Fermat's principle of least time

Fermat’s principle of least time states that light from a point source takes the path to a destination which takes the least total time. If we place an obstacle in a path that is not the shortest, the brightness at the destination will be negligibly affected.

Nearby paths have a small difference in distance traveled, which leads to a phase difference. The distance traveled as a function of path taken is a polynomial with local minimum at the shortest path.

The phases of nearby paths are shown as arrows in the complex plane. Due to the phase differences, the contribution from nearby paths sum to roughly zero.

We can think of the light wave as taking every possible path from source to destination, but all paths other than those near the minimum interfering destructively. In quantum mechanics, we can apply the same reasoning to particles. A particle “exists” where its underlying wave constructively interferes. Otherwise, destructive interference makes the probability of finding the particle near zero. This is applied in angular momentum quantization in the Bohr atom.