[Physics] the real interpretation of Planck’s constant and what are its origins

blackbodyphotonsphysical constantsquantum mechanicsunits

In the physics texts I have read and from other online information, I gather that Planck's constant is the quantum of action or that it is a constant specifying the ratio of the energy of a particle to its frequency. However, I'm still not understanding exactly what it is?

From other things I have read, I understand that Planck did a "fit" of data concerning others' experiments and came up with this value; exactly what other data exactly did he fit to arrive at this really small value? Or maybe he did it some other way? Perhaps an answer concerning its origins will help me understand my first question better?

Best Answer

In point particle classical mechanics, the action $S$ is the time integral of the Lagrangian $L$

$$S=\int Ldt$$

You can check its dimensions are of $[ML^2T^{-2}][T]=[ML^2T^{-1}]$ this is, energy times time. The constant ratio is due to the energy $E$ and frequency $\nu$ relation for photons:

$$E=h\nu \Rightarrow h=\frac{E}{\nu}$$

The "fit" that you are talking about comes of the blackbody radiation spectrum. If we use as variables temperature $T$ and frequency $\nu$ in classical physics we have two laws:

High frequency law: Wien's law $$I(\nu,T)=\frac{2h\nu^3}{c^2}e^{-\frac{h\nu}{kT}}$$

Low frequency law: Rayleigh-Jeans law $$I(\nu,T)=\frac{2h kT\nu^2}{c^2} $$

There is no intermediate frequency law. Planck assumed that radiative energy is quantized via $E=h\nu$ and interpolated the energy fitting for an expression of the type

$$I(\nu,T)=F(\nu,T)e^{g(\nu,T)}$$

that should satisfy both limits ($\nu \approx 0, h\nu >> kT$). Finally he obtained

$$I(\nu,T)=\frac{2h\nu^3}{c^2}\frac{1}{1-e^{\frac{h\nu}{kT}}} $$

However there is a much more nicer and physical derivation of Planck's law due to Einstein that you can find in Walter Greiner Quantum Mechanics an Introduction chapter 2