39.1.3 A Photon Analysis of Interference

Now let’s look at the double-slit results from a photon perspective. We know, from experimental evidence, that the interference pattern is built up photon by photon. The bottom portion of Figure 39.1 shows the pattern made on a detector after the arrival of the first few dozen photons. It is clearly a double-slit interference pattern, but it’s made, rather like a newspaper photograph, by piling up dots in some places but not others.

The arrival position of any particular photon is unpredictable. That is, nothing about how the experiment is set up or conducted allows us to predict exactly where the dot of an individual photon will appear on the detector. Yet there is clearly an overall pattern. There are some positions at which a photon is more likely to be detected, other positions at which it is less likely to be found.

If we record the arrival positions of many thousands of photons, we will be able to determine the probability that a photon will be detected at any given location. If 50 out of 50,000 photons land in one small area of the screen, then each photon has a probability of $50 / 50, 000 = 0.001 = 0.1 %$ of being detected there. The probability will be zero at the interference minima because no photons at all arrive at those points. Similarly, the probability will be a maximum at the interference maxima.

FIGURE 39.3a shows a narrow strip with width $δ x$ and height $H$ . (We will assume that $δ x$ is very small in comparison with the fringe spacing, so the light’s intensity over $δ x$ is very nearly constant.) Think of this strip as a very narrow detector that can detect and count the photons landing on it. Suppose we place the narrow strip at position $x$ . We’ll use the notation $N (in δ x at x)$ to indicate the number of photons that hit the detector at this position. The value of $N (in δ x at x)$ varies from point to point. $N (in δ x at x)$ is large if $x$ happens to be near the center of a bright fringe; $N (in δ x at x)$ is small if $x$ is in a dark fringe.

The figure labeled ("a") illustrates the interference pattern of photons and a narrow strip placed on the pattern.

The figure labeled (b) illustrates the interference pattern of photons and two narrow strips placed on the pattern.

Full Alternative Text

Suppose $N_{tot}$ photons are fired at the slits. The probability that any one photon ends up in the strip at position $x$ is

Prob(in δ x at x) = lim_{N_{tot} \to \infty} \frac{N (in δ x at x)}{N_{tot}}

(39.6)

As FIGURE 39.3b shows, Equation 39.6 is an empirical method for determining the probability of the photons hitting a particular spot on the detector.

Alternatively, suppose we can calculate the probabilities from a theory. In that case, the expected value for the number of photons landing in the narrow strip when it is at position $x$ is

N (in δ x at x) = N \times Prob(in δ x at x)

(39.7)

We cannot predict what any individual photon will do, but we can predict the fraction of the photons that should land in this little region of space. $Prob(in δ x at x)$ is the probability that it will happen.