space is described by the Hamiltonian... 

However, in all the rest of their approach, Robertson and Yarwood consider the slow mode Q as a scalar obeying simply classical mechanics, because they neglect the noncommutativity of Q with its conjugate momentum P. As a consequence, the logic of their approach is to consider the fluctuation of the slow mode as obeying classical statistical mechanics and not quantum statistical mechanics. Thus we write, in place of Eq. (138), the corresponding classical formula ... 

The phase space for three-dimensional motion of a single particle is defined in terms of three cartesian position coordinates and the three conjugate momentum coordinates. A point in this six-dimensional space defines the instantaneous position and momentum and hence the state of the particle. An elemental hypothetical volume in six-dimensional phase space dpxd Pydpzdqxdqydqz, is called an element, in units of (joule-sec)3. For a system of N such particles, the instantaneous states of all the particles, and hence the state of the system of particles, can be represented by N points in the six-dimensional space. This so-called /r-space, provides a convenient description of a particle system with weak interaction. If the particles of a system are all distinguishable it is possible to construct a 6,/V-dimensional phase space (3N position coordinates and 3N conjugate momenta). This type of phase space is called a E-space. A single point in this space defines the instantaneous state of the system of particles. For / degrees of freedom there are 2/ coordinates in /i-space and 2Nf coordinates in the T space. 

A canonically conjugate momentum may be obtained from (32) in the usual way as the derivative of the Lagrangian,... 

The Kramers-type equation corresponding to Eq. (l.l) and adapted in Ref. 2 to the microcanonical case for a system with coordinate q and its conjugate momentum p is... 

Fig. 7. The product of uncertainties of the coordinate and the conjugated momentum of the phonon 1 for 77 = 0.

We consider a system of N particles. To keep the notation simple, we have chosen only to consider the coordinate g and conjugate momentum p, and we neglect all the other coordinates and momenta in all the derivations to follow. The straightforward generalization to include all coordinates and momenta will be discussed toward the end of the appendix. 

In quantum mechanics, the independent variables q and p of classical mechanics are represented by the Hermitian operators q and p with the following matrix elements in the Cartesian coordinate basis q), here just written for the coordinate g and the conjugate momentum pp... 

The discussion has so far been restricted to one degree of freedom, that is, one coordinate and its conjugate momentum, to make the notation simple. When we extend... 

When quantizing the theory, the variable x and is conjugate momentum p become operator obeying the commutation rule ... 

Because of the complex nature of the variable u, the conjugate momentum v is given by13... 

In analogy with the one-dimensional analysis, the Jj are defined over complete periods of the orbit in the (qj,Pj) plane, Jj = ptdq j. If one of the separation coordinates is cyclic, its conjugate momentum is constant. The corresponding orbit in the (qj,Pj) plane of phase space is a horizontal straight line, which may be considered as the limiting case of rotational periodicity, for which the cyclic qj always has a natural period of 2-k, and Jj = 2irpj for all cyclic variables. 

Coordinate position operator = q Conjugate momentum operator = p [q,p] = ih... 

Here, q are the position coordinate operators of the oscillators of the bath and p are the conjugate momentum, whereas m, and m, are the corresponding reduced masses and angular frequencies. At last, the q, parameters describe the strength of the coupling between the H-bond bridge (described by the coordinate Q) and the oscillators of the bath. 

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