The Narrow Wave Packet Approach

Let us first consider the evolution of a narrow wave packet|q, piinSwith average position q and average momentumpat some time t0. Given our assumptions

that for a general wave function, md 2_hxˆi dt2 =−h ∂V(ˆx) ∂ˆx i (2.137)

dictates that such a wave packet will approximately satisfy the stronger condi- tion md 2_hxˆi dt2 ≈ − ∂V(hxˆi) ∂hˆxi (2.138)

and that the wave packet therefore will follow an approximately classical tra- jectory as long as it remains sufficiently narrow. More specifically, it will follow the classical trajectory whose position and momentum at timet0 areqandp.

If the wave packet follows a classical trajectory, then by equivariance, es- sentially all of the Bohmian trajectories associated with that wave packet will also be approximately classical; by the phrase ‘essentially all of the Bohmian trajectories,’ I am referring to an ensemble of possible Bohmian trajectories cor- responding to different possible initial configurations. Thus, it seems initially that the systemSbeing in a narrow wave packet suffices to ensure that Bohmian trajectories are classical. Among others, Bowman has been a strong advocate of this approach, although, as I discuss below, he begins to incorporate effects of the environment after considering this result for the base of isolated systems. However, narrow wave packets constitute only a very restricted subset of possible solutions to the Schrodinger equation, and the most general solution will not necessarily be narrowly peaked in both position and momentum space. The most general solution will rather be a superposition of narrow wave packets, of the form

|Ψi=

Z

dq dp α(q, p)|q, pi, (2.139) where each|q, pitraverses its own classical trajectory.

Bohmian configuration. For example, consider the simple case whereS consists of a single center of mass with a free Hamiltonian ˆHS =

ˆ

P2

2m; the mass may be macroscopic, though this does not affect my conclusion in this instance. Let the wave function of this system initially take the form of two spatially separated wave packets moving toward each other with opposite average momenta, so that they overlap at some point in time and then pass through each other:

|Ψi=√1

2[|q1, pi+|q2,−pi]. (2.140) Initially, the set of Bohmian trajectories associated in the ensemble with each of these packets will, by equivariance, follow the same classical path that its wave packet follows. However, this will cease to be true when the packets overlap. Because Bohmian trajectories associated with a single pure state can never cross, when the packets overlap and pass through each other, the trajectories will not be able to follow suit. Instead, they will reverse direction and leave the region of overlap in the packet in which they did not begin. This reversal of direction represents a highly non-classical effect on the trajectories inS, since if the trajectories were classical, they would simply follow a straight line path with the same wave packet all the way through.

We can see more generally that this sort of non-classical behavior on the part of the Bohmian trajectory will occur whenever the expansion (2.139) of the wave function in terms of spatially localized wave packets (e.g., coherent states) contains wave packets that are initially separated and later come to overlap in configuration space. Ifα(q, p) andα(q0, p0) are non-zero for any two initially non-overlapping wave packets|q, piand|q0, p0iwhose future evolutions cause them to overlap in configuration space at some point in time, the Bohmian trajectories associated with |Ψi will become non-classical. Thus, non-classical Bohmian trajectories can result from a very wide variety of wave functions, even when the mass is large.

configuration, we cannot explain the emergence of classical trajectories at the macroscopic scale in terms of an isolated set of macroscopic degrees of freedom without excluding a very broad class of solutions to the Schrodinger equation as viable physical descriptions of the system in question. Since one would have to exclude any wave functions that contain a pair of wave packets that inter- sect in configuration space, the set of wave functions that one must discard will depend heavily on the dynamics of the particular system, which makes such an exclusion seem especially ad hoc.

Bowman has suggested invoking environmental decoherence - and therefore abandoning the assumption of isolation - as the explanation for a restriction to narrow wave packets; while the analysis that I provide below agrees with this approach, Bowman’s analysis overlooks certain subtleties involving the need specifically for configuration space decoherence, and is less comprehensive in that it does not consider the stucture of the total pure state of the closed system

SE, but only that of the mixed state of the open systemS.

As in the case of Bare/Everett theory, another problem with attempting to model classical behavior using an isolated macroscopic system is that the assumption of isolation is highly unrealistic. While microscopic environmental degrees of freedom are often ignored in classical descriptions of macroscopic sys- tems, the extreme succeptibility of macroscopic superpositions in these systems to entanglement with their environment requires us to consider the effect of the environment when we are examining such systems at the quantum level. In section 2.5.4, we shall see in more detail what sort of effect interaction with the environment can have on such a system.