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§ 11. Planck

In Planck's paper, "On the Law of Distribution of Energy in the Normal Spectrum" (1901), Planck is supporting Maxwell's electromagnetic theory of light then attempts to support Lenard's assumption that the color of light determines the energy of a light particle.

"In any case the theory requires a correction, and I shall attempt in the following to accomplish this on the basis of the theory of electromagnetic radiation which I developed." (Planck, Intro).

"In my last article4 I showed that the physical foundations of the electromagnetic radiation theory, including the hypothesis of "natural radiation", withstand the most severe criticism" (Planck, Intro).

"It seemed to me that I have found one such condition in the form of statement, immediately then considered by me as plausible, that by the infinitesimal irreversible alteration of the near thermal equilibrium being system of N uniform, just in stationary radiation field placed resonators, the bound up with it alteration of the total entropy SN = NS depends only on their total energy UN = NU and their alteration but not on the energy U of particular resonators." (Planck, Intro).

Planck justifies Maxwell's theory that Lenard's photoelectric effect contradicts by deriving an energy element (hγ) that is dependent on the frequency using the black body that emits light and radio waves but Planck's derivation is based on coupling the black body surface electrons' kinetic energies with the black body light emissions using Boltzmann's thermodynamic entropy that results in the energy element (hγ) represented with the units of the kinetic energy. Planck is supporting Maxwell's theory by structurally unifying light with electromagnetic radio waves but the black body is also emitting electrons yet Maxwell's electromagnetic theory of light is based on Faraday's induction effect that is not luminous nor is induction an ionization effect.

..................................................................................................................................................................................................................................................................................................................................................................................................................................

Planck's energy element (hv) is derived using Boltzmann's thermodynamic entropy equation,

S = k log R................................................................................................21

that is used with Planck's blackbody electron (resonator) kinetic energy distribution ratio (Planck, § 3),

R = (N + P)N + P/ NN · PP...........................................................................22

to form (Planck, § 5)

SN = k{N + P) log (N + P) - N log N - P log P)............................................23

Using UN = NU and UN = Pe, equation 23 becomes,

S = k{(1 + U/e) log (1 + U/e) - U/e log U/e}.................................................24

Equation 24 is represented as,

S = f(U/e)....................................................................................................25

The blackbody photon entropy function is derived (Planck, § 8),

S = f(U/γ)...................................................................................................26

Using equations 25 and 26, a proportionality is formed,

e α γ...........................................................................................................27

Planck's energy element is derived using equation 27.

"§10. If we apply Wien's displacement law in the latter form to equation (6) for the entropy S, we then find that the energy element e must be proportional to the frequency v, thus:

e = hγ"....(Planck, § 10)..............................................................................28

"Now it is evident that any distribution of the P energy elements among the N resonators can result only in a finite, integral, definite number. Every such form of distribution we call, after an expression used by L. Boltzmann for a similar idea, a "complex." If one denotes the resonators by the numbers 1, 2, 3, ... N, and writes these side by side, and if one sets under each resonator the number of energy elements assigned to it by some arbitrary distribution, then one obtains for every complex a pattern of the following form:

Table 2

1 2 3 4 5 6 7 8 9 10

__________________________________

7 38 11 0 9 2 20 4 4 5

(Planck, § 3). Planck uses the kinetic energies depicted in Table 2 in the derivation of the energy element (hγ) since Planck's constant h = 6.6 × 10-34 m2 kg/s contains the unit of the mass (kg) yet light is composed of massless light particles. Plus, Planck's derivation uses Boltzmann's closed system thermodynamics entropy S = k log [V/Vo] (equ 21) that represents the change in the initial V and final Vo closed volumes of a gas in thermodynamic equilibrium that conflicts with the open system of the black body radiation effect which cannot form the equilibrium of Boltzmann's entropy (equ 21).

.........................................................................................................................................................................................................................................................................................................................................................................................................................................

