Broglie-Bohm formulation of quantum mechanics

Recently, there has been a renewed interest in the de Broglie-Bohm formulation of quantum mechanics as a numerical tool to perform multidimensional wave packet calculations [13-15]. It has also been used to visualize the motion of quantum mechanical wave packets by trajectories and to study the transition from quantum mechanics to classical mechanics [16-18]. Carlsen and Goscinski [16], for instance, have studied fractional and full revivals of circular Rydberg wave packets in the hydrogen atoms using this formulation. 

In what follows we shall show that the de Broglie-Bohm formulation can also be used to establish a hybrid quantum/classical scheme to treat the dynamics of systems with a large number of degrees of freedom in which a few need to be described quantum mechanically. 

Since the method to mix quantum and classical mechanics to be presented can be considered as an approximate method derived from the de Broglie-Bohm formulation of quantum mechanics, this completely equivalent perspective of quantum mechanics will be briefly reviewed. To this end, we consider a two-dimensional Hilbert space. 

Note that considering two dimensions is no restriction to what will be shown below, actually, x, X can be viewed as collective variables one of which will comprise all quantum degrees of freedom while the other all classical ones. Writing the wavefunction as X, t) = R x, X, t)exp (i5(x, X, t)/h), with R, S being real, the Schrddinger equation can be recast in terms of a continuity equation. 

Hence one sees that the phase of ilf x,X,t) can be viewed as an action function, a solution to the Hamilton-Jacobi equation with an additional 

---

Next, consider how the energy conserving MF force is formulated in Bohmian mechanics. De Broglie [73,74] and Bohm [64,65] express the wave fimction in the polar form = R q)exp iS q)/h) and rewrite the expectation value of the quantum Hamiltonian as... 

---