Hamilton-Jacobi

Section VI shows the power of the modulus-phase formalism and is included in this chapter partly for methodological purposes. In this formalism, the equations of continuity and the Hamilton-Jacobi equations can be naturally derived in both the nonrelativistic and the relativistic (Dirac) theories of the electron. It is shown that in the four-component (spinor) theory of electrons, the two exha components in the spinor wave function will have only a minor effect on the topological phase, provided certain conditions are met (nearly nonrelativistic velocities and external fields that are not excessively large). 

The aim of this section is to show how the modulus-phase formulation, which is the keytone of our chapter, leads very directly to the equation of continuity and to the Hamilton-Jacobi equation. These equations have formed the basic building blocks in Bohm s formulation of non-relativistic quantum mechanics [318]. We begin with the nonrelativistic case, for which the simplicity of the derivation has... 

Again, the summation convention is used, unless we state otherwise. As will appear below, the same strategy can be used upon tbe Dirac Lagrangean density to obtain the continuity equation and Hamilton-Jacobi equation in the modulus-phase representation. 

Variationally deriving with respect to a leads to the Hamilton-Jacobi equation... 

The result of interest in the expressions shown in Eqs. (160) and (162) is that, although one has obtained expressions that include corrections to the nonrelativistic case, given in Eqs. (141) and (142), still both the continuity equations and the Hamilton-Jacobi equations involve each spinor component separately. To the present approximation, there is no mixing between the components. 

The terms before the square brackets give the nonrelativistic part of the Hamilton-Jacobi equation and the continuity equation shown in Eqs. (142) and (141), while the term with the squaie brackets contribute relativistic corrections. All terms from are of the nonmixing type between components. There are further relativistic terms, to which we now turn. 

In Eq. (168), the first, magnetic-field term admixes different components of the spinors both in the continuity equation and in the Hamilton-Jacobi equation. However, with the z axis chosen as the direction of H, the magnetic-field temi does not contain phases and does not mix component amplitudes. Therefore, there is no contribution from this term in the continuity equations and no amplitude mixing in the Hamilton-Jacobi equations. The second, electric-field term is nondiagonal between the large and small spinor components, which fact reduces its magnitude by a further small factor of 0 particle velocityjc). This term is therefore of the same small order 0(l/c ), as those terms in the second line in Eqs. (164) and (166) that refer to the upper components. 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

Caustics The above formulae can only be valid as long as Eq. (9) describes a unique map in position space. Indeed, the underlying Hamilton-Jacobi theory is only valid for the time interval [0,T] if at all instances t [0, T] the map (QOi4o) —> Q t, qo,qo) is one-to-one, [6, 19, 1], i.e., as long as trajectories with different initial data do not cross each other in position space (cf. Fig. 1). Consequently, the detection of any caustics in a numerical simulation is only possible if we propagate a trajectory bundle with different initial values. Thus, in pure QCMD, Eq. (11), caustics cannot be detected. 

An alternative procedure is the dynamic programming method of Bellman (1957) which is based on the principle of optimality and the imbedding approach. The principle of optimality yields the Hamilton-Jacobi partial differential equation, whose solution results in an optimal control policy. Euler-Lagrange and Pontrya-gin s equations are applicable to systems with non-linear, time-varying state equations and non-quadratic, time varying performance criteria. The Hamilton-Jacobi equation is usually solved for the important and special case of the linear time-invariant plant with quadratic performance criterion (called the performance index), which takes the form of the matrix Riccati (1724) equation. This produces an optimal control law as a linear function of the state vector components which is always stable, providing the system is controllable. 

The equation for S is recognized as a Hamilton-Jacobi equation for a mechanical action. The solution can be written in terms of a Lagrangian L, for the nuclear motions, introducing the path Q t, Qin,Q), starting initially at positions Qin and ending at Q at time t, and the corresponding generalized velocities Q. The result is... 

Let us first start with the simple ID problem of the symmetric potential V —x) = V x) with 2 equivalent potential minima ix. In our treatment, the principal exponential factor VTo(x) does not depend on the energy E and thus the Hamilton-Jacobi equation does not change. Putting... 

Hamilton-Jacobi equation, molecular systems, modulus-phase formalism, 262-265 Lagrangean density correction term, 270 nearly nonrelativistic limit, 269... 

The book contains very little original material, but reviews a fair amount of forgotten results that point to new lines of enquiry. Concepts such as quaternions, Bessel functions, Lie groups, Hamilton-Jacobi theory, solitons, Rydberg atoms, spherical waves and others, not commonly emphasized in chemical discussion, acquire new importance. To prepare the ground, the... 

This equation has been used by Sundstrom and coworkers [151] and adapted to the analysis of femtosecond spectral evolution as monitored by the bond-twisting events in barrierless isomerization in solution. The theoretical derivation of Aberg et al. establishes a link between the Smoluchowski equation with a sink and the Schrodinger equation of a solute coupled to a thermal bath. The reader is referred to this important work for further theoretical details and a thorough description of the experimental set up. It is sufficient to say here that the classical link is established via the Hamilton-Jacobi equation formalism. By using the standard ansatz Xn(X,t)= A(X,i)cxp(S(X,t)/i1l), where S(X,t) is the action of the dynamical system, and neglecting terms in once this... 

This result is self-consistent with the demonstration [15] that the IFE can be described through % by using the Hamilton-Jacobi equation for one electron in... 

The sum is over all classical paths connecting R1 and R in time t, S is the classical action along such paths, and Det denotes the determinant which ensures unitarity of the propagator K up to second-order variations in S. The classical action is a solution of the Hamilton-Jacobi equation,... 

We assume, furthermore, that the interaction takes place during very short intervals of time. In this approximation, there is only one classical path for which we solve the Hamilton-Jacobi equation. In the vicinity of the collision diameter (where the spectroscopically significant interactions take place) we have, to a good approximation (Poll 1980)... 

An obvious approach to the problem was to re-examine the theories of mechanics that failed to predict the observed frequencies and bring them into line with observation. It is therefore not surprising to find that both new theories were modified versions of classical Hamilton-Jacobi theory, adapted so as to incorporate the newly discovered quantum features of atomic spectra. To understand the type of reasoning that produced the quantum theory, it is necessary to make a short excursion into classical HJ theory. Some readers may prefer to skip this section on first reading. For an in-depth discussion more enthusiastic readers are referred to the authoritative text of Goldstein... 

The Hamilton-Jacobi form of the classical equations of motion has been shown to have provided the basis for the quantum-mechanical formulations according to Sommerfeld, Heisenberg, Schrodinger and Bohm. Each of these formulations inspired its own peculiar interpretation of quantum effects, despite their common basis. Each of the different points of view still has its adherents and the debates about their relative merits continue. Closer scrutiny shows that the Sommerfeld and Heisenberg systems assume quanta to be particles in the classical sense, although Heisenberg considered electronic positions to be fundamentally unobservable. 

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