Ordinarily, we think of heavier* things being bigger (larger) than lighter objects. In everyday life, that scale seems to hold. Humans are heavier than ants, the earth is heavier than humans, the sun is heavier than the earth. It’s a natural assumption because we imagine things are made up of composite materials (cells, atoms, electrons, protons). It is natural to think of larger things as being made out of constituent parts known as building blocks.

But the scale doesn’t always hold firm. Although we don’t ordinarily encounter such exceptions in our day to day life, they do exist. A neutron star, for example, is much smaller than the sun, and yet far more massive (a teaspoon of a neutron star would weigh 4 billion tons!). The proton is more than 1,800 times massive than the electron. Yet it is smaller than an electron. It turns out that this intuitive scale is merely a rule of thumb, not as an iron-clad law of nature

Wait did you say the proton is smaller than the electron? I don’t understand. Don’t we always see a big old proton surrounded by tiny electrons? What’s going on?

Individual particles are not the particles of our classical understanding. Quantum mechanics came along and radically upended our view of reality at subatomic scales. What we call particles are actually excitations of fields. The world is made up of fields, and particles arise out of the interaction of those fields. And what are fields made of? Fields are fields, that’s all, but mathematically they can be described as values at every point in space. Fields can come in all kinds of shapes and forms (spinnor fields, vector fields, scalar fields).  

At the quantum level, these fields are described by equations for the wavefunction. For every particle of energy of certain momentum, there is an associated frequency and wavelength. The wavefunction gives the probability of measuring a particle as proportional to the square of the magnitude of the particle’s wavelength (the Schrödinger equation obeying the uncertainty principle).

Given the wavelike nature of these fields, how can we describe a particle’s mass at the quantum level? The relationship between the properties of mass and physical constraints for a subatomic particle is described by its Compton wavelength. The Compton wavelength is given by the formula:

λ = ћ/mc

where ћ is the reduced Planck’s constant, m is the mass, and c is the speed of light. Setting h-bar and c equal to 1 (natural units), we can simplify the formula for the Compton wavelength as equal to 1/m. This also simplifies the underlying relationship between mass and wavelength. As mass goes up, the wavelength is smaller. When measured, you get the following results:

The Compton wavelength for an electron is 2 x 10^-10 cm.

The Compton wavelength for a proton is 1 x 10^-13 cm

Thus the Compton wavelength is shorter for a proton than an electron because the relationship is determined by 1 over the mass.

In quantum mechanics, heavier particles are smaller in some real sense. This has to do with energy scales (energy = mass correlation). The Compton wavelength determines the smallest size the wave function can be and still safely predict there is only one particle. Think of it as the smallest box we can squeeze a particle into and still stay within a 1-particle regime. Any smaller and energy levels would give rise to a greater number of particles.

In other words you can’t make things out of individual particles that are smaller than their Compton wavelength. There is a minimum size you can squeeze the wave function down to and still call it a single particle.

The idea that we can shrink bigger, heavier things via some miniaturizing device into smaller, lighter things (while fundamentally still being considered the same thing) isn’t just science fiction. It’s forbidden by the laws of physics and the rules of quantum mechanics. 😿😿

Sadly, this also means that Ant-Man will never be real!  You can’t shrink down a person by shrinking the individual atoms. Either you must remove individual particles (to keep the person light) or the person will have to become more massive (and thus not a very agile, deft, or effective superhero, more like a superhero that keeps falling through floors).

(*Mass of course is the true measure of the amount of stuff that goes into creating a thing. Weight is a function of gravity, hence we feel lighter in an elevator going down and heavier going up.)