The Hidden Reality_ Parallel Universes and the Deep Laws of the Cosmos - Brian Greene [118]
Figure 8.8 (a) An initial probability wave shape can be decomposed as the union of two simpler shapes. (b) The evolution of the initial probability wave can be reproduced by evolving the simpler pieces and combining the results.
Figure 8.9 An electron’s probability wave, at a given moment, is spiked at Thirty-fourth Street and Broadway. A measurement of the electron’s position, at that moment, confirms that it is located where its wave is spiked.
But linearity not only makes quantum calculations manageable; it’s also at the heart of the theory’s difficulty in explaining what happens during a measurement. This is best understood by applying linearity to the act of measurement itself.
Imagine you’re an experimentalist with great nostalgia for your childhood in New York, so you’re measuring the positions of electrons that you inject into a miniature tabletop model of the city. You start your experiments with one electron whose probability wave has a particularly simple shape—it’s nice and spiked, as in Figure 8.9, indicating that with essentially 100 percent probability the electron is momentarily sitting at the corner of Thirty-fourth Street and Broadway. (Don’t worry about how the electron got this wave shape; just take it as a given.)* If at that very moment you measure the electron’s position with a well-made piece of equipment, the result should be accurate—the device’s readout should say “Thirty-fourth Street and Broadway.” Indeed, if you do this experiment, that’s just what happens, as in Figure 8.9.
It would be extraordinarily complicated to work out how Schrödinger’s equation entwines the probability wave of the electron with that of the trillion trillion or so atoms that make up the measuring device, coaxing a collection of the latter to arrange themselves in the readout to spell “Thirty-fourth Street and Broadway,” but whoever designed the device has done the heavy lifting for us. It’s been engineered so that its interaction with such an electron causes the readout to display the single definite position where, at that moment, the electron is located. If the device did anything else in this situation, we’d be wise to exchange it for a new one that functions properly. And, of course, Macy’s notwithstanding, there’s nothing special about Thirty-fourth and Broadway; if we do the same experiment with the electron’s probability wave spiked at the Hayden Planetarium near Eighty-first and Central Park West, or at Bill Clinton’s office on 125th near Lenox Avenue, the device’s readout will return these locations.
Let’s now consider a slightly more complicated wave shape, as in Figure 8.10. This probability wave indicates that, at the given moment, there are two places the electron might be found—Strawberry Fields, the John Lennon memorial in Central Park, and Grant’s Tomb in Riverside Park. (The electron’s in one of its dark moods.) If we measure the electron’s position but, in opposition to Bohr and in keeping with the most refined experiments, assume that Schrödinger’s equation continues to apply—to the electron, to the particles in the measuring device, to everything—what will the device’s output read? Linearity is the key to the answer. We know what happens when we measure spiked waves individually. Schrödinger’s equation causes the device’s display to spell out the spike’s location, as in Figure 8.9. Linearity then tells us that to