The heart of quantum mechanics – Probability distribution

The position of a photon on the screen is determined by chance. Let us, for example, consider the red pipe in which 6 out of 150 balls have landed. The probability P(x) in this pipe is 6/150= 4%. The position x of the photon can only be determined with an accuracy of δx, which corresponds to the width of the individual pipes. For any given point x within the pipe, we can only specify a probability density of ρ(x) = P(x)/δx.

In any given location x, the number of balls is equal to N_S times P(x). The ball distribution in the pipes is thus a representation of the probability distribution P(x) of each individual ball. As the ball distribution is proportional to the intensity distribution, it follows that I(x) δx is proportional to N ρ(x) δx equals N P(x) – for both the single slit S and the double slit D.

If we place the two curves of the single and double slit that emerge from the ball distribution on top of one another, we see that additional minima arise once a second slit is opened.

So how do we get from the intensity distribution I(x) to the probability density ρ(x) of an individual photon? As I(x) is proportional to ρ(x), we only need to adjust the scale. In other words, the scale of the y-axis changes, but not the shape of the curves. As we have only observed N/2 photons at the double slit, the corresponding curve on the y-axis is doubled when the scale is adjusted. The areas underneath the curves correspond to the overall probability of 100%, as the photon will impact upon one point or another on the screen.