# What’s the Wavelength?

Lately I’ve been playing a little game of What’s the Wavelength? The question is certainly a bit evocative. Wavelength could refer to many things: a favorite radio station or, metaphorically extended, a favorite anything. It might even evoke an old news meme, although the supposed question posed that time was about frequency (which is just the inverse of wavelength).

Wavelength might even apply to one’s political, social, sexual, musical, or whatever, alignment, but in this case I mean it literally and physically. Under quantum mechanics — our best description of small-scale physical reality — everything manifests as a wave. That means everything has a wavelengththe de Broglie wavelength.

I’ve been curious about it for a couple of reasons.

Firstly, I’m just curious about how the value changes for different objects and situations, in particular what the domain range is. (Both of the inputs and outputs. What ranges of masses and velocities lead to what ranges of wavelengths?)

Secondly, I’m wondering if the extreme values for classical objects says anything helpful about the nature of reality. Specifically, whether it could be related to a putative Heisenberg Cut — a dividing line between the quantum world and the classical one.

The question is an urgent one in that identifying a Heisenberg Cut might help answer a key question in quantum mechanics — what happens during “measurement”?

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The formula de Broglie gave us is simple enough:.

$\displaystyle\lambda=\frac{h}{mv}$

An object’s wavelength — lambda (λ) — is equal to the Planck constant (h) divided by the object’s momentum (mv).

It’s the last item, momentum, the object’s mass, m, times its velocity, v, that’s the input. Those are the two free variables that determine an object’s wavelength.

With photons, wavelength (λ) and frequency (f) are linked through the speed of light in the formulas f=c/λ and λ=c/f. This is because photons are massless and travel at the speed of light.

Matter waves have mass and propagate more slowly, so frequency isn’t linked like that. Instead, the formula is f=E/h, where E is the object’s energy (note that wavelength doesn’t enter into it).

(With photons, energy is also linked to frequency/wavelength. A photon of a given frequency always has the same energy. Add energy to a photon, and its frequency increases (and its wavelength decreases). Energy is separate in massive objects; two objects with the same mass and velocity can have different energies even though their wavelengths are the same. The energy difference would create a frequency difference, though.)

The Planck constant is very small, 6.62…×10-34, so the momentum of the object must also be very small or the wavelength will be extremely short. That means the mass and/or velocity of the object must be very small.

Note that an object with zero velocity (or zero mass) has an undefined wavelength, but as either quality approaches zero, the wavelength approaches infinity.

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Some reference points:

EM Type Wavelength (m) Freq. (Hz) Energy (Ev)
Gamma Rays 1×10-12 (1 pm) 300×1018 1.24×106
X-Rays 1×10-9 (1 nm) 300×1015 1.24×103
Blue Light 470×10-9 (470 nm) 637×1015 2.64
Red Light 680×10-9 (680 nm) 440×1015 1.82
Microwaves 1.0 meter 300×106 1.24×10-6

Here I’ve included the photon frequencies (and energies) even though the notion of frequency won’t really apply to the de Broglie wavelengths. The key thing to note is that the upper range of EM radiation has wavelengths with 12 decimal points. I skimped on listing the lower range because, as you’ll see, we’ll be far beyond the gamma ray range — the extremes of EM.

Three other reference points:

1. The Planck Length is 1.616255×10-35 meters. It’s thought by many to be the shortest possible distance. The physics we know has no meaning at smaller scales.
2. The charge radius of the proton is 8.40×10-16 meters. (Remember the proton is made of quarks and gluons, which seen as particles are points without known size. Seen as waves, they are much less localized and presumably would have size.)
3. A hydrogen atom has a (Van der Waals) radius of 1.20×10-10 meters (much bigger than a proton). It’s smaller than x-rays, but bigger than gamma rays. (Its single electron, like the quarks, is a point without size if seen as a particle. It’s otherwise seen as a probability cloud surrounding the nucleus and then largely accounts for the atom’s size.)

Contrast all this with a one-kilogram instrument sitting very still. We can’t give it a velocity of precisely zero (for several reasons), so let’s say, due to wind or not being exactly level, it’s moving at one pico-meter/second (a speed at which it would take 57-million years to go one mile).

Then:

$\displaystyle\lambda=\frac{{6.626}\!\cdot\!{10}^{-34}}{{1.0}\times{10}^{-12}}={6.626}\!\cdot\!{10}^{-22}\;\mathrm{m}$

Our one-kilogram instrument has a de Broglie wavelength with 22 decimal points — ten orders of magnitude shorter than gamma rays. And that’s for something that only weighs a kilogram hardly moving.

