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By Root 2086 0

that line would be. The reality is, he couldn’t. With each passing year, experimenters confirm that Schrödinger’s equation works, without modification, for increasingly large collections of particles, and there’s every reason to believe that it works for collections as hefty as those making up you and me and everything else. Like floodwaters slowly rising from your basement, rushing into your living room, and threatening to engulf your attic, the mathematics of quantum mechanics has steadily spilled beyond the atomic domain and has succeeded on ever-larger scales.

So the way to think about the problem is this. You and I and computers and bacteria and viruses and everything else material are made of molecules and atoms, which are themselves composed of particles like electrons and quarks. Schrödinger’s equation works for electrons and quarks, and all evidence points to its working for things made of these constituents, regardless of the number of particles involved. This means that Schrödinger’s equation should continue to apply during a measurement. After all, a measurement is just one collection of particles (the person, the equipment, the computer …) coming into contact with another (the particle or particles being measured). But if that’s the case, if Schrödinger’s math refuses to bow down, then Bohr is in trouble. Schrödinger’s equation doesn’t allow waves to collapse. An essential element of the Copenhagen approach would therefore be undermined.

So the third question is this: If the reasoning just recounted is right and probability waves don’t collapse, how do we pass from the range of possible outcomes that exist before a measurement to the single outcome the measurement reveals? Or to put it in more general terms, what happens to a probability wave during a measurement that allows a familiar, definite, unique reality to take hold?

Everett pursued this question in his Princeton doctoral dissertation and came to an unforeseen conclusion.

Linearity and Its Discontents

To understand Everett’s path of discovery, you need to know a little more about Schrödinger’s equation. I’ve emphasized that it doesn’t allow probability waves to suddenly collapse. But why not? And what does it allow? Let’s get a feel for how Schrödinger’s math guides a probability wave as it evolves through time.

This is fairly straightforward, because Schrödinger’s is one of the simplest kinds of mathematical equations, characterized by a property known as linearity—a mathematical embodiment of the whole being the sum of its parts. To see what this means, imagine that the shape in Figure 8.7a is the probability wave at noon for a given electron (for visual clarity, I will use a probability wave that depends on location in the one dimension represented by the horizontal axis, but the ideas are general). We can use Schrödinger’s equation to follow the evolution of this wave forward in time, yielding its shape at, say, one p.m., schematically illustrated in Figure 8.7b. Now notice the following. You can decompose the initial wave shape in Figure 8.7a into two simpler pieces, as in Figure 8.8a; if you combine the two waves in the figure, adding their values point by point, you recover the original wave shape. The linearity of Schrödinger’s equation means that you can use it on each piece in Figure 8.8a separately, determining what each wave fragment will look like at one p.m., and then combine the results as in Figure 8.8b to recover the full result shown in Figure 8.7b. And there’s nothing sacred about decomposition into two pieces; you can break the initial shape up into any number of pieces, evolve each separately, and combine the results to get the final wave shape.

Figure 8.7 (a) An initial probability wave shape at one moment evolves via Schrödinger’s equation to a different shape (b) at a later time.

This may sound like a mere technical nicety, but linearity is an extraordinarily powerful mathematical feature. It allows for an all-important divide-and-conquer strategy. If an initial wave shape is complicated, you are free to divide it