We need one last definition. Recall that the mass of a wire or string of a length $L$ can be expressed in terms of the linear mass density $\mu$ as $m=\mu L$. Similarly, the charge along a length $L$ of wire can be expressed in terms of the linear charge density $\text{λ}$ as $Q=\text{λ}L$. If the length had been very short—in which case we might have denoted it as $\delta x$—and if the density varied from point to point, we could have written

where $\mu \left(x\right)$ and $\text{λ}\left(x\right)$ are the linear densities at position $x$. Writing the mass and charge this way separates the role of the density from the role of the small length $\delta x$.

Equation 39.11 looks similar. Using the mass and charge densities as analogies, as shown in FIGURE 39.4, we can define the probability density $P\left(x\right)$ such that

FIGURE 39.4 The probability density is analogous to the linear mass density.

In one dimension, probability density has SI units of $\text{m}{}^{-1}$. Thus the probability density multiplied by a length, as in Equation 39.12, yields a dimensionless probability.

By comparing Equation 39.12 to Equation 39.11, you can see that the photon probability density is directly proportional to the square of the light-wave amplitude:

The probability density, unlike the probability itself, is independent of the width $\delta x$ and depends on only the amplitude function.

Although we were inspired by the double-slit experiment, nothing in our analysis actually depends on the double-slit geometry. Consequently, Equation 39.13 is quite general. It says that for any experiment in which we detect photons, the probability density for detecting a photon is directly proportional to the square of the amplitude function of the corresponding electromagnetic wave. We now have an explicit connection between the wave-like and the particle-like properties of the light.