Planck's black body intensity equation is derived using the complete equation of the photon entropy function S = f(U/γ) that is derived using the electron entropy (equ 24),

S = f(U/e) = k{(1 + U/e) log (1 + U/e) - U/e log U/e}.................................................29

Planck energy element (e = hγ) is used in equation 29 to form,

S = f(U/γ) = k{(1 + U/hγ) log (1 + U/hγ) - U/hγ log U/hγ}..............................................30

Planck derives the energy element using S = f(U/e) and S = f(U/γ) but only the complete equation of the entropy function S = f(U/e) is derived before the energy element is derived.

........................................................................................................................................................................................................................................................................................................................................................................................................................................

Planck's black body intensity equation is derived by differentiating equation 30 using,

1/T = dS/dU.........................................................................................31

to form,

U = hv/(ehv/kT - 1) ............................................................................32

Equation 32 is used to derive Planck's black body intensity equation,

u = 8πhv3/(ehv/kT - 1) .......................................................................33

Planck's black body intensity equation is derived using Boltzmann's thermodynamic entropy (equ 21) that is not luminous.

In Planck's paper, "On the Law of Distribution of Energy in the Normal Spectrum" (1901), Planck is supporting Maxwell's electromagnetic theory of light then attempts to support Lenard's assumption that the color of light determines the energy of a light particle.

"In any case the theory requires a correction, and I shall attempt in the following to accomplish this on the basis of the theory of electromagnetic radiation which I developed." (Planck, Intro).

"In my last article4 I showed that the physical foundations of the electromagnetic radiation theory, including the hypothesis of "natural radiation", withstand the most severe criticism" (Planck, Intro).

"It seemed to me that I have found one such condition in the form of statement, immediately then considered by me as plausible, that by the infinitesimal irreversible alteration of the near thermal equilibrium being system of N uniform, just in stationary radiation field placed resonators, the bound up with it alteration of the total entropy SN = NS depends only on their total energy UN = NU and their alteration but not on the energy U of particular resonators." (Planck, Intro).

Planck justifies Maxwell's theory that Lenard's photoelectric effect contradicts by deriving an energy element (hγ) that is dependent on the frequency using the black body that emits light and radio waves but Planck's derivation is based on coupling the black body surface electrons' kinetic energies with the black body light emissions using Boltzmann's thermodynamic entropy that results in the energy element (hγ) represented with the units of the kinetic energy. Planck is supporting Maxwell's theory by structurally unifying light with electromagnetic radio waves but the black body is also emitting electrons yet Maxwell's electromagnetic theory of light is based on Faraday's induction effect that is not luminous nor is induction an ionization effect.

..................................................................................................................................................................................................................................................................................................................................................................................................................................

Planck's energy element (hv) is derived using Boltzmann's thermodynamic entropy equation,

S = k log R................................................................................................21

that is used with Planck's blackbody electron (resonator) kinetic energy distribution ratio (Planck, § 3),

R = (N + P)N + P/ NN · PP...........................................................................22

to form (Planck, § 5)

SN = k{N + P) log (N + P) - N log N - P log P)............................................23

Using UN = NU and UN = Pe, equation 23 becomes,

S = k{(1 + U/e) log (1 + U/e) - U/e log U/e}.................................................24

Equation 24 is represented as,

S = f(U/e)....................................................................................................25

The blackbody photon entropy function is derived (Planck, § 8),

S = f(U/γ)...................................................................................................26

Using equations 25 and 26, a proportionality is formed,

e α γ...........................................................................................................27

Planck's energy element is derived using equation 27.

