Probability of arriving at a point

The model I am describing on the following pages seems very strange. However, we shall see that it is a very powerful model as it can be used to explain so many things that other models can't. When we get very small, the world is a very weird place!

A photon is at point A at some particular time. At some time in the future it might be at point B (or it might be somewhere else). What is the probability that the photon will arrive at B?

Imagine the photon has a little clock with one hand. We call this a phasor. The phasor arrow spins round at the same frequency as the frequency of the photon.

Choose a path from A to B (a straight line is the most obvious). If the phasor is pointing up at the start, what direction does it point in after the photon has traveled the chosen path? ( at 3 x 10⁸ m/s )

There are many possible paths from A to B. In fact there are an infinite number. Repeat the process above for every possible path, each time making a note of where the phasor arrow ends up after the journey. We will end up with lots and lots of arrows.

The next step is to add together all the arrows to find the resultant. The size of this arrow represents the amplitude of the "probability density function" at point B. ( don't worry too much about what this means ) If we square the size of the resultant arrow we get a value which is proportional to the probability we are looking for.

Now adding together an infinite number of arrows will take quite a bit of time! Fortunately we need only consider the arrows for the most obvious paths. The most obvious paths will be quite similar in the time they take so their phasor arrows point in similar directions. Less obvious paths take random amounts of time so their phasor arrows will cancel each other out.

Probability is proportional to (magnitude of resultant)²