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Courtesy links: If photons have no mass, how can they have momentum? How is it possible photons have no mass but have energy?

I saw two other questions asked about why photons have momentum and energy even though they do not have a rest mass. I now wanted to ask about Planck's constant. An object's action has dimensions of $M L^2 T^{-1}$. I do not see this being followed by photons. A constant action of $6.6 \cdot 10^{-31} \text{g} \, \text{m}^2 \text{s}^{-1}$, called Planck's constant $h$ has some connection with photons. The energy of photons is $E = h f$, where $f$ is the frequency of the photons. Where does this constant come from?

Arunabh
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h is a "fundamental" constant, like the speed of light, c.

Its size sets the dimension (scale) of quantum phenomena, whose cornerstone is the (then, a century ago) counterintuitive noncommutativity relation $[x,p]=i\hbar$.

Phenomena with action much-much-much larger than ℏ normally don't reflect this peculiar feature, and are then described by classical mechanics, an approximate, "easy" theory that dominated our description of our world for centuries, before the discovery of QM, and much of engineering to this day.

The characteristic dimensions of photons, are its wavelength (λ) and its momentum (p) understood to be inversely related to each other a century ago, $$ p\lambda = h . $$
This relation leads to the small scale of the photon momentum emitted in atomic energy level transitions, since $E= c p = h c/\lambda= h f$ .

In engineering units (SI), describing planets and mosquitoes, h is small, dramatizing the fact that quantum behavior is elusive and took delicate technology and precision to explore it when the time came a century + ago.

In short, h is the fundamental action dimension "atom" of our present description of the world. It didn't come from anyplace: our world comes from it.

Cosmas Zachos
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