39.2 Connecting the Wave and Photon Views

The wave model of light describes the interference pattern in terms of the wave’s intensity $I (x)$ , a continuous-valued function. The photon model describes the interference pattern in terms of the probability $Prob(in δ x at x)$ of detecting a photon. These two models are very different, yet Figure 39.1 shows a clear correlation between the intensity of the wave and the probability of detecting photons. That is, photons are more likely to be detected at those points where the wave intensity is high and less likely to be detected at those points where the wave intensity is low.

The intensity of a wave is $I = P / A$ , the ratio of light power $P$ (joules per second) to the area $A$ on which the light falls. The narrow strip in Figure 39.3a has area $A = H δ x$ . If the light intensity at position $x$ is $I (x)$ , the amount of light energy $E$ falling onto this narrow strip during each second is

E (in δ x at x) = I (x) A = I (x) H δ x = H I (x) δ x

(39.8)

The notation $E (in δ x at x)$ refers to the energy landing on this narrow strip if you place it at position $x$ .

From the photon perspective, energy $E$ is due to the arrival of $N$ photons, each of which has energy $h f$ . The number of photons that arrive in the strip each second is

N (in δ x at x) = \frac{E (in δ x at x)}{h f} = \frac{H}{h f} I (x) δ x

(39.9)

We can then use Equation 39.6, the definition of probability, to write the probability that a photon lands in the narrow strip $δ x$ at position $x$ as

Prob(in δ x at x) = \frac{N (in δ x at x)}{N_{tot}} = \frac{H}{h f N_{tot}} I (x) δ x

(39.10)

Equation 39.10 is a critical link between the wave model and the photon model. It tells us that the probability of detecting a photon is proportional to the intensity of the light at that point and to the width of the detector.

As a final step, recall that the light intensity $I (x)$ is proportional to $| A (x) |^{2}$ , the square of the amplitude function. Consequently,

Prob(in δ x at x) \propto | A (x) |^{2} δ x

(39.11)

where the various constants in Equation 39.10 have all been incorporated into the unspecified proportionality constant of Equation 39.11.

In other words, the probability of detecting a photon at a particular point is directly proportional to the square of the light-wave amplitude function at that point. If the wave amplitude at point A is twice that at point B, then a photon is four times as likely to land in a narrow strip at A as it is to land in an equal-width strip at B.

NOTE Equation 39.11 is the connection between the particle perspective and the wave perspective. It relates the probability of observing a particle-like event—the arrival of a photon—to the amplitude of a continuous, classical wave. This connection will become the basis of how we interpret the results of quantum-physics calculations.