some key concepts like payloads and detachable stages. Toy models can be fun and useful. Similarly, in trying to understand complicated concepts or equations, it's good to have toy models. A good toy model captures some sense of the real thing but is small enough that we can wrap our minds around it.
In the next few paragraphs I’ll show you a toy model of quantum reality. It’s a vastly simplified model, but I think it’s just intricate enough to suggest the vastness of quantum reality. The main point is that quantum reality is REALLY, REALLY BIG. We’ll build up a toy model that describes social life among the spins of just five particles—and discover that it fills out a space of thirty-two dimensions.
Start with one quantum particle that has a minimal unit of spin. We abstract away—that is, ignore—all its other properties. The resulting object is what is called a quantum bit, or qubit. (For sophisticated: A cold electron trapped in a definite spatial state, say by appropriate electric fields, is effectively a qubit.) The spin of a qubit can point in different directions. We’ll write
|↑〉
for the state in which the spin of the qubit is definitely up, and
↓〉
for the state in which the spin is definitely down.
The qubit can also be in states where the spin points sideways, and that's were the fun begins. It's exactly here, at this juncture, that the central weirdness of quantum mechanics comes into play.
The sideways-pointing states are not new, independent states. These sideways-pointers, and all other states of the qubit, are combinations of the states |uparrowrangle and |downarrowrangle we already have.
Specifically, for instance, the east-spinning state is
→rangle = 1/(root(2))|uparrowrangle+1/(root(2))|downarrowrangle
The state where the spin definitely points east is an equal mixture of north and south. If you measure the spin in the horizontal direction, you'll always find that it points east. But if you measure the spin in the vertical direction, you're equally likely to find that it points north or south. That's the meaning of this strange equation. In more detail, the rule for computing the probability of finding a given result (up or down) when you measure the spin in the vertical direction is that you take the square of the number that multiplies the state with that result. Here, for example, the number 1/root(2) multiplies the spin up state, so the probability of finding spin up is (1/root(2))2 = 1/2.
This example illustrates, in miniature form, the ingredients that enter into the description of a physical system according to quantum theory. The state of the system is described by its wave function. You’ve just seen the wave functions for three specific states. The wave function consists of a set of numbers multiplying each possible configuration of the object being described. The number multiplying a
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