6: Amplitudes and Probabilities
- Page ID
- 56758
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The classical physics represented by Newton’s Laws is deterministic. The equations tell you that if a particle is here and its speed is exactly just this much, then it will be over here and moving this fast later. It gives us a picture of a clockwork universe, where everything future possible measurement is completely determined by the current state of the system.1
In quantum physics, as we have seen, this is not the case. If you have an electron whose spin has been measured to be pointing along the \(\ +z\) axis, then the best statement you can make about the \(\ x\) projection of the electron’s spin angular momentum is a probabilistic one: there is a 50% chance you’ll measure \(\ x\) spin along the \(\ +x\) direction, and a 50% chance you’ll measure \(\ x\) spin along the \(\ −x\) direction. What’s more, this probabilistic nature is not simply due to our lack of knowledge. Statistics is an entire branch of mathematics used to estimate what we know and determine our confidence in what we know when we have imperfect information. While statistics does apply to quantum mechanics, most of the time statistics is employed in practice the probabilities come not from a fundamental probability, but from lack of perfect knowledge about the state of the system, or because the system itself contains individuals who vary. In quantum mechanics, this probabilistic nature runs more deeply, even though each and every electron is identical. Whereas in classical physics, we may never be able to make perfect measurements, but the theory underneath them is able to presume perfectly determined quantities. In quantum mechanics, the theory needs to be able to handle the calculation and propagation of these probabilities.
1In fact, chaos theory has shown us that nonlinearities even in classical physics place a limit on the predictability of those systems. However, the laws themselves are deterministic.
- 6.1: Complex Numbers
- Before we begin, however, we need briefly to review complex numbers. Complex numbers are intrinsic to quantum mechanics, and indeed the entire theory wouldn’t work if we didn’t use complex numbers as part of it.
- 6.2: Amplitudes
- What makes quantum mechanics so different from the propagation of uncertainty in classical physics is that it’s not directly the probabilities that propagate, but rather these things called amplitudes. Suppose you constructed something like a SternGerlach machine, and propagated the system through it using the rules of classical physics.