Probability of taking a path.
As well as calculating the probability that a photon will arrive at a particular place we can also look at the probability of what path it will take to get there. We get some interesting results when we do this.
Consider a photon traveling from A to B, reflecting off a mirror during its journey. Here are a number of possible paths.
I have also included the phasor arrows that we would get for the end of each path. Notice that the arrows for the paths with waypoints around the middle of the mirror add together. Phasor arrows for paths near the edges of the mirror are all over the place.
We can see why if we study a graph of the trip time for each path.
| Paths near the ends of the mirror | These take the longest time. Also, the time they take differs the most (the graph is steepest). This is why the phasor arrows for these paths point in very different directions. |
| Paths near the middle of the mirror | These take the least time. Also, the time they take differs very little. This is why their phasor arrows point in similar directions. |
Conclusion - paths near the middle of the mirror are the most probable.
In general, paths that take the least amount of time are the most probable.
At last, we know why light travels in straight lines to get from one point to another (probably!)
Here's an example of where taking the quickest path is the best option.
Does this situation remind you of anything to do with light?