Making sense of quantum physics requires going back and forth between
- imagining a system in physical space, as you and I would see it if it were visible, and
- imagining the same system in the space of possibilities, where quantum wave functions are defined.
One can’t, for instance, understand the implications of quantum interference for what we observe in physical space — such as the strange behavior of the double-slit experiment — without looking at what is happening in the space of possibilities. That’s because the interference is an interference between possibilities.
I’ve explained what these two spaces are in this article. In this one I’ll give you some more tools for understanding how to relate the two spaces, focusing on the simplest illustrative case of two particles moving on a line. The methods used serve as a good introduction to an upcoming post.
Imagine two objects moving on a one-dimensional line — physical space — that we’ll call the x-axis. One, marked in purple, has position x1; the other, in blue, has position x2. Defining some point on the x-axis to be the origin x=0, let’s start with a situation where the first object is five meters to the left of the origin (we’ll write that as x1 =-5,) while the second object is at x2 =-1. This is drawn in physical space in Fig. 1a. In the space of possibilities, however, we have a two dimensional space, with one axis being x1, the possible positions of the purple object, and the other being x2, the possible positions of the blue object. The system of the two objects — i.e., the two objects considered together — is located at the point (x1,x2) = (-5, -1), as shown in Fig 1b; the system is indicated by the star, and its coordinates are indicated by the dashed lines.
The orange dashed line which crosses the plot diagonally is an interesting one. At every point on that line, x1 = x2 — which means that if the system were found there, the two objects would be in the same location, somewhere on the x-axis. In other words, when the system reaches that line, the objects coincide and potentially collide, and something interesting may happen.
In that regard, Figs. 2a and 2b show the moment when the purple object, having traveled to the right from where it was in Fig. 1a, collides with the blue object. Now the star lies precisely on the diagonal line, at the point (-1,-1).
The collision can have an effect on the objects, but exactly what effect depends on the details of the interactions between them. As one example, perhaps the purple object bounces off the blue object, so that the purple object moves back to the left (to smaller x1) while the blue object now starts moving to the right (to larger x2). Such are the motions in physical space. In the space of possibilities, it is different. With the objects moving to smaller x1 and to larger x2, the star moves to the left and up — toward negative x1 and positive x2 . This is shown in Figs. 3a and 3b.
Or, as a different example, suppose the purple object disturbs the blue object but is able to continue on to the right as before. Then the objects are moving to larger x1 and to larger x2 — both of them to the right in physical space — which means the star moves to the right and upward in the space of possibilities, as in Fig. 4.
Fig. 5 shows an animation of this last scenario; notice how things change following the collision in both physical space and in the space of possibilities. Hopefully these examples will help you translate more easily between these two spaces; we’ll be doing this translation many times.
