We assume that the derivation of Planck's radiation law in QM can be carried out without the need for Planck's postulate of quantization of the quantum oscillator energy (E(ph)=n.h.f, where n=1.2, 3 ...etc)
A. Einstein in 1916
was excited when he discovered that he could rigorously derive Planck's law for black body radiation by combining Einstein's laser equation (Amplification of Light by Stimulated Emission of Radiation), the classical statistics of Boltzmann and the Bohr relation hf=E2-E1.
We now have two seemingly different derivations, Plancks and Einstein, for the same radiation law.
The question arises: which of the two derivations is more revealing?
Planck's law for the spectral electromagnetic energy I(λ) radiated per unit volume by a cavity of a black body BB in the wavelength interval λ and (λ + Δλ) can be written in terms of Planck's constant (h), speed of light (c), Boltzmann constant (k) and absolute temperature (T):
I(Lambda,T)= 2 h c^2/Lambda^5 *{ 1/[exp hc/ Lambda.K.T)]-1} . . . . . . (1)
Where Lambda is the wavelength of the emitted radiation.
Max Planck derived equation (1) via his postulate of quantizing the energy of the quantum atomic oscillator:
E(osc)=n.h.f , i.e.
E(osc)=hf,2hf,3hf as non-continuous but discrete or quantized energy values.
Planck's hypothesis led to a revolutionary correction of the classic Rayleigh Jeans formula and the Wiens formula for BB radiation.
However, it is rarely mentioned that A. Einstein intuitively assumed that Planck's postulate of quantizing the energy of the quantum atomic oscillator was inherently a statistical assumption and that the derivation of Planck's radiation law could be carried out by replacing it with other suitable statistical approaches.
Einstein announced his proof shortly after his discovery of the laser theory of atomic spectral lines which belongs entirely to A. Einstein.
A brief explanation to clarify the connection between Einstein's atomic LASER and Planck's atomic radiation formula can be presented as follows:
The history of the laser undoubtedly began in 1916 with Albert Einstein, who assumed that there were three distinct processes essentially involved in the formation of an atomic spectral line:
Spontaneous emission and absorption (well known before Einstein) and stimulated emission of SER radiation (discovered by Einstein).
Consider three quantized electron levels 1, 2, 3 where the intermediate level 2 is a meta-stable state.
Obviously, the existence of a meta-stable state E2,N2 between E1,N1 and E3,N3 is essential for population inversion and light amplification where we obtain two coherent photons out of one.
But how did Einstein find his SER coefficient?
Another thought experiment was added to his two remarkable thought experiments on the theories of general and special relativity.
consider a cavity with two-level atomic systems N2 and N1 and the radiation density of frequency f0 is u (f0) is in thermodynamic equilibrium.
1-The classic Boltzmann statistic determines the N2/N1 ratio.
Einstein assumed that atomic quantum energy levels at thermal equilibrium obey classical Boltzmann statistics, viz.
N2 / N1 = exp - (E2-E1) / kT.
2-With f0 or f obeys the Bohr relation, namely,
f = (E2-E1)/ h ....(Bohr relation)
3- The absorption rate for the upper level N2 must be equal to the emission rate. The laser equation immediately follows,
dN1/dt] absorption = dN2/dt] spontaneous. emission + B21 u (f0) N2 . . . (2)
B21 is the Einstein coefficient of stimulated emission of radiation.
Note that B12 and B21 are inherently statistical in nature. SO,
U(f) = (A12/B21) / ((B12/B21) N1/N2 - 1). . . . . . (3)
Eq. 3 is the laser equation.
Obviously, the Einstein coefficients of spontaneous and stimulated emission B12 and B21 are equal, viz.
B12 = B21.
Finally, Einstein himself was very excited when he observed that Planck's quantization law for black body radiation Eq(1) could be derived from his laser equation 2 by the substitution
A21=8 π h f ^3 B21/ c^3.. . . . . . (4)
Equation (4) is the formula for the number of standing waves in a 3D blackbody box used by M. Planck and later by A. Einstein.
It is worth mentioning that as an award from the scientific society, the name of the laser is still the acronym of Einstein's discovery.
Amplification of light by stimulated emission of radiation: L-A-S-E-R.
Now, unanswered questions arise:
i-Is there a clear proof or reason why the classical Boltzmann distribution can be applied to electromagnetic standing waves (especially in the proof of Planck's radiation law)?
ii-Is electromagnetic radiation in all physical situations really quantified and measurable?
iii-Think about the meaning of the Schrödinger equation, is it complete and whether it relates to modeling tools or nature itself.
iv-Is the fact that Planck's constant h is non-zero an inherent property of a quantum universe, i.e. is Planck's constant an inherent property of the universe or a useful tool to study
Interaction of electromagnetic radiation with matter?
In fact, A. Einstein and E. Schrödinger argued against over-amplification of the Heisenberg uncertainty principle in the Shrödinger equation, such that SE serves the Heisenberg uncertainty and not the other way around.
In De Broglie's theory, there is a wave function which is a mathematical and physical construct.
Particles move and move and wave-particle duality is a fact proven by the double slit experience.
In the example of the double slit experiment, the particle passes through a single slit and the associated wave passes through both.
To be continued.
The answer to your question is yes. This is precisely what Louis de Broglie did in his 1924 thesis, which is what caused Schrödinger to develop his wave equation. De Broglie's derivation of Planck's quantum of action from the structure of the Bohr atom is analyzed and described in Section 6 of this article:
Article Introduction to Synchronized Kinematic and Electromagnetic Mechanics
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