# Probability and probability amplitude.

1. Feb 2, 2016

Quoted from reddit, an explanation for quantum superposition :

By analogy, you could say that the traveling electron is in a superposition of paths. Some are direct (close to what is called the "classical path" -- the path that a particle would take according to classical mechanics). Some are indirect, going to the moon and back before arriving at its destination.

But, when you measure the electron, you cause the superposition to collapse, just as before, and the electron is forced to take on a single path.

Now, the crazy thing is, all of the paths have the same probability. The probability is a positive number between 0 and 1 that says how probable it is that a certain path will be chosen. A probability of 0.5 means that the path has a 50% chance of being chosen. The crazy thing I'm saying here, though, is that all of the paths have the same probability, which sounds crazy.

Quantum mechanics has a weird quantity, though, called the probability amplitude. The amplitude is like the square root of the probability. Since it's a square root, it need not be positive, and it doesn't even have to be real. A probability amplitude can be any complex number, as long as the square of its modulus is between 0 and 1.

And it turns out that, even though all of the paths have an equal probability, they do not all have the same probability amplitude. Specifically, they differ by a phase. When you add up the probability amplitudes for each path, you find two things:

a) The paths that are close to the "classical" path tend to re-enforce one another. b) The paths that stray far from the classical path tend to cancel each other out.

So in the end, the probability of finding the particle very far away from the classical path is very small. Small particles like electrons can stray a little bit, but it's not too far before the probability becomes too small. Heavier objects, like a grain of sand, can't stray by any measurable amount before the probability becomes near-zero, which is why those objects seem to behave in a classical way.

Someone pointed out that the one that stays the same within all possible paths should be the probability rather than the probability amplitude. By the relationship of (probability amplitude)² = (probability), if the probability amplitudes are equal within all possible paths, how does the probabilty cancellation mechanism (within similar paths) work? How can equal values cancel each other out and produce different values for their squares (probability)?

2. Feb 2, 2016

### zonde

All probability amplitudes are summed (as vectors) and only one final amplitude is then squared. Amplitudes cancel each other out when they have opposite phases.
I can recommend you Richard Feynman's book "QED: The strange theory of light and matter". It's accurate but it describes the the math in an alternative layman friendly way.

3. Feb 2, 2016