Of Particular Significance

Yes, Standing Waves Can Exist Without Walls

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 03/19/2024

After my post last week about familiar and unfamiliar standing waves — the former famous from musical instruments, the latter almost unknown except to physicists (see Chapter 17 of the book) — I got a number of questions. Quite a few took the form, “Surely you’re joking, Mr. Strassler! Obviously, if you have a standing wave in a box, and you remove the box, it will quickly disintegrate into traveling waves that move in opposite directions! There is no standing wave without a container.”

Well, I’m not joking. These waves are unfamiliar, sure, to the point that they violate what some readers may have learned elsewhere about standing waves. Today I’ll show you animations to prove it.

When a Standing Wave Loses Its Box

The animations below show familiar and unfamiliar standing waves inside small boxes (indicated in orange). The boxes are then removed, leaving the waves to expand into larger boxes. What happens next is determined by straightforward math; if you’re interested in the math, see the end of this post.

Though the waves start out with the same shape, they have different vibrational frequencies; the unfamiliar wave vibrates ten times faster. Each wave vibrates in place until the small box is taken away. Then the familiar wave instantly turns into two traveling waves that move in opposite directions at considerable speed, quickly reaching and reflecting off the walls of the new box. Nothing of the original standing wave survives, except that its ghost is recreated for a moment when the two traveling waves intersect.

The unfamiliar wave, however, has other plans. It continues to vibrates at the center of the box for quite a while, maintaining its coherence and only slowly spreading out. As the traveling waves from the familiar standing wave are hitting the walls of the outer box, the unfamiliar wave is still just barely tickling those walls. Only at the very end of the animation is this wave even responding of the presence of the box.

A familiar standing wave vibrates within a small box. When the small box is removed, the wave decomposes into traveling waves that reflect off the walls of the larger box. Animation made using Mathematica.
Same as at left, but for an unfamiliar standing wave. For the same shape, it initially has a higher frequency, and it spreads much more slowly when the smaller box is removed. Animation made using Mathematica.

To fully appreciate this effect, imagine if I’d made the ratio between the two waves’ frequencies one thousand instead of ten. Then the unfamiliar wave would have taken a thousand times longer than the familiar wave to completely spread across its box. However, I didn’t think you’d want to watch such boring animations, so I chose a relatively small frequency ratio.

Now let’s put in some actual numbers, to appreciate how impressive this becomes when applied to real particles.

Photons and Electrons in Boxes

Let’s take an empty box (having removed the air inside it) whose sides are a tenth of a meter (about three inches) long. If I put a standing-wave photon (a particle of light) into it, that wave will have a frequency of 3 billion cycles per second. That puts it in the microwave range.

If I then release the photon into a box a full meter across, the photon’s wave will turn into traveling pulses, as my first animation showed. Moving at the speed of light, the pulses will reach the walls of the larger box in about 1.5 billionths of a second (1.5 nanoseconds.) This is what we are taught to expect: without the walls, the standing wave can’t survive.

But if I put a standing-wave electron in a box a tenth of a meter across, it will have a frequency of 800 billion billion cycles per second. That’s not a typo — I really do mean 800 Billion-Billion, which is enormously faster vibration than for a microwave photon.

Correspondingly, when the electron is released from its original box to a larger one a meter across, it will simply remain vibrating at the center of the box, in an extreme version of the second animation. The edges of the electron’s wave will expand, but no faster than a few millimeters per second. The amount of time it will take for its vibrating edges to reach out to the edges of the new box will be well over a minute.

From the electron’s perspective, vibrating once every billionth of a trillionth of a second, this spreading takes almost forever. It’s a long time even for a human physicist. Most experiments on freely floating electrons, including those that measure an electron’s rest mass, take much less than a second. For many such measurements, the fact that an unconstrained electron is gradually spreading is of little practical importance.

Atoms are Boxes Too

Thus standing waves can exist without walls for a quite a while, if they are sufficiently broad to start with. The word broad is important here. From smaller boxes, or from atoms, the spreading is more rapid; an electron liberated from a tiny hydrogen atom can grow to the size of a room in the blink of an eye. The larger the electron’s initial container, the wider the electron’s initial standing wave will be, and the more slowly it will spread.

This pattern might remind you of the famous and infamous uncertainty principle. And well it should.

For the math behind this, read this article (the fourth of this series); the familiar waves satisfy what I called Class 0 wave equations, while the unfamiliar ones satisfy Class 1 wave equations. If you read to the end of the series, you’ll see the direct connection of these two classes of waves with photons and electrons, and more generally with particles of zero and non-zero rest mass.

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10 Responses

  1. It’s seems hard to believe that (clasically at least) the two pictures wouldn’t just be time-scaled copies of eachother’s movie. What breaks the symmetry?

