Momentum 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 angular momentum conjugate to each of these three angular coordinates (each is the... 

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... 

Here, En is the energy of the nth quantum state of the transition state for all coordinates but X and Q, the reaction coordinate, UC(X) is the potential energy in the transition state at the point X and at the position of minimum potential energy with respect to all other coordinates in the transition state, and px is the momentum conjugate to x. 

Here, as in the above, q is the pathlength along the curve (38), p = J 2[Eq — U-(q)] is the linear momentum conjugated to q, U-(q) is the lower-sheet potential energy expressed by equation (35), a, b, c, and d are the turning points in the double-well potential of Fig. 4 ordered from left to right, and o> is the one-well frequency determined in equation (22). Finally, substituting equation (41) in equation (40), we find [7,8] ... 

Here Pa(a = 6, e) is the momentum conjugate to Qa. In the absence of spin-orbit interaction, the e vibration does not mix the orbital components of the 4T2 g and we have vibrational potential energy surfaces consisting of three separate ( disjoint ) paraboloids in the two-dimensional (2D) space of the Qe and Qe coordinates of the e vibration. The Jahn-Teller coupling leads only to a uniform shift (—ZsPJX = — V2/2fia>2 = —Sha>) of all vibronic levels. 

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

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