Bohmian Mechanics on Scattering Theory

Abstract

The quantum equilibrium distribution tells us the probability for a system to be in a certain configuration at a given time t. That is the basis for the quantum formalism of POVMs, PVMs, and self-adjoint observables on a Hilbert space. In this last chapter we shall return to the beginning of it all, namely to Born's 1926 papers [1, 2], in which he applies Schrödinger's wave equation to a scattering situation. In this application , Born recognized the importance of the quantum equilibrium distribution ρ = |ψ| 2 as the distribution of the random position of the particle after scattering. Curiously though, the application of quantum mechanics to scattering situations comes along with a shift of emphasis on the meaning of the quantum equilibrium distribution. In scattering theory, the crossing probability of spacetime surfaces becomes meaningful. This is naively clear when one pictures the scattering situation as in Fig. 16.1. There are detectors surrounding the scattering potential. The question is: What is the probability that the detector will click? Prior to answering that one must first clarify the following question: Is the time at which a detector clicks a fixed given time, i.e., a time the experimenter can choose? Intuitively and correctly, the answer is no. The time is random. The detector clicks when the particle arrives at the detector surface and crosses it. So both the where and the when of the detection event are random. It is immediately clear that these are questions which Bohmian mechanics is tailored-made to answer, since the notion of where and when the particle crosses a surface is a natural one when trajectories exist. It is another matter to find a closed formula for the crossing probability. A nice formula can be found when one considers the scattering regime, which is a space regime where the particles move essentially along straight lines. There are plenty of books on scattering theory, and we shall not invent scattering theory anew. But true to our intention to provide a clear ontological picture and true to our maxim expressed by Melville: While you take in hand to school others, and to teach them by what name a whale-fish is to be called in our tongue leaving out, through ignorance, the letter H, which almost alone maketh the signification of the word, you deliver that which is not true.-HACKLUYT