  2. I think the issue likely to trip people is the question of just HOW the wave is… waving. Math is fine and dandy, but humans do like an intuitive idea of something. You can go on about the math of Class 0 wave equations,but people are going to fall back on their intuition of a ball on a spring or wave on a rope. There’s an idea that the rope pulls back when displaced, and idea of how the forces involved cause the cycle of peak-flat-trough. You can manipulate a model with your hands,even FEEL the way the wave propagates.

    But the Class 1 waves… how do they work? What makes the maximum peak displacement begin to lessen? What forces are acting on this class 1 rope to make it behave thus? A familiar standing wave, perfectly sensible,the rope pinned to the walls,the forces involved. But the unfamiliar? WHY doesn’t it just breakup? What is holding it together and allowing it to spread so slowly? It’s hard for the mind to grasp that concept.

    1. I agree; that’s why two explicit examples of how Class 1 waves work already appear in the book’s chapter 20.2, as part of showing how the Higgs field gives mass to particles. [They are illustrated in Figures 48 (see also Figure 47) and in Figure 50, which are already online here, though you may need the explanation in the book to see how they’re connected with the Higgs field.]

      I plan to give more examples here, but each one requires a substantial amount of preparatory work on my part to design animations for them, and I simply haven’t had time yet.

  3. It works for me, reminding us to use our understanding of a photon and the electromagnetic field to then understand the electron and its electron field, bearing in mind that there are also fundamental differences; which you’ve already mentioned in various articles. For more technically ambitious interested readers, Matt wrote an article on the photon, with comments still left open: https://profmattstrassler.com/technical-zone/the-triplet-the-w-mass/5-two-gauge-symmetries/

  4. Just a quick thanks for your posts. I’ve been following you for a number of years and really appreciate the insight you give

  5. Very interesting, and thanks for addressing the wall issue! That’s a lovely simulation of a photon-based boxed-in qubit.

    I note that for a single-photon resonator, the two waves you show bouncing back and forth are necessarily entangled states: Only one can hold the photon under a detection test, yet both give a 50% probability of holding the electron. This issue of “existence entanglement” was the very one that got Einstein in trouble at the Solvay meetings when he publicly pointed out that Born’s then-new model seemed to allow for multiple particles to pop up at multiple places, violating energy conservation.

    Do you agree that single-photon resonators act as directional qubits when released? Your nice simulation certainly seems to suggest that when combined with energy quantization.

    You completely lost me on your second part. Why is your simulator violating Schrödinger’s wave equation? The wave scaling is noise. Is your simulator introducing “decoherence” (detection, much as in a cloud chamber) by default? That’s bad programming, if so.

    Experimentally, and ignoring the easier case of metallic band charge-splitting (e.g., Peierls transition, polyacetylene fractional charge solitons), it would be incredibly difficult, but not impossible, to get an electron to split into two well-defined, diverging probability nodes. But particle splitting is a “thing” in the literature, especially in some more exotic solid state scenarios.

    1. Terry — I don’t understand most of this comment. The post is what the post says it is.

      I am not showing a simulation of a photon-based boxed-in qubit. There is no entanglement of states. These are not wave functions. Nothing here is not related to existence entanglement. The first animation is simply a solution to the Maxwell equation, the equation that the electromagnetic field satisfies, as do its quanta (individual photons).

      As for the second animation, thank goodness my simulator violates Schrödinger’s wave equation! As Schrödinger was fully aware, his equation is an approximation, one that (1) assumes that all motions are slow compared to the speed of light, and (2) drops the mc^2 term that is central in understanding what particles really are and how they work. If I showed you the solution to that equation, that would give you a picture of electrons from the 1920s, one that was already obsolete by the 1930s. I’m trying to move you out of the last century, and you’re fighting me tooth and nail.

      When you fix Schrödinger’s wave equation to account for relativity — in particular, remembering that E = mc^2 for an electron at rest — then you obtain exactly the equation that the simulator is solving: the class 1 equations of https://profmattstrassler.com/articles-and-posts/particle-physics-basics/fields-and-their-particles-with-math/waves-classical-equation-of-motion/ , which are the ones that the electron field and its quanta (i.e. electrons) actually satisfy.

      [There is still a Schrödinger-like equation here for the wave function of the field, but that wave function is a function of an infinite number of variables, and doesn’t look anything like my simulation.]

      As for your final paragraph, I have no idea what you are talking about; I don’t see how any of these issues are relevant to this post.

  6. So, is it correct to say in an atom there is a force well, [gravity (force) well, cold (temperature at zero) well, null (fields cancelling to zero) well], that keeps, “traps” the various waves, quarks, and electrons oscillating about the geometric center?

    These forces is what is meant as “falling”. If so, falling from what to what, energy and/or curvature of space-time?

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