Canonical Hamilton

Finally, a very interesting quantum feature is obtained from considering the classical canonical Hamilton equations rewritten with Poisson parentheses... 

As first noted by Dirac [85], the canonical equations of motion for the real variables X and P with respect to J Pmf are completely equivalent to Schrddinger s equation (28) for the complex variables d . Moreover, it is clear that the time evolution of the nuclear DoF [Eq. (32)] can also be written as Hamilton s equations with respect to M mf- Similarly to the equations of motion for the mapping formalism [Eqs. (89a) and (89b)], the mean-field equations of motion for both electronic and nuclear DoF can thus be written in canonical form. 

In order to obtain the Hamiltonian for the system of an atom and an electromagnetic wave, the classical Hamilton function H for a free electron in an electromagnetic field will be considered first. Here the mechanical momentum p of the electron is replaced by the canonical momentum, which includes the vector potential A of the electromagnetic field, and the scalar potential O of the field is added, giving [Sch55]... 

Action-angle variables can also be introduced for certain types of motion in systems with many degrees of freedom, providing there exists one or more sets of coordinates in which the HJ equation is completely separable. If only conservative systems are considered Hamilton s characteristic function should be used. Complete separability means that the equations of canonical transformation have the form... 

To quantize the dynamics of the particles first requires that we express the velocities of the particles in terms of canonical momenta. In the presence of electromagnetic fields, the canonical momenta are not merely m dx-Jdt). Rather, in order to incorporate Lorentz s velocity-dependent forces into Hamilton s formulation of classical mechanics, the canonical momenta are given by [2]... 

It should be noted that the classical equations of motion (usually in the Hamilton s canonical form) are solved by various numerical methods and the... 

There exists a special type of coordinate transformation in phase space, called a canonical transformation, which transforms the original system variables (q,p) to new system variables q, p ) = q, q 2, , q p, P2, , p ) while retaining the structure of Hamilton s equations of motion, that is,... 

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. 

These are Hamilton s canonical equations. One may determine the temporal behaviour of a classical system with N degrees of freedom by solving Lagrange s N second-order differential equations with the constants of integration being fixed by the IN initial values of the coordinates and Velocities which determine the initial state of the system, or by solving Hamilton s IN first-order equations for the same initial state. 

Earlier in this section it was commented on how the minimal-coupling QED Hamiltonian is obtained from fhe classical Lagrangian function. A few words are in order regarding the derivation of the multipolar Hamiltonian (6). One method involves the application of a canonical transformation to the minimal-coupling Hamiltonian [32]. In classical mechanics, such a transformation renders the Poisson bracket and Hamilton s canonical equations of motion invariant. In quantum mechanics, a canonical transformation preserves both the commutator and Heisenberg s operator equation of motion. The appropriate generating function that converts H uit is propor-... 

We shall place ourselves in the Hamiltonian framework. We consider a 2n-dimensional phase space F endowed with canonical coordinates qi,..., qn,pi,..., pn. The flow in the phase space is determined by a smooth Hamiltonian function F[(q,p,t) via the Hamilton s equations... 

The canonical transformations are characterised by the fact that they leave invariant the form of the equations of motion, or the stationary character of the integral [(6) of 5] expressing Hamilton s principle. This raises the question whether there are still other invariants in the case of canonical transformations. This is in fact the case, and we shall give here a series of integral invariants introduced by PoincarA1 We can show that the integral... 

In this section, we are concerned with the canonical equations of the radiation field. We consider the fact that the electromagnetic wave is a transverse wave, and convert it into the form of Hamilton kinetic equations which are independent of the transformation parameter. In this process we will reach the conclusion that the radiation field is an ensemble of harmonic oscillators. During this process we will stress the concepts of vector potential and scalar potential. The equations of an electromagnetic wave in the vacuum are summarized as follows ... 

The derivation of the Hamiltonian resembles the standard procedure the classical Lagrange function is constructed first, then it is used to express the classical Hamilton function and then quantisation is applied by substituting the canonical variables for corresponding quantum-mechanical operators. There are two additional requirements the Hamiltonian should be symmetric with respect to the interchange of two electrons, and it should be Hermitian. 

In the quantum mechanical case we start with a Hamilton operator H which we assume to be obtained from the Weyl quantization of a classical Hamilton function H(q, p). Like in the previous section, q = pi, p2, , pf) and p = pi, p2,. .., pd) denote the canonical coordinates and momenta, respectively, of a Hamiltonian system with d DoEs. Eor convenience, we again choose atomic units, so that q and p are dimensionless. We denote the... 

As shown in Appendix B, it can easily be proved for any transforms described by the functional form of Eq. (2.15), that if z(0) are canonical, z(s) are also canonical (and vice versa), as the time evolution of any Hamiltonian system is regarded as a canonical transformation from canonical variables at an initial time to those at another time, maintaining the structure of Hamilton s equations. 

Such a system generally does not have analytically integrable equations of motion. However, we may apply Hamilton s equations of motion, solve them numerically, and thus generate a unique trajectory for each set of initial conditions we choose. The resulting dynamics generally exhibits a variety of interesting phenomena. First, the frequency of motion in each mode is no longer a constant [as would be the case if we had f(q, qi) = 0] but depends on the instantaneous values of the canonical coordinates ( p, ) ... 

First we consider a system with two degrees of freedom (N = 2). Suppose we have two closed curves yi and y2 phase space, both of which encircle a tube of trajectories generated by Hamilton s equations of motion. These curves can be at two sequential times (tj, or they can be at two sequential mappings on a Poincare map These curves are associated with domains labeled (Dj, D2), which are the projections of the closed curves upon the coordinate planes (pj, qj. Because both the mapping and the time propagation are canonical transformations, the integral invariants (J-j,. 2) are preserved (constant) in either case. There are two of them, of the form ... 

Similarly, it can be shown that Eq. [55] is a statement of the preservation of phase space volume under propagation by Hamilton s equations of motion, that is, Liouville s theorem. It is important to note that the Poincare integral invariants are also preserved under a canonical transformation of any kind and not just the propagation of Hamilton s equations. 

The first step in the study of collision dynamics is to assume that nuclear motion obeys tha laws of classical mechanics. This approximation is expected to give, at least qualitatively, a correct description of the collision between heavy particles at high (relative) velocities. The most appropriate formalism for such a description is based on the Hamilton canonical equations of motion... 

More complicated is the treatment of the collision of an atom A with a diatomic molecule BC. In this case the canonical equations of motion (14.11) must be used for a system of three atoms A,B,C with 6 nuclear degrees of freedom (in a center-of-mass coordinate system). The Hamilton function for the initial configuration (large separation of atom A from molecule BC) is conveniently written in the form... 

---