The tunnel effect is not only a quantum mechanical phenomenon but rather a statistical phenomenon which precisely obeys modern statistical techniques.
Statistics from modern physics techniques (classical physics plus transition probability) show that quantum tunneling exists and that its description formula by the classical Schrödinger PDE is also correct.
The difference is that the description of classical physics techniques is understandable, while Schrödinger's classical PDE is not.
This is indeed the case.
Consider Planck's law of radiation: does it belong to quantum or classical physics?
It was a “battle for the world” at the time.
This particular unanswered question where classical mechanics and quantum mechanics meet and interact in an equation is still of great importance and helps in understanding both subjects. 1- Consider Max Planck's formula, which replaced the complete failure of those of Wien and Rayleigh Jeans. , for the black body thermal radiation density u (f, T):
du(f,T) = {(2hf^3/c^2). 1 / (e ^( hf /kB.T) -1)} df. . . . . . . . (1)
with h = Planck, s constant and f = radiation frequency, kB = classical statistical constant of Boltzmann and T = classical thermodynamic temperature. A first look at the exponent hf/kB.T shows that the numerator hf is pure quantum mechanics QM while the de-numerator is purely classical physics without forgetting that a quantum temperature is not yet defined in a unique way.
And now should we call the equation. 1 Classic, Quantum or half and half?
2- What QM interpretation is introduced into equation 1? Einstein QM interpretation or Bohr-Copenhagen QM interpretation? Keep in mind that Einstein in 1916 rigorously proved equation.1. without it being necessary to assume energy oscillators quantified by Max Planck,
E= hf, 2hf, 3hf, or quantum numbers n=1,2, 3, N, ... large infinite number.
We can proceed to study the famous law of Max Planck called the law of black body radiation.
Planck's well-known formula for black body radiation,
du(f,T) = {(2hf^3/c^2). 1 / (e ^( hf /kB.T) -1)} df. . . . . . . . (2)
with h = Planck, s constant and f = radiation frequency, kB = classical statistical constant of Boltzmann and T = classical thermodynamic temperature.
A close look at equation 1 shows that it is made up of two parts, the first to be known (2 h f^3/c^2) or a constant. f^3/c^2.* It follows directly from the second corollary of Laplacian's theorem, product of the statistical theory of Cairo techniques,
Corollary 2 says that
[1],The volume integral of ∇2 U(x,y,z,t) over the closed volume V for any source/sink distribution inside V is equal to the closed surface integral of U.C.
It is clear that C is the speed of the wave in the medium considered.
C= 330 m s-1 for the sound wave and 3E8 m s-1 for the EMW. In mathematical language [1]
,∫∫∫closed volume ∇2 U(x,y,z,t) dV = ∫∫closed surface U(x,y,z,t).C dA
Corollary 2 implies, (L/λ)^3=number of harmonic oscillators of wavelength λ and frequency f=C/λ. In other words, the density number of harmonic oscillators of frequency f is given by a constant * f^3/ c^2
o be completed.
1-Quantum mechanics-The complex and untold story. ResearchGate, IJISRT journal.
It is quite striking that Planck's revolutionary hypothesis (2), which led to the famous Planck's law, on the stationary distribution of harmonic quantum oscillators for black body radiation, can be removed.
Answer II Continued.
Laplacian theorem in mathematical language [1],
∫∫∫closed volume ∇2 U(x,y,z,t) dV = ∫∫closed surface U(x,y,z,t).C dA
therefore implies:
(L/λ)^3=number of harmonic oscillators of wavelength λ and frequency f=C/λ.
In other words, the density number of harmonic oscillators of frequency f is given by a constant * f^3/c^2.Which is the proof of the first part of the so-called Planck's law.
In order to proceed with the derivation of the second part of Planck's formula for black body radiation, we propose the following:
The spectral density of electromagnetic energy U(λ,T) radiated per unit volume by a cavity of a black body BB in the wavelength interval λ and (λ + dλ) can be written in terms of a constant Planck (h), speed of light (c), Boltzmann constant (k) and absolute temperature (T):
I(λ,T)= 2 h c^2/λ^5 *{ 1/[exp hc/ λ.K.T)]-1} . . . . . . (1)
Where λ 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, that is to sayMi(Osc.)= hf, 2hf, 3hf, . . infinity. . . . (2)
in the form of non-continuous but discrete or quantized energy values.
