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 wavelength — the 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”?
The formula de Broglie gave us is simple enough:.
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.
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|
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:
- 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.
- 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.)
- 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).
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.
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:
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:
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:
(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:
Definitively sub-Planck Length!
We can get really crazy and see what the de Broglie wavelength of the Earth is:
Using a mass of almost 6×1024 kg and an orbital velocity of almost 30 km/s. Now we’ve gotten seriously small!
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.)
An aside about the Planck constant, h: It’s a unit of angular momentum in terms of cycles per second. Its value is:
Note the units of Joules/cycle. Physicists often use the reduced version of the constant, known as h-bar. Its value is:
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:
See the Wikipedia Planck units page for other constants derived from h.
Stay waving, my friends! Go forth and spread beauty and light.