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Before we knew about quantum physics, humans thought that if we had a system of two small objects, we could always know where they were located — the first at some position x1, the second at some position x2. And after Isaac Newton’s breakthroughs in the late 17th century, we believed that by combining this information with knowledge of the objects’ motions and the forces acting upon them, we could calculate where they would be in the future.

But in our quantum world, this turns out not to be the case. Instead, in Erwin Schrödinger’s 1925 view of quantum physics, our system of two objects has a wave function which, for every possible x1 and x2 that the objects could have, gives us a complex number Ψ(x1, x2). The absolute-value-squared of that number, |Ψ(x1, x2)|2, is proportional to the probability for finding the first object at position x1 and the second at position x2 — if we actually choose to measure their positions right away. If instead we wait, the wave function will change over time, following Schrödinger’s wave equation. The updated wave function’s square will again tell us the probabilities, at that later time, for finding the objects at those particular positions.

The set of all possible object locations x1 and x2 is what I am calling the “space of possibilities” (also known as the “configuration space”), and the wave function Ψ(x1, x2) is a function on that space of possibilities. In fact, the wave function for any system is a function on the space of that system’s possibilities: for any possible arrangement X of the system, the wave function will give us a complex number Ψ(X).

Drawing a wave function can be tricky. I’ve done it in different ways in different contexts. Interpreting a drawing of a wave function can also be tricky. But it’s helpful to learn how to do it. So in today’s post, I’ll give you three different approaches to depicting the wave function for one of the simplest physical systems: a single object moving along a line. In coming weeks, I’ll give you more examples that you can try to interpret. Once you can read a wave function correctly, then you know your understanding of quantum physics has a good foundation.

For now, everything I’ll do today is in the language of 1920s quantum physics, Schrödinger style. But soon we’ll put this same strategy to work on quantum field theory, the modern language of particle physics — and then many things will change. Familiarity with the more commonly discussed 1920s methods will help you appreciate the differences.

Complex Numbers

Before we start drawing pictures, let me remind you of a couple of facts from pre-university math about complex numbers. The fundamental imaginary number is the square root of minus one,

which we can multiply by any real number to get another imaginary number, such as 4i or -17i. A complex number is the sum of a real number and an imaginary number, such as 6 + 4i or 11 – 17i.

More abstractly, a complex number w always takes the form u + i v, where u and v are real numbers. We call u the “real part” of w and we call v the “imaginary part” of w. And just as we can draw a real number using the real number line, we can draw a complex number using a plane, consisting of the real number line combined with the imaginary number line; in Fig. 1 the complex number w is shown as a red dot, with the real part u and imaginary part v marked along the real and imaginary axes.

Figure 1: Two ways of representing the complex number w, either as u + i v or as |w|eiφ .

Fig. 1 shows another way of representing w. The line from the origin to w has length |w|, the absolute value of w, with |w|2 = u2 + v2 by the Pythagorean theorem. Defining φ as the angle between this line and the real axis, and using the following facts

• u = |w| cos φ
• v = |w| sin φ
• eiφ = cos φ + i sin φ

we may write w = |w|eiφ , which indeed equals u + i v .

Terminology: φ is called the “argument” or “phase” of w, and in math is written φ = arg(w).

One Object in One Dimension

We’ll focus today only on a single object moving around on a one-dimensional line. Let’s put the object in a “Gaussian wave-packet state” of the sort I discussed in this post’s Figs. 3 and 4 and this one’s Figs. 6 and 7. In such a state, neither the object’s position nor its momentum [a measure of its motion] is completely definite, but the uncertainty is minimized in the following sense: the product of the uncertainty in the position and the uncertainty in the momentum is as small as Heisenberg’s uncertainty principle allows.

We’ll start with a state in which the uncertainty on the position is large while the uncertainty on the momentum is small, shown below (and shown also in Fig. 3 of this post and Fig. 6 of this post.) To depict this wave function, I am showing its real part Re[Ψ(x)] in red and its imaginary part Im[Ψ(x)] in blue. In addition, I have drawn in black the square of the wave function:

• |Ψ(x)|2 = (Re[Ψ(x)])2 + (Im[Ψ(x)])2

[Note for advanced readers: I have not normalized the wave function.]

Figure 1: For an object in a simple Gaussian wave packet state with near-definite momentum, a depiction of the wave function for that state, showing its real and imaginary parts in red and blue, and its absolute-value squared in black.

But as wave functions become more complicated, this way of doing things isn’t so convenient. Instead, it is sometimes useful to represent the wave function in a different way, in which we plot |Ψ(x)| as a curve whose color reflects the value of φ = arg[Ψ(x)] , the argument of Ψ(x). In Fig. 2, I show the same wave function as in Fig. 1, depicted in this new way.

Figure 2: The same wave function as in Fig. 1; the curve is the absolute value of the wave function, colored according to its argument.

As φ cycles from 0 to π/4 to π/2 to 3π/4 and back to 2π (the same as φ = 0), the color cycles from red to yellow-green to cyan to blue-purple and back to red.

Compare Figs. 1 and 2; its the same information, depicted differently. That the wave function is actually waving is clear in Fig. 1, where the real and imaginary parts have the shape of waves. But it is also represented in Fig. 2, where the cycling through the colors tells us the same thing. In both cases, the waving tells us that the object’s momentum is non-zero, and the steadiness of that waving tells us that the object’s momentum is nearly definite.

Finally, if I’m willing to give up the information about the real and imaginary parts of the wave function, and just want to show the probabilities that are proportional to its squared absolute value, I can sometimes depict the state in a third way. I pick a few spots where the object might be located, and draw the object there using grayscale shading, so that it is black where the probability is large and becomes progressively lighter gray where the probability is smaller, as in Fig. 3.

Figure 3: The same wave function in Figs. 1 and 2, here showing only the probabilities for the object’s location; the darker the grey, the more likely the object is to be found at that location.

Again, compare Fig. 3 to Figs. 1 and 2; they all represent information about the same wave function, although there’s no way to read off the object’s momentum using Fig. 3, so we know where it might be but not where it is going. (One could add arrows to indicate motion, but that only works when the uncertainty in the momentum is small.)

Although this third method is quite intuitive when it works, it often can’t be used (at least, not as I’ve described it here.) It’s often useful when we have just one object to worry about, or if we have multiple objects that are independent of one another. But if they are not independent — if they are correlated, as in a “superposition” [more about that concept soon] — then this technique usually does not work, because you can’t draw where object number 1 is likely to be located without already knowing where object number 2 is located, and vice versa. We’ve already seen examples of such correlations in this post, and we’ll see more in future.

So now we have three representations of the same wave function — or really, two representations of the wave function’s real and imaginary parts, and two representations of its square — which we can potentially mix and match. Each has its merits.

How the Wave Function Changes Over Time

This particular wave function, which has almost definite momentum, does indeed evolve by moving at a nearly constant speed (as one would expect for something with near-definite momentum). It spreads out, but very slowly, because its speed is only slightly uncertain. Here is its evolution using all three representations. (The first was also shown in this post’s Fig. 6.)

I hope that gives your intuition some things to hold onto as we head into more complex situations.

Two More Examples

Below are two simple wave functions for a single object. They differ somewhat from the one we’ve been using in the rest of this post. What do they describe, and how will they evolve with time? Can you guess? I’ll give the full answer tomorrow as an addendum to this post.

Two different wave functions; in each case the curve represents the absolute value |Ψ(x)| and the color represents arg[Ψ(x)], as in Fig. 2. What does each wave function say about the object’s location and momentum, and how will each of them change with time?