-5

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 Bhattacharya
  • 191
  • 2
  • 15
  • 1
    https://physics.stackexchange.com/a/730230/292464 read this to understand energy in relativistic cases. As for where planck's constant came from , read about ultraviolet catastrophe. –  Oct 07 '22 at 15:51
  • 4
    Planck’s constant is the quantum of action. Asking “where it comes from” is asking why action is quantized. We can imagine a universe in which it isn’t, but it would look nothing like our universe. – Ghoster Oct 07 '22 at 15:53
  • 1
    Planck's constant has some connection with photons? Have you learned that the energy $E$ and frequency $f$ of a photon are related by $E=hf$? – Ghoster Oct 07 '22 at 15:58
  • @Ghoster I know that equation. – Arunabh Bhattacharya Oct 07 '22 at 15:59
  • 1
    Then you know that there is a connection with photons. Planck’s constant lets you find how energetic a photon is, based on its frequency. – Ghoster Oct 07 '22 at 16:00
  • If it bothers you that the dimensions of action include an $M$ but photons have no mass, then write the dimensions as $ET$ instead. – Ghoster Oct 07 '22 at 16:03
  • 2
    planck's constant was discovered in the context of photons, but is a lot more fundamental a quantity than just "governing how the energy of photons is quantized" – Zo the Relativist Oct 07 '22 at 19:56
  • 3
    I think it is fair to ask "Where does Planck's constant come from?", but the answer is a trivial "We don't know." right now, at least not on the level of first principles. – FlatterMann Oct 07 '22 at 22:09
  • 1
    At the beginning Planck's constant was a hypothetical constant to explain the radiation of a black body (ultraviolet catastrophe)...., We can say that Planck's constant is the product of Milikan's constant (experimentally deduced from the photoelectric effect) times the electric charge of the electron: h=ke, with the relation: eV=hf-W. https://en.wikipedia.org/wiki/Ultraviolet_catastrophe ** https://en.wikipedia.org/wiki/Photoelectric_effect – The Tiler Oct 08 '22 at 07:40
  • 1
    @TheTiler Waiting for an actual answer to this question while I have a bounty. – Arunabh Bhattacharya Oct 11 '22 at 15:50
  • 1
    @arunabh-bhattacharya a good analogy for h is to imagine a very small mass M swinging on a string at a constant velocity V. In this analogy MV+h. Now double the rotation speed and MV=2h. 10 times the rotation speed and MV=10h. Rotation equals frequency so a photon with a frequency of 2,000,000Hz has twice the energy of a photon with a frequency of 1,000,000Hz. – Bill Alsept Oct 13 '22 at 07:02
  • @Billal Why (and how) is V constant? What is spinning? – PM 2Ring Oct 14 '22 at 06:46
  • @PM2Ring At the start of the analogy the mass is swinging (circling) on a one meter long string, one revolution per second. This establishes V and the mass is too small to measure, but it is there. The revolutions per second is an oscillation that represents the frequency of a photon. If you increase the revolutions per second or increase the frequency of the photon the energy increases. A photon oscillating with a frequency of 480,000,000,000,000 cps is perceived as red. A photon oscillating with a frequency of 680,000,000,000,000 cps is perceived as blue. Both photons propagate at speed c. – Bill Alsept Oct 14 '22 at 15:21

1 Answers1

2

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
  • 62,595