FWIW, a photon that wavelength has a frequency of 4.527×1029 Hz. Since its energy is linked to frequency by E=hf, its energy would be almost two peta-electron-volts (which in the grand scheme of things isn’t a huge amount of energy for an object with mass, but it’s gang-busters for a photon).

If a one-kilo mass sitting very still has such a short wavelength, then higher masses, or higher velocities, can only make it all that much shorter. A 100 kg mass sitting equally still increases the numbers by two orders of magnitude. The wavelength now has 24 decimal digits.

It gets even more interesting if we don’t sit still.

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By interesting, I mean even smaller wavelengths.

Let’s stick with the one-kilo test instrument for another minute. Suppose it’s moving along at a leisurely one meter per second. That’s a hair over 2.2 mph, which is an easy walking speed. (In my morning walks, I’m shooting for a target rate of 4 mph, but I’ve been having a hard time getting above about 3.6 mph.)

With 1 kg at 1 m/s, we have:

$\displaystyle\lambda=\frac{{6.626}\!\cdot\!{10}^{-34}}{{1.0}\times{1.0}}={6.626}\!\cdot\!{10}^{-34}\;\mathrm{m}$

And now the wavelengths are getting down near the Planck Length.

We can get there by motorizing our 1 kg package so it can tool along at 45 m/s — a hair over 100 mph. Then the wavelength is:

$\displaystyle\lambda=\frac{{6.626}\!\cdot\!{10}^{-34}}{{1.0}\times{45.0}}={1.47}\!\cdot\!{10}^{-35}\;\mathrm{m}$

Which is just a bit shorter than the Planck Length. If we replace the 1 kg instrument with a 100 kg person (going 100 mph) then the wavelength is two orders of magnitude smaller (-37), which is definitely sub-physics as we know it.

Even very light objects can have very short wavelengths if they move fast enough. Consider a 30-06 rifle bullet, a standard 165 grain slug with a muzzle velocity of 2,800 ft/s:

$\displaystyle\lambda=\frac{{6.626}\!\cdot\!{10}^{-34}}{{1.06918}\!\cdot\!{10}^{-2}\times{8.5344}\!\cdot\!{10}^{2}}={7.2616}\!\cdot\!{10}^{-35}\;\mathrm{m}$

(Because 165 grain is about 10 grams, and 2,800 ft/s is about 850 m/s.) In this case we’re just a tiny bit longer than the Planck Length.

How about everyone’s favorite, the International Space Station? It has a mass of almost a half-million kilos and moves at almost eight kilometers per second:

$\displaystyle\lambda=\frac{{6.626}\!\cdot\!{10}^{-34}}{{4.19725}\!\cdot\!{10}^{5}\times{7.66}\!\cdot\!{10}^{3}}={2.0609}\!\cdot\!{10}^{-43}\;\mathrm{m}$

Definitively sub-Planck Length!

We can get really crazy and see what the de Broglie wavelength of the Earth is:

$\displaystyle\lambda=\frac{{6.626}\!\cdot\!{10}^{-34}}{{5.972}\!\cdot\!{10}^{24}\times{2.978}\!\cdot\!{10}^{4}}={3.7255}\!\cdot\!{10}^{-63}\;\mathrm{m}$

Using a mass of almost 6×1024 kg and an orbital velocity of almost 30 km/s. Now we’ve gotten seriously small!

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The ontology of the de Broglie wavelength is unclear (and therefore debated). But experiments confirm that increasingly large massive objects demonstrate the kind of diffraction and interference — the same wave behavior — as quantum particles.

My interest lies in what role this wavelength might play in helping to identify what divides quantum behavior from classical behavior. For instance, one might theorize that some length limit, perhaps the Planck Length, plays a role.

Is there, with regard to wavelength, something analogous to the the Planck constant itself — something that prevents the equivalent of the ultraviolet catastrophe in black body radiation?

In any event, I do find the wavelength regime of de Broglie wavelength interesting and perhaps even suggestive.

The wave-particle duality is a deeply embedded pebble in the shoe of quantum mechanics. Given quantum field theory, what we’ve called “particles” are currently seen as wave packets in a quantum field — one field for each particle type. On some level there is no such thing as a particle (i.e. some ball of “stuff” with properties), there are only waves. A key oddity of QM is that these waves manifest in points — perhaps more properly, in point-like interactions.

