Classical physics says momentum is mass times velocity, $p = mv$. Quantum mechanics says momentum is Planck’s constant divided by wavelength, $p = \hbar/\bar{\lambda}$ — a formula that applies even to light, which has no rest mass at all. Are these two different concepts sharing one name, or one concept written two ways? Expressed in Planck units, the question answers itself.
Two formulas, one name
A textbook introduces momentum through the classical formula, and the formula suggests a story: momentum is a massive body in motion. But the quantum formula computes momentum for photons, which have no mass and whose velocity never varies. For radiation, momentum is set entirely by wavelength. If momentum is one physical concept, the two formulas must describe the same thing in different words — and in three short steps, we can show that they do.
Step 1: Write the formulas in universal form
The key is the dimensional structure of Planck’s constant. Expressed in Planck units, $\hbar = l_{\mathrm{P}}\, m_{\mathrm{P}}\, c$ — a length, a mass, and a velocity. Substituting this into the quantum formula puts momentum in universal form:
Every momentum in nature is the Planck-scale momentum $m_{\mathrm{P}}c = 6.52$ kg·m/s — roughly the momentum of a thrown baseball — reduced by one dimensionless ratio, $l_{\mathrm{P}}/\bar{\lambda}$. The ratio does two things at once: it converts the wavelength from arbitrary human units into an invariant number, the same in every unit system; and it represents a physical property of the particle — how concentrated its wave is, compared against nature’s own length scale.
The same substitution puts the two characteristic wavelengths of a massive particle into universal form. The reduced Compton wavelength, fixed by the particle’s rest mass:
and the reduced de Broglie wavelength, set by its state of motion:
Step 2: Read off two relationships
The universal forms make two relationships easy to see. First, a particle’s rest mass and its Compton wavelength are one ratio viewed from opposite ends:
The heavier the particle, the shorter its Compton wavelength; one number expresses both facts. Second, dividing the two wavelength formulas shows that the de Broglie wavelength stretches beyond the Compton wavelength by exactly the factor by which the particle’s velocity falls below $c$ (in the nonrelativistic regime):
Step 3: Rebuild the classical formula
Now substitute both relationships into $p = m_0 v$:
The Compton wavelength cancels, and the classical formula lands exactly on the quantum one. Read factor by factor: the rest mass supplies the momentum scale of the particle’s Compton wavelength, and the velocity rescales that wavelength out to the de Broglie wavelength of the particle’s actual state of motion. What the multiplication produces — in both formulas — is an inverse wavelength. The two formulas are one formula.
Check it with numbers
For the ground-state electron in hydrogen ($v = 2.188 \times 10^{6}$ m/s, $\bar{\lambda}_{\mathrm{dB}} = 5.292 \times 10^{-11}$ m, the Bohr radius):
Three routes, one number.
Same wavelength, same momentum
The unification carries a concrete consequence: any two particles with the same de Broglie wavelength carry the same momentum, regardless of rest mass or velocity. Compare the ground-state electron with a photon of the same reduced wavelength:
| Property | Electron (ground state) | Photon |
|---|---|---|
| Reduced wavelength | $5.292 \times 10^{-11}$ m | $5.292 \times 10^{-11}$ m |
| Rest mass | $9.109 \times 10^{-31}$ kg | 0 |
| Velocity | $\alpha c = 2.188 \times 10^{6}$ m/s | $c$ |
| Momentum | $1.993 \times 10^{-24}$ kg·m/s | $1.993 \times 10^{-24}$ kg·m/s |
Same wavelength, same momentum — though one particle is massive and slow, and the other massless and at light speed. Their energies differ, because energy depends on how fast the momentum is delivered. That is the subject of What is energy?
What momentum is
One way to read all of this: momentum is the wavelength side of a particle’s energy — a capacity to deliver energy, set by how concentrated the particle’s wave is — while velocity is the rate at which that capacity is delivered. On this reading the classical formula is not wrong; it is a recipe. Rest mass and velocity are the bulk handles we can measure for matter, and their product computes the one quantity that matters for matter and radiation alike: the inverse wavelength.
The reading can be expressed in one more identity. Dividing momentum by $c$ leaves a quantity in the mass dimension, $m = p/c$, and in universal form this mass is inverse wavelength directly:
This holds for matter and radiation alike; to return to conventional momentum units, multiply by $c$.
Made my day!