Planck's hypothesis led to a revolutionary correction of the classic Rayleigh Jeans formula and Wiens' law for BB radiation.
However, it is rarely mentioned that the author, among other physicists, intuitively ignored Planck's postulate on energy quantization and proposed a brilliant alternative to derive Planck's formula of black body radiation [1 ].
we propose the following:
Planck's law for the spectral electromagnetic energy U(λ,T) radiated per unit volume by a cavity of a black body BB in the wavelength interval λ and (λ + dλ) can be written as terms of Planck's constant (h), speed of light (c), Boltzmann's constant (k) and absolute temperature (T):
I(λ,T)= 2 h c^2/Lambda^5 *{ 1/[exp hc/ λ.K.T)]-1} . . . . . . (3)
Where λ is the wavelength of the emitted radiation.
Max Planck derived equation (19) via his postulate of quantizing the energy of the quantum atomic oscillator:
E(osc)= hf, 2hf, 3hf, . . infinity. . . . (4)
in the form of non-continuous but discrete or quantized energy values.
Equation 4 proposed by Max Planck is by nature a statistical hypothesis and the calculation of Planck's radiation law could be carried out by replacing his proposal with other suitable statistical approaches.
Albert Einstein discovered the laser theory of atomic spectral lines in 1916, which presents the appropriate statistical approach required [1].
Consider two quantized electron levels 1, 2 subject to spontaneous emission and absorption (well known before Einstein) and stimulated emission of SER radiation (discovered by Einstein).
Another thought experiment was added to his two remarkable thought experiments on the theories of general and special relativity.
He considered a cavity with two-level atomic systems N2 and N1 filled with a radiation energy density of frequency f where u(f) is in thermodynamic equilibrium with the occupation of the energy levels.
The absorption rate of the upper level N2 must be equal to its emission rate (Continuity of the number of particles in steady state).
The rate equation for the Einstein laser is:
U(f) dN1 / dt] absorption = dN2 / dt] spontaneous emission + B21 u (f) N2,
SO,
U(f) . B21 N1 = A12 N2 + U(f) B12 N2
B21 is the Einstein coefficient of stimulated emission of radiation where it is logical to assume B21=B12.
The above equation reduces to,
U(f)= A12 / { B21 N1/N2 - B12 .1}
Or,
U(f)= A12/B12 / {N1/N2 - 1}
By replacing N1/N2 = Exp -(E2-E1)/h
which is called Bohr relation, we obtain,
U(f)= A12/B12 / { Exp -(E2-E1)/h - 1} . . . . . (5)
Eq. 5 is called the laser equation.
.Finally, one should be excited when we observed that Planck's law of quantization of black body radiation can be obtained from equation 5 by the simple substitution,
A21=8 π h f ^3 B21/ c^3.. . . . . . (6)
Equation (6) is the formula for the number of standing waves in a 3D blackbody box derived by Max Planck and later by A. Einstein.
#The interesting fact that the exact value of Planck's constant h = 6.63 E-34 J s is an inherent property of a quantum universe deserves to be further studied.
1-Useless quantum mechanics, ResarchGate, IJJISRT journal.
It is quite striking that Schrödinger's revolutionary PDE, which led to the quantitative explanation of the phenomena of
Reflection and transmission (sometimes called tunneling) can be completely ignored and replaced with B-matrix string statistics.
However, it is rarely mentioned that the author, among other physicists, intuitively ignored Schrödinger's PDE on energy quantization and proposed a brilliant alternative to derive a reflection and transmission formula on the coefficients of a quantum energy barrier (R, T), as shown in Figure 1.
we propose the following stationary solution:
Zone I:
U(x) = E eV - R E eV
Zone II:
U(x) = E Exp (-a x)eV
from x=0 to x=dx
Note that the exponential decay in zone II comes from the fact that this is the case for this initial value problem imposed by the statistics of matrix B.
Zone III:
U(x) = E Exp (-a dx)eV
from x=dx to x=infinity
And apply the continuity equation to the two boundaries y=0 and y=dx.
To be continued.
Please take a look at:
Useless Quantum Mechanics – The Complex and Untold Story III
ResearchGate, IJISRT journal.
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