(Which means their wave-function must always collapse to a position eigenstate. For anything to have a definite position, its wave-function must collapse.)

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An aside about the Planck constant, h: It’s a unit of angular momentum in terms of cycles per second. Its value is:

$\displaystyle{h}={6.62607015}\!\cdot\!{10}^{-34}\;\mathrm{J}\!\cdot\!\mathrm{Hz}^{-1}$

Note the units of Joules/cycle. Physicists often use the reduced version of the constant, known as h-bar. Its value is:

$\displaystyle\hbar=\frac{h}{2\pi}={1.054571817}\!\cdot\!{10}^{-34}\;\mathrm{J}\!\cdot\!\mathrm{s}$

Now the units are in Joules/radian, a measure of angular frequency.

Note also that the Planck constant and the Planck Length are not the same thing:

$\displaystyle\ell_{p}=\sqrt{\frac{\hbar{G}}{c^3}}={1.616255}\!\cdot\!{10}^{-35}\;\mathrm{m}$

See the Wikipedia Planck units page for other constants derived from h.

Stay waving, my friends! Go forth and spread beauty and light.

The canonical fool on the hill watching the sunset and the rotation of the planet and thinking what he imagines are large thoughts. View all posts by Wyrd Smythe

#### 9 responses to “What’s the Wavelength?”

• Wyrd Smythe

Happy Birthday, JMS, where ever you are. I will always remember you!

• Wyrd Smythe

For those hopelessly American or British, a kilogram is about 2.2 pounds.

• Wyrd Smythe

According to Google, the average weight of a car is 2,871 pounds (just over 1300 kg). Zipping down the freeway at a nice legal 55 MPH (just over 24 m/s):

$\displaystyle\lambda=\frac{{6.626}\!\cdot\!{10}^{-34}}{{1.302264}\!\cdot\!{10}^{3}\times{2.45872}\!\cdot\!{10}^{1}}={2.06942}\!\cdot\!{10}^{-38}\;\mathrm{m}$

Sub-Planck Length, baby!

• SelfAwarePatterns

I think the first time I ever came across the Planck length was in a book (or maybe a magazine article) on interstellar travel, and speculation on what happens if the Lorentz length contraction factor reaches a point where the ship is shorter than the Planck length.

But most of what I’ve read since then has cautioned that the significance of the Planck length, in terms of whether it’s the pixel size of reality, remains highly speculative.

The items you considered do raise an interesting (disturbing?) possibility, that if there is a Heisenberg cut, maybe it doesn’t kick in until well into the macroscopic level. Schrodinger’s cat may exist in a superposition after all, but eventually all but one of the outcomes wink out. The idea that a conscious entity may disappear in the collapse would be one of the most disconcerting scenarios. It means that, in all probability, most versions of you reading this (or me writing it) are about to wink out.

• Wyrd Smythe

No doubt observers would see massive amounts of energy coming from an impossibly flat region of space moving at nearly light speed! That reminds me of something that was going around a while back: the idea of going so fast your mass increases to the point of forming a black hole. Except, of course, the mass increase is just apparent to outside observers — it’s an observational effect — so no black hole forms.

(For some reason it reminds me of that one you sometimes see about whether an airplane could take off from a conveyer belt going at take-off speed.)

Yes, as you say, the Planck Length is best seen as the length scale where our physics breaks down. It’s not that smaller isn’t possible, but that we have no idea what’s involved with physics that small. (As I’ve said many times, I’ve long held out hope that, although matter/energy obviously is quantized, maybe spacetime isn’t and GR is a generally accurate description. Unfortunately I’ve seen strong arguments that spacetime can’t be smooth, and I’m finding MOND more and more reasonable, so GR is looking a little tarnished. But I’m absolutely down with the idea of smaller length scales.)

The “pixilation” at that scale might be real enough, but only due to physical laws rather than literal quantizing of spacetime. We’re so far away from probing that scale that who knows!

The cat (I believe) is constantly collapsed by the environment including itself. Its own heat is more than enough to insure decoherence and, thus, no possible superposition. I’ve mentioned that, IMO, the three most pressing QM questions involve superposition, interference, and entanglement, and I’m very interested in experiments testing the upper limits of those. I’m struck by how they all require a lot of isolation from the environment. Size, I’m sure, plays a role — as interactions increase, coherence is lost — but the environment clearly does. I suspect the Cut, if there be one, is based on multiple factors.

