We assume that the derivation of Planck's radiation law in QM can be performed without the need for Planck's postulate of quantum oscillator energy quantization (E(ph)=n.h.f?)
by appropriate statistical approaches.
We recall the derivation of A. Einstein in 1916 by the appropriate application of the reciprocity rule on stimulated emission of radiation which led to the discovery of the LASER in 1966, 50 years later.
Planck's law for the energy Eλ 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} . . . . . . Eq (1)
Where Lambda is the wavelength of the emitted radiation.
Equation (1) is based on Planck's postulate of energy quantization 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 led to a revolutionary correction of the classical Rayleigh Jeans formula and the Wiens formula.
A. Einstein intuitively assumed that Planck's postulate of quantizing the energy of the quantum atomic oscillator is inherently a statistical assumption and that the derivation of Planck's radiation law can be done by replacing it by other appropriate statistical approaches.
Einstein announced his proof shortly after his discovery of the laser theory of atomic spectral lines which still belongs to A. Einstein.
A preliminary 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 started in 1916 with Albert Einstein, who assumed that there were three distinct processes essentially involved in forming an atomic spectral line:
Spontaneous emission and absorption (well known before Einstein) and the stimulated emission of radiation SER (discovered by Einstein).
Only fifty years later, in the early 1960s, the first laser was produced when experimental physicists were able to reach the so-called inverted population in a three-level / state laser inside a gas discharge tube (gas discharge induces pumping from lower energy states to higher states by electron collisions) in their laboratories.
Consider three quantized electron levels 1, 2, 3 where level 2 in between is a meta stable state .
Obviously the existence of a Meta stable E2,N2 in between E1,N1 and E3,N3 is essential for population inversion and light amplification where you obtain two coherent photons out of one.
But how did Einstein discover the coefficient of SER?
Another thought experiment was added to his two thought experiments of general and special theories of 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.
i-The Boltzmann factor determines the N2/N1 ratio.
Einstein assumed that atomic quantum energy levels at thermal equilibrium obey Boltzmann's classical statistics, viz.
N2 / N1 = exp - (E2-E1) / kT.
With f0 or f = (E2-E1)/ h ....(N. Bohr,s relation )
ii- The absorption rate for the upper level N2 must be equal to the emission rate. The laser rate equation follows,
dN1/dt] uptake = dN2/dt] spontaneous. emission + B21 u (f0) N2
B21 is Einstein's 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). . . . . . (2)
Eq. 2 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 so excited when he observed that Planck's quantization law for blackbody radiation Eq(1) can be derived from equation 2 by the substitution
A21=8 Pi h f ^3 B21/ c^3.. . . . . . (3)
Equation (3) conforms to the formula for the number of standing waves in a 3D blackbody box used by M. Planck and later by A. Einstein.+
+Note:
As a recognition award from the scientific society, the name of the laser is still an acronym for the discovery of Einstein.
Light Amplification by Stimulated Emission of Radiation : L-A-S-E-R .
I ran the text of the answer by Rana Hamza Shakil
through https://gptzero.me. The site identified the text as "Your text is likely to be written entirely by AI"Rana Hamza Shakil is a fake, a person who passes off ChatGPT answers as his own, without caring whether they are correct or not. He has answered over 900 questions on RG, from a wide range of disciplines, with most answers clearly identified as generated by AI.
- Badges
- Science topic
- Similar topics
- Filing