The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [122]
Let’s start simply—or, as simply as we can. Consider measuring an electron that has a spiked probability wave, as in Figure 8.9. (Again, don’t worry about how it got this wave shape; just take it as a given.) As noted earlier, to tell the first story of even this measurement process in detail is beyond what we can do. We’d need to use Schrödinger’s math to figure out how the probability wave describing the positions of the huge number of particles that constitute you and your measuring device joins with the probability wave of the electron, and how their union evolves forward in time. My undergraduate students, many of whom are quite able, often struggle to solve Schrödinger’s equation for even a single particle. Between you and the device, there are something like 1027 particles. Working out Schrödinger’s math for that many constituents is virtually impossible. Even so, we understand qualitatively what the math entails. When we measure the electron’s position, we cause a mass particle migration. Some 1024 or so particles in the device’s display, like performers in a crisply choreographed halftime show, race to the appropriate spot so that they collectively spell out “Thirty-fourth Street and Broadway,” while a similar number in my eyes and brain do whatever’s required for me to develop a firm mental grasp of the result. Schrödinger’s math—however impenetrable explicit analysis of it might be when faced with so many particles—describes such a particle shift.
To visualize this transformation at the level of a probability wave is also far beyond reach. In Figure 8.9 and others in that sequence, I used two axes, the north-south and east-west street grid of our model Manhattan, to denote the possible positions of a single particle. The probability wave’s value at each location was denoted by the wave’s height. This already simplifies things because I’ve left out the third axis, the particle’s vertical position (whether it’s on the second floor of Macy’s, or the fifth). Including the vertical would have been awkward, because if I’d used it to denote position, I’d have no axis left for recording the size of the wave. Such are the limitations of a brain and a visual system that evolution has firmly rooted in three spatial dimensions. To properly visualize the probability wave for roughly 1027 particles, I’d need to include three axes for each, allowing me to account mathematically for every possible position each particle could occupy.* Adding even a single vertical axis to Figure 8.9 would have made it difficult to visualize; to contemplate adding a billion billion billion more is, well, silly.
But a mental image of the key ideas is important; so, however imperfect the result, let’s give it a try. In sketching the probability wave for the particles making up you and your device, I’ll abide by the two-axis flat-page limit but will use an unconventional interpretation of what the axes mean. Roughly speaking, I’ll think of each axis as comprising an enormous bundle of axes, tightly grouped together, which will symbolically delineate the possible positions of a similarly enormous number of particles. A wave drawn using these bundled axes will therefore lay out the probabilities for the positions of a huge group of particles. To emphasize the distinction between the many-particle and single-particle situations, I’ll use a glowing outline for the many-particle probability wave, as in Figure 8.13.
Figure 8.13 A schematic depiction of the combined probability wave for all the particles making up you and your measuring device.
The many-particle and single-particle illustrations have some features in common. Just as the spiked wave shape in Figure 8.6 indicates probabilities that are sharply skewed (being almost 100 percent at the spike’s location and almost 0 percent everywhere else), so the peaked wave in Figure 8.13 also denotes sharply skewed probabilities.