Single Planck $h$ constants Planck developed his black body radiation theory assuming that atoms treated as simple harmonic oscillators can stay in states of very much defined energy. If normal frequency of such oscillator is $\nu$, then the energy levels are the multiples of $h \nu$ (that is $E_n = n h \nu$, forgetting about zero-point vibrations). From my understanding, here $h$ serves just a proportionality constant.
Later, Einstein stated that light can exist in quanta (photons). For each electromagnetic wave of frequency $\nu$ the minimal energy is again $h \nu$. He then very successfully explained the photoelectric effect with this approach. Here, again, $h$ is a proportionality constant.
My question is that why in these two cases $h$ is (or should be?) the same constant? What is the relation between these two $h$'s in two approaches. Why did this evolve this way? I mean from black body radiation experiments and later photoelectric effect measurements one can derive Planck constants, and see they are indeed the same (within some uncertainty). But this does not solve my problem of these $h$'s being assumed to be the same. I clearly miss some link between these ideas. Many thanks for those who can explain these in detail or point to relevant literature on the topic.
 A: There are three pillars of experiments that forced quantum mechanics at first as a phenomenological theory and then as a more formal theory of physics with principles and postulates  and differential equations.

*

*atomic spectra


*black body radiation


*the photoelectric effect
Bohr's atom tied up the observations by assuming quantized energy levels for the atoms, using h explicitly in the arbitrarily imposed quantization of angular momentum that allowed for stable energy levels. (See this answer of mine).
Then Schrodinger's equation introduced the wave equations and after that the theory of quantum mechanics took off.
So even though new students are introduced to the theory, the development of the theory was laborious, and strongly dependent on fitting observations and measurements. The single constant was forced by the data.
A: Einstein was inspired by Plank’s quantum hypothesis. Plank proposed that in order to explain the black body spectrum, one had to assume that the black body absorbed and emitted only quantised energy of radiation. Plank did not believe in the atomic model (at the time at least) and did not investigate further.
Einstein on the other hand was a firm believer of the atomic model and saw that at the time there was a discrepancy in nature. Matter was made up of discrete chunks called atoms. But radiation (light) comprised of waves, thanks to Maxwell. So Einstein, wanting a unified nature tried to quantise light. Where Plank proposed that light was absorbed/emitted as packets, Einstein took it a step further and claimed that light itself was made of packets.
Once he did so, he could utilise the established machinery of atomic calculations directly to light and he showed that it lead directly to Plank’s formula for the blackbody spectrum. So he showed that his hypothesis was consistent with established observations.
Next he sought for unexplained problems to test his hypothesis on. One such unsolved mystery was the photoelectric effect. And he applied his hypothesis and made predictions that were verified by experiments much later.

To summarise, Plank had successfully established his formula for the blackbody spectra by assuming quantised emission/absorption. Einstein came up with a better theory where light itself was quantised. This was consistent with Plank’s formula and predicted something that couldn’t be predicted by Plank’s hypothesis, the photoelectric effect. This is why the same constant appears in both cases. Because the underlying theory is the same.