• SelfAwarePatterns

I don’t think I’d ever heard of the airplane on conveyer belt thing. My first reaction was, well, why wouldn’t it work? But then I googled it and found out the conveyer belt is moving in the opposite direction. Ah, right. Yeah, that wouldn’t work at all. (This reminds me of an old TV sci-fi movie where Lee Majors is in a hypersonic airplane accidentally in orbit, that has to somehow make an reentry back into the atmosphere, but he has no heat shield. Solution: follow a space shuttle as it reenters. Um, no. In both cases, the active ingredient is being overlooked.)

On the cat, decoherence is definitely an environmental thing. (It’s actually the definition of decoherence.) It seems like the collapse, if it happens, could be during or anytime after decoherence. In principle, it should be possible to take into account the decoherent interactions to detect any remaining interference, if they’re there. (Some experiments have apparently even been able to recohere a few particles, to reverse the process, as long as it hadn’t gone too far.) Of course, doing it with any substantial environment in practice is a whole other matter.

• Wyrd Smythe

If you look deeper into the plane-on-a-conveyer thing, you’ll find it’s one of those internet battle things people can never resolve. Part of the issue is that it isn’t a well-posed problem, and there is an ambiguity regarding the conveyer that people see different ways.

It helps to focus on that airplanes are driven by their engines, not their wheels. It may be easiest to see with a prop plane. The props chew into the air like a screw into wood. They pull/push the plane forward just as a boat prop does a boat. The wheels just spin freely. Jet engines are all push, but imagine a horizontal rocket on freely turning wheels.

Imagine the plane at a stop. The conveyer is also stopped. As the engines begin to push the plane, the wheels begin to turn, and the conveyer begins to move. The ambiguity here is the speed of the conveyer. Doe it match the wheel speed or the plane speed? If the latter, then it ends up going backwards at increasing speed to match the plane moving forward and the wheels just spin twice as fast as they normally would. But if the conveyer attempts to match wheel speed, then due to the plane’s added speed, the conveyer is forced to go faster, which makes the wheels go faster, which makes the conveyer go faster,… until something gives (or the plane gets off the ground).

Mythbusters tried this with a long sheet of plastic, a motorized spindle to wind it up, and an ultra-light for super low take-off speed. It’s a hard experiment to accomplish and as I recall they got mixed results, but generally confirmed that the plane can take off.

“It seems like the collapse, if it happens, could be during or anytime after decoherence.”

Yep. And (I think) superposition is never an issue for the cat because the particle detector system is also large enough, and in an environment, to be decohered. It’s the radioactive particle that might be seen as in a superposition, or even the state of the radioactive sample, maybe.

There seems a nuance here. Every atom in the sample has some chance of decaying and emitting a particle. (In some versions of this, only a single atom is monitored, which is a bit magic, but one sometimes sees the phrasing, “if a certain atom decays…”.) But even a single atom can decay at any time during the experiment interval; multiple atoms just mean more possibilities. So the superposition is really of all possible moments an atom can decay and, assuming a macro-sized sample, all the atoms, so it’s really a very large superposition of things.

(Including various failure modes and rare occurrences. An instance where the power fails or where an asteroid hits the lab. MWI seems to suggest all such possible realities are included in the superposition.)

But all that nuance aside, what might be a legit quantum superposition at the decay level results in an emitted particle that’s superposed with not being present. When that single particle hits the decohered particle detector and interacts with the first 10, 100, 1000, 10000, etc particles it encounters — all of which are decohered — any quantum superposition is very quickly lost.

To the point I believe it cannot be said the detector is ever in superposition, let alone the cat. The quantum superposition of particle-not-particle is nearly instantly lost upon encountering the decohered detector. It’s simply not possible for the detector to be in a superposition of detecting-not-detecting. (Or if so, for only such a brief instant so as not to matter. There might well be some instant initial uncertainty to the detector as to whether it detected, but it would quickly collapse under the weight of all the decoherence.)

• SelfAwarePatterns

Hmmm. Yeah, I was making (additional) assumptions about how the conveyer belt works. I can see it being a difficult problem due to ambiguity. Clarification seems like most of the issue.

To me, that feels like an analogy for most philosophical problems.

• Wyrd Smythe

Like angels dancing on the head of a pin. Or Mary’s Room, for that matter. Thought experiments that aren’t practical or, in some cases, even really possible.

As I mentioned, it helps to focus on the engines (especially with props). They’re gonna push the bird forward no matter what silliness is going on with the wheels. (Other than actually tying them down.)