Conjugate coordinates and momenta

Let (p, q) be one set of canonically conjugate coordinates and momenta (the old variables) and (P,Q) be another such set (the new variables).13 (P, Q, p and q are IV-dimensional vectors for a system with N degrees of freedom, but for the sake of clarity multidimensional notation will not be used the explicitly multidimensional expressions are in most cases obvious.) In classical mechanics P and Q may be considered as functions of p and q, or inversely, P and Q may be chosen as the independent variables with p and q being functions of them. To carry out the canonical transformation between these two sets of variables, however, one must rather choose one old variable and one new variable as the independent variables, the remaining two variables then being considered as functions of them. The canonical transformation is then carried out with the aid of a generating function, or generator, which is some function of the two independent variables, and two equations which express the dependent variables in terms of the independent variables.13... 

In order to use statistical mechanics to calculate the macroscopic properties introduced only by spatial averaging one needs to perform enses le averages over states. For present purposes classical mechanics will suffice using the phase space of conjugate coordinates and momenta qi and pi with a time dependent probablity density function z (qi p t). The meao expectation value of a function A (qi. p. t) is denoted by A(t) (q pj t) f(qj Pj t)> where the angle brackets now refer to the ensemble average over the qj and pj. 

To obtain the rotational factor in the classical molecular partition fimction of a diatomic or linear polyatomic gas, we must find the conjugate coordinates and momenta as discussed in Appendix E. We consider a rigid rotor. In spherical polar coordinates with r = (fixed) and with y = 0 (fixed) the Lagrangian is... 

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. 

The state of a classical system is specified in terms of the values of a set of coordinates q and conjugate momenta p at some time t, the coordinates and momenta satisfying Hamilton s equations of motion. It is possible to perform a coordinate transformation to a new set of ps and qs which again satisfy Hamilton s equation of motion with respect to a Hamiltonian expressed in the new coordinates. Such a coordinate transformation is called a canonical transformation and, while changing the functional form of the Hamiltonian and of the expressions for other properties, it leaves all of the numerical values of the properties unchanged. Thus, a canonical transformation offers an alternative but equivalent description of a classical system. One may ask whether the same freedom of choosing equivalent descriptions of a system exists in quantum mechanics. The answer is in the affirmative and it is a unitary transformation which is the quantum analogue of the classical canonical transformation. 

Generalized momentum operators as defined by Eq. (2.77) can be used in wave mechanical as well as in matrix mechanical formulations. It ensures that the operators are Hermitian, and that momenta, 7r, conjugated to generalized coordinates, qh fulfil commutation relations similar to the canonical relations of Cartesian coordinates and momenta,... 

The remaining aspect of a trajectory simulation is choosing the initial momenta and coordinates. These initial conditions are chosen so that the results from an ensemble of trajectories may be compared with experiment and/or theory, and used to make predictions about the chemical system s molecular dynamics. In this chapter Monte Carlo methods are described for sampling the appropriate distributions for initial values of the coordinates and momenta. Trajectories may be integrated in different coordinates and conjugate momenta, such as internal [7], Jacobi [8], and Cartesian. However, the Cartesian coordinate representation is most general for systems of any size and the Monte Carlo selection of Cartesian coordinates and momenta is described here for a variety of chemical processes. Many of... 

For a diatomic molecule the Cartesian Hamiltonian can be transformed to a Hamiltonian dependent on relative (or internal) and center-of-mass coordinates and momenta. The relative coordinates x, y, z and their conjugate momenta are defined by... 

There are a number of formulations of classical mechanics, each providing different insights into its nature. For example, Hamilton s method, used here, describes dynamics in terms of trajectories in generalized coordinates and momenta. Consider an M degrees of freedom system with system Hamiltonian H(q, p), where (q, p) is a complete set of M conjugate generalized coordinates and momenta. The time evolution of the system is given by Hamilton s equations,... 

The action integral, equation (29), can also be evaluated by using a Fourier series representation of the coordinates and momenta. In the simplest version the normal coordinates Qj and their conjugate momenta Pj are represented by... 

Write the Hamiltonian of a particle in terms of a) Cartesian coordinates and momenta, and b) spherical coordinates and their conjugate momenta. Show that Hamilton s equations of motion are invariant to the coordinate transformation from Cartesian to polar coordinates. 

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. 

We consider here the relation between volume elements in phase space in particular, the relation between dqdp and dQdP, where dq = dq dqn refers to Cartesian coordinates in a laboratory fixed coordinate system, dQ = dQ dQn refers to normal-mode coordinates, and p and P are the associated generalized conjugate momenta. 

The term density matrix arises by analogy to classical statistical mechanics, where the state of a system consisting of N molecules moving in a real three-dimensional space is described by the density of points in a 6N-dimensional phase space, which includes three orthogonal spatial coordinates and three conjugate momenta for each of the N particles, thus giving a complete description of the system at a particular time. In principle, the density matrix for a spin system includes all the spins, as we have seen, and all the spatial coordinates as well. However, as we discuss subsequently we limit our treatment to spins. For simplicity we deal only with application to systems of spin % nuclei, but the formalism also applies to nuclei of higher spin. 

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