39.3 The Wave Function

Now let’s look at the interference of matter. Electrons passing through a double-slit apparatus create the same interference patterns as photons. The pattern is built up electron by electron, but there is no way to predict where any particular electron will be detected. Even so, we can establish the probability of an electron landing in a narrow strip of width $δ x$ by measuring the positions of many individual electrons.

A photograph of an electron interference fringes. — Electrons create interference fringes.

Full Alternative Text

For light, we were able to relate the photon probability density $P (x)$ to the amplitude of an electromagnetic wave. But there is no wave for electrons like electromagnetic waves for light. So how do we find the probability density for electrons? We have reached the point where we must make an inspired leap beyond classical physics. Let us assume that there is some kind of continuous, wave-like function for matter that plays a role analogous to the electromagnetic amplitude function $A (x)$ for light. We will call this function the wave function $ψ (x)$ , where $ψ$ is a lowercase Greek psi. The wave function is a function of position, which is why we write it as $ψ (x)$ .

To connect the wave function to the real world of experimental measurements, we will interpret $ψ (x)$ in terms of the probability of detecting a particle at position $x$ . If a matter particle, such as an electron, is described by the wave function $ψ (x)$ , then the probability $Prob(in δ x at x)$ of finding the particle within a narrow region of width $δ x$ at position $x$ is

Prob(in δ x at x) = | ψ (x) |^{2} δ x = P (x) δ x

(39.14)

That is, the probability density $P (x)$ for finding the particle is

P (x) = | ψ (x) |^{2}

(39.15)

With Equations 39.14 and 39.15, we are defining the wave function $ψ (x)$ to play the same role for material particles that the amplitude function $A (x)$ does for photons. The only difference is that $P (x) = | ψ (x) |^{2}$ is for particles, whereas Equation 39.13 for photons is $P (x) \propto | A (x) |^{2}$ . The difference is that the electromagnetic field amplitude $A (x)$ had previously been defined through the laws of electricity and magnetism. $| A (x) |^{2}$ is proportional to the probability density for finding a photon, but it is not directly the probability density. In contrast, we do not have any preexisting definition for the wave function $ψ (x)$ . Thus we are free to define $ψ (x)$ so that $| ψ (x) |^{2}$ is exactly the probability density. That is why we used $=$ rather than $\propto$ in Equation 39.15.

FIGURE 39.5 shows the double-slit experiment with electrons. This time we will work backward. From the observed distribution of electrons, which represents the probabilities of their landing in any particular location, we can deduce that $| ψ (x) |^{2}$ has alternating maxima and zeros. The oscillatory wave function $ψ (x)$ is the square root at each point of $| ψ (x) |^{2}$ . Notice the very close analogy with the amplitude function $A (x)$ in Figure 39.1.

The figure consists of four parts that illustrate the double-slit experiment with electrons.

FIGURE 39.5 Full Alternative Text

NOTE $| ψ (x) |^{2}$ is uniquely determined by the data, but the wave function $ψ (x)$ is not unique. The alternative wave function $ψ' (x) = - ψ (x)$ —an upside-down version of the graph in Figure 39.5—would be equally acceptable.

FIGURE 39.6 is a different example of a wave function. After squaring it at each point, as shown in the bottom half of the figure, we see that this wave function represents a particle most likely to be detected very near $x = - b$ or $x = + b$ . These are the points where $| ψ (x) |^{2}$ is a maximum. There is zero likelihood of finding the particle right in the center. The particle is more likely to be detected at some positions than at others, but we cannot predict what its exact location will be at any given time.

The figure consists of two parts that illustrate the wavefunction versus position and the probability density versus position graphs.

FIGURE 39.6 Full Alternative Text

NOTE One of the difficulties in learning to use the concept of a wave function is coming to grips with the fact that there is no “thing” that is waving. There is no disturbance associated with a physical medium. The wave function $ψ (x)$ is simply a wave-like function (i.e., it oscillates between positive and negative values) that can be used to make probabilistic predictions about atomic particles.

39.3 The Wave Function

FIGURE 39.5 The double-slit experiment with electrons.

FIGURE 39.6 The square of the wave function is the probability density for detecting the electron at position $x$ .