Action angle variables

W. H. Miller, Semi-classical theory for non-separable systems construction of good action-angle variables for reaction rate constants, Faraday Disc. Chem. Soc. 62, 40 (1977). 

Formally the unperturbed Hamiltonian is equivalent to the Hamiltonian of the hydrogen atom in constant homogenious electric field. Chaotic dynamics of hydrogen atom in constant electric field under the influence of time-periodic field was treated earlier (Berman et. al, 1985 Stevens and Sundaraml987). To treat nonlinear dynamics of this system under the influence of periodic perturbations we need to rewrite (1) in action-angle variables. Action can be found using its standard definition ... 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

In nonlinear classical dynamics it is convenient to express the Hamiltonian in action-angle variables. The total Hamiltonian H can then be resolved as... 

The zero-order Hamiltonian is a function of the actions alone. It therefore corresponds to uncoupled modes whose actions are conserved (since dljdt = - <)H/(), ). From Section 7.5 on we will express the classical limit of algebraic Hamiltonians in terms of variables i = 1,..., n. These are related to the action-angle variables by t, = I112 exp(/0), = I112 exp(-i0). Loosely... 

We begin with the simple case of one-dimensional problems described algebraically by U(2). The coset space for this case is just a single complex variable, which we call We denote the complex conjugate by 2,. These variables can be interpreted in terms of the position (q) and momentum (p) variables in phase space. Equivalently the t, variables can be related to the action-angle variables /,0 introduced in Section 3.4. To be more precise... 

There is, however, an important conceptional difference between the two approaches. On the quasi-classical level, this difference simply manifests itself in the initial conditions chosen for the electronic DoF. Let us consider an electronic two-level system that is initially assumed to be in the electronic state vl/]). In the mean-field formulation, the initial conditions are initial distribution in (90), is given by pgj = 6 Ni — 1)6 N2). In the mapping formalism, on the other hand, the initial electronic state vl/]) is represented by the first oscillator being in its first excited state and second oscillator being in its ground state [cf. Eq. (91)]. This corresponds to the... 

A further simplification of the semiclassical mapping approach can be obtained by introducing electronic action-angle variables and performing the integration over the initial conditions of the electronic DoF within the stationary-phase approximation [120]. Thereby the number of trajectories required to obtain convergence is reduced significantly [120]. A related approach is discussed below within the spin-coherent state representation. 

In integrable systems, the periodic orbits are not isolated but form continuous families, which are associated with so-called resonant tori. In action-angle variables, the Hamiltonian depends only on the action variables, similar to the Dunham expansion, ... 

In order to give a more quantitative interpretation, we analyzed the classical motion. The classical Hamiltonian is a function of the following action-angle variables Hci(JJ2,73, J4, J, L4, L5, 0 - 03, 0j- 02 - 204, 04 - 03, X4 Xs)> so that (3.13) are also classical constants of motion. As another consequence, the dynamics can be reduced to a 4F subsystem that determines the motion of the 7F system. This 4F subsystem contains itself many subsystems with a smaller number of active degrees of freedom. 

Concerning the Poincare surface of section, it should be noticed that a sort of quantum surface of section can be constructed by intersection of the Wigner or Husimi transform of the eigenfunctions expressed in the quantum action-angle variables of the effective Hamiltonian, which can provide a comparison with the classical Poincare surfaces of section (e.g., in acetylene). 

In order to keep the expressions transparent we, once again, restrict the discussion to the dissociation of the triatomic molecule ABC into products A and BC. Furthermore, the total angular momentum is limited to J = 0. In this chapter we consider the vibration and the rotation of the fragment molecule simultaneously. The corresponding Hamilton function, i.e., the total energy as a function of all coordinates and momenta, using action-angle variables (McCurdy and Miller 1977 Smith 1986), reads... 

Askar, A. and Rabitz, H. (1984). Action-angle variables for quantum mechanical coplanar scattering, J. Chem. Phys. 80, 3586-3595. 

Berry, M.V. and Tabor, M. (1977). Calculating the bound spectrum by path summation in action-angle variables, J. Phys. A 10, 371-379. 

Smith, N. (1986). On the use of action-angle variables for direct solution of classical nonreactive 3D (Di) atom-diatom scattering problems, J. Chem. Phys. 85, 1987-1995. 

The most important problem that has been solved in detail by using action-angle variables is the Kepler problem of planetary motion. The details of the analysis are not important in the present context, but the form of the Hamiltonian for rotation in a central potential V r) = —k/r, obtained as... 

This success set the scene for Sommerfeld s "royal road to quantization", that only required solution of the classical problem using action-angle variables and replacing the J-variables with integral multiples of h. 

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... 

Equation (21) makes it clear that the action-angle variable representation directly addresses the oscillation frequencies without looking into the details of the dynamics. Indeed, as shown below, the action-angle variable representation plays a key role in understanding important qualitative features of Hamiltonian dynamics. 

Next, it is straightforward to use action-angle variables, with the aim of writing K as the normal form... 

Let us now lift this disk D2 into phase space. To do so, one must go back to the sphere equation, Eq. (37). There are several ways of depicting a 3-sphere one is particularly appropriate here [24]. The sphere is dynamically composed of two identical harmonic oscillators without explicit coupling, but whose total energy is a constant, hs3 > 0. Let us thus transform the Hamiltonian (37) in action angle variables, where N,Iy are the actions of the two oscillators and Q, 0, are the two associated angles. Since... 

The canonical transformation to action-angle variables can now be stated exphcitly. With p = dF2/dx and 0 = dF2/dI we obtain... 

Written in action-angle variables, the Hamiltonian H reads... 

Here and (]) are the conjugate action-angle variables, with counted fiom the resonance action, and Djjy(Q) is the interaction parameter that is generated Irom the interaction energy l>(f,n) at the 1 resonance t>i v(Q) = l>(e,f2) with... 

To accomplish this we transform to action-angle variables (x, v —> K, ). Recall that the action K is related to the energy E by... 

It is convenient to use a Langevin starting point, so we begin with Eq. (14.39). We transform to action-angle variables (W, ) ... 

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