Hamilton-Jacobi quantum equation

Bohm [4] demonstrated that the close parallel of the classical Hamilton-Jacobi (HJ) equation (T3.4) with Schrodinger s equation provides a logical point of departure for a causal account of quantum events. A wave function in polar form with real amplitude R and phase S,... 

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... 

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. 

David Bohm gave new direction to Madelung s proposal by using the decomposition of the wave equation for a radically new interpretation of quantum theory. He emphasized the similarity between the Madelung and Hamilton-Jacobi equations of motion, the only difference between them being the quantum potential energy term,... 

In order to obtain a more compact formulation of the mixed quantum-classical equations we use a Hamilton-Jacobi-like formalism for the propagation of the quantum degree of freedom as in earlier studies [23], A similar approach has been introduced by Nettesheim, Schiitte and coworkers [54, 55, 56], TTie formalism presented here is based on recent investigations of the present authors [23], This formalism can be summarized as follows. Starting from the Hamiltonian Eqn. (2.2) and averaging over the x- and y-mode, respectively, gives... 

The integrals Jk (12) introduced here appear to be suitable for the formulation of quantum conditions in the form Jk=nji. By definition, however, they are associated with a co-ordinate system (q, p) in which the Hamilton-Jacobi equation is separable it is therefore essential that we should next examine the conditions under which this co-ordinate system is uniquely determined by the condition of... 

Suffixes p and a both refer to accidentally degenerate variables.) This is a partial differential equation of the Hamilton-Jacobi type. It does not admit of integration in all cases, and the method fails, therefore, for the determination of the motion for arbitrary values of the Jfc s. We can show, however, as in the example of 44, that the motions for which the wp° s are constant to zero approximation, and remain constant also to a first approximation, are stationary motions in the sense of quantum theory. 

By introducing the quantum postulate huo = E — V, the energy in excess of a constant potential, and the velocity of a matter wavefront, in Hamilton-Jacobi formalism (Goldstein, 1980), c = JT/2m, these equations (in 3D) transform into the familiar set of Schrodinger equations ... 

The replacement of this expansion back into the quantum Hamilton-Jacobi equation provides the successive equations for the various orders of h ... 

However, this approximation clearly breaks down at the classical turning points E = V(x), i.e., at the points were the classical particle (or the manifestation of the wave packet) stops, since p(x) = 0 there, and turns due to the potential equalizes its (eigen-value) energy. This is the WKB approximation holds for the cases where is much smaller than equivalently with the requirement that quantum Hamilton-Jacobi equation to collapse on its classical variant through the action condition ... 

---