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.
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.]
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.
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.
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.
How to Interpret the First Example
This figure represents a state of completely unknown position (that is why the absolute value of the wave function is the same at every location, implying the square of the absolute value, which gives the probability of finding the object at a particular point, is also the same at all points.) It also is a state of definite momentum (which is why the argument of the wave function, shown in color, changes simply and cyclically across the curve.) These features are reflected in the three different representations of the wave function’s behavior over time, which show steady motion to the left, always with unknown position; in the third representation, we see that the since the object may be anywhere in space at any given time, the motion across space is not visible.
How to Interpret the Second Example
This figure represents a state of somewhat known position (that is why the absolute value of the wave function peaks at that location) but the color of the curve, representing the argument of the wave function, is the same everywhere, which indicates that the momentum, while indefinite, is close to zero. The three different representations of the wave function’s behavior over time show how the indefinite motion leads the particle have to have equal probability to go either left or right, though on average it goes nowhere. This causing the wave function to spread out symmetrically as we become more uncertain about where the particle is located. In the second representation, note that the cycling of the colors is reversed on the right side compared to the left side; that’s because if the particle is to the right of the initial peak, it is most likely moving to the right, while the reverse is true if it is to the left of the initial peak.
15 Responses
Well, I did manage to get the first puzzle correct; easy enough since you had given that explicit example in a previous post. The uniform color of the second puzzle confused me. I recognize what ought to have been “obvious.” Energy relates to the cycle frequency, and, the asymmetry between position and momentum involves the relationship of momentum to energy. So negligible momentum ought to correspond with uniform coloration in the representation.
Two questions:
Since elementary wavicle masses are described using energy equivalents, can one mentally associate those energies with the “definite momentum” / “unknown position” single wavicle representations?
It seems that the “Lagrangian” for quantum field theory is somewhat different from how one understands Lagrangian mechanics. One speaks of a “free Lagrangian,” but, its usage may not have classical parameters. In particular, it may not rely upon spatial (and temporal?) parameters. One mention of this which I found suggested that it “acts” on the wave equation. I don’t want to drag you too far of topic, but is this naively correct? Does the uncertainty principle for position and momentum force one to understand the “Lagrangian” as an operator acting on a wave equation? Or, does reference to a “Langrangian” suggest exactly what it means for Lagrangian mechanics?
If too much, ignore the second question.
lol. Energy equivalents for wavicle masses are intended to correspond with intrinsic mass. It should have nothing to do with momentum intended to correspond with motion energy. Sorry about the first question.
Your questions are good ones, but you are running ahead of me. So far this week we are still doing 1920s quantum physics, and there are no wavicles yet, only particles described by wave functions. Let me get to quantum field theory later this week and next week, and then wavicles will appear. At that point I think your questions will start to receive answers, and so you’ll probably want to rephrase them at that time.
How does nature create massive particles, fermions, from massless waves, bosons? Do you have a post(s) explaining how symmetry breaking, (Higgs field) creates mass or gives mass to electromagnetic waves?
How does the Daric fermion fit in with the Higgs field?
Jonathan Gorard, a researcher at Princeton originally from Cambridge, has been working on micro black holes as the “building blocks” of all matter. So, instead of the usual mass/spring/damper cell, replace it with tiny black holes and run models to see if energy can be trapped and create fermions, particles.
Are you aware of any such research and could you direct me to any. Gorard is just a young man but has there been such research in the past?
I am sure Gorard will have difficulty with obtaining photons as massless black holes.
Unless you can “trapped” photons creating a momentum matrix that is the structure black holes. Rest mass is not the only way to create black holes and rest mass is not the only way to create gravity. As long as the structure can curve spacetime gravity will ensure and this could be the mechanism of the strong force to trap the photons and create quarks.
Maybe the nucleus of black holes is completely empty space at 0 deg Kelvin and all the mass is at the horizon in the form of a highly dense, very small wavelength, photons. (momentum matrix).
How does energy, photons, get trapped?
dear matt , first thank you for your blogs whose value is invaluable. Question: when analysing the double slit experiment, the probability of finding an electron at a certain point in the screen is obtained by calculating the wave function for the field to obtain the shape of it and from there calculate the position of the electron?
thank you
In quantum mechanics of the 1920s, the wave function is describing the electron itself, not the electron field of which the electron is a ripple. It is a function (usually called Psi) of the electron’s position (x,y,z). The probability of finding the electron at a point (x,y,z) on the screen is proportional to |Psi(x,y,z)|^2 .
In quantum field theory, the story is vastly more complicated, and to my knowledge no one has ever formulated the double slit experiment in this language, though this is something I am looking into. There may be simpler variants of the double-slit experiment that can be worked out in detail in quantum field theory. In that case, yes, the wave function has to tell you… well, not the shape of the field really, but *the probability of the field taking on one shape or another.* From this one can in principle work out the probabilities for the energy and charge of the state containing one electron to be stored in one region or another. But it’s not an easy task. We have easier questions to address first.
Thanks Matt, and one more curiosity: you told us that the wave function is in principle one for the whole universe, but when you calculate the probability of an electron being in a certain place is there a method to simplify the wave function and not consider for example the influence of vega on the electron in your lab? Thank you very much
Roughly: It is the experimenter’s job to assure that the experimental system is sufficiently isolated during the process of the experiment, with the following effects: (1) the part of the wave function that describes the experimenter’s reality approximately factors into two pieces, a wave function for the experimental system times a wave function for everything else, and (2) the same is true for the Schrodinger equation (in jargon, the evolution operator factors into two pieces, which means the Hamiltonian breaks into a sum of two pieces) for the duration of the experiment, so that (3) this factorization holds for the duration of the entire experiment.
But there are many subtle issues with experimental measurements, and I absolutely do not want to discuss them yet; it is far too early.
Very helpful post. One request, could you clarify what each axis represents in your diagrams? From the previous posts it seemed that x-axis is location but y-axis is the Wavicle’s probability of either being found at a particular position or of having a certain momentum (unclear which one in which graphs). Thank you.
Thanks for the question. In all three graphs, the horizontal axis is the x-axis, the possible location of the object described by the wave function. The vertical axis differs from diagram to diagram. In the first diagram with the red, blue and black curves, it gives the actual value of the real part, the imaginary part, and the squared absolute-value of the wave function Psi(x). In the second with the multi-colored curve, it gives the absolute value of Psi(x). In the third there is no vertical axis required.
None of the axes refer to momentum; the possible behavior of the particle’s momentum has to be inferred from the waving shape of the real part and imaginary part curves in the first image, or from the varying colors in the second image. The momentum dependence cannot be seen in the third image.
You are a wonderful educator and communicator, Dr. Strassler. In fact, you are a Feynman-esque communicator. Thank you very much for your book, and these posts, too. Due to my limited intellect, I’d be fibbing to suggest that I understand everything. Do know, however, that in my case at least, you are reaching a cohort of folks who are passionate about science but who lack formal education in these topics. We very much want to learn, and you are providing that education. Again, thank you.
Hi Matt, thanks for your efforts to explain these things. Just as feedback, I found the Fig 3 representation rather confusing at first, since on a first impression it seemed to be showing something akin to a two-slit interference pattern, implying that the particle could only be at the dots and not in-between them. Perhaps making it a solid bar, of varying shading, might avoid giving this impression?
It’s a fair concern, and you could do the solid bar in some cases, including this one. (And it would give you exactly the same information as the black curve in Fig. 1.) But in more complex situations, often a solid bar won’t work, so it’s good to know how to read this representation. In fact this representation will be crucial when we get to quantum field theory.