"§10. If we apply Wien's displacement law in the latter form to equation (6) for the entropy S, we then find that the energy element e must be proportional to the frequency v, thus:

e = hγ"....(Planck, § 10)..............................................................................28

"Now it is evident that any distribution of the P energy elements among the N resonators can result only in a finite, integral, definite number. Every such form of distribution we call, after an expression used by L. Boltzmann for a similar idea, a "complex." If one denotes the resonators by the numbers 1, 2, 3, ... N, and writes these side by side, and if one sets under each resonator the number of energy elements assigned to it by some arbitrary distribution, then one obtains for every complex a pattern of the following form:

Table 2

1 2 3 4 5 6 7 8 9 10

__________________________________

7 38 11 0 9 2 20 4 4 5

(Planck, § 3). Planck uses the kinetic energies depicted in Table 2 in the derivation of the energy element (hγ) since Planck's constant h = 6.6 × 10-34 m2 kg/s contains the unit of the mass (kg) yet light is composed of massless light particles. Plus, Planck's derivation uses Boltzmann's closed system thermodynamics entropy S = k log [V/Vo] (equ 21) that represents the change in the initial V and final Vo closed volumes of a gas in thermodynamic equilibrium that conflicts with the open system of the black body radiation effect which cannot form the equilibrium of Boltzmann's entropy (equ 21).

.........................................................................................................................................................................................................................................................................................................................................................................................................................................

Planck's black body intensity equation is derived using the complete equation of the photon entropy function S = f(U/γ) that is derived using the electron entropy (equ 24),

S = f(U/e) = k{(1 + U/e) log (1 + U/e) - U/e log U/e}.................................................29

Planck energy element (e = hγ) is used in equation 29 to form,

S = f(U/γ) = k{(1 + U/hγ) log (1 + U/hγ) - U/hγ log U/hγ}..............................................30

Planck derives the energy element using S = f(U/e) and S = f(U/γ) but only the complete equation of the entropy function S = f(U/e) is derived before the energy element is derived.

........................................................................................................................................................................................................................................................................................................................................................................................................................................

Planck's black body intensity equation is derived by differentiating equation 30 using,

1/T = dS/dU.........................................................................................31

to form,

U = hv/(ehv/kT - 1) ............................................................................32

Equation 32 is used to derive Planck's black body intensity equation,

u = 8πhv3/(ehv/kT - 1) .......................................................................33

Planck's black body intensity equation is derived using Boltzmann's thermodynamic entropy (equ 21) that is not luminous.

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And, in Planck's opinion it was a property of the walls of the blackbody that caused the quantization. I believe it was Einstein when he derived the photo-electric effect that first openly considered the quantization to be a property of the light itself.

-Dan

What do the walls have to do with the quantization since the particle are imediately quantized when they are formed since light is produce without a black body. Example, a glowing to piece of iron produces light and does not require a black body cavity to produce light particles.

Yes, but remember we are looking back on the blackbody problem with the knowledge of Quantum Theory. No one could explain why the light seemed to have a quantized energy (\(\displaystyle E = h \nu\)). Einstein's use of quanta of light energy in the photoelectric effect was sticking his neck out pretty far. As it happened it was a good risk.

What do the walls have to do with the quantization since the particle are imediately quantized when they are formed since light is produce without a black body. Example, a glowing to piece of iron produces light and does not require a black body cavity to produce light particles.

And again, Einstein got inspiration from the Lorentz transformation but eventually derived them from a different source. SR is

-Dan

I thought you were trying to talk about SR as being derived from EM only because EM contains the Lorentz transformation as a symmetry? Did I misunderstand you?Thank you for agreeing!

-Dan

A symmetry is an operation performed that leaves the state of the object the same as the original. Consider the symmetries of a square. Let (1, 0 ), (0, 1 ), (-1, 0 ), and (0, -1 ) be the vertices of the square. We can rotate it by 90, 180, and 270 degrees and still end up with the same square. (A "rotation" of 0 degrees is the identity of the group because it leaves everything alone.) We can also make a reflection about the x-axis, y-axis, and over the lines y = x and y = -x. The collection of these symmetries form a group known as \(\displaystyle D_8\).

In Quantum theory these symmetries tend not to be discrete, like in the example above, but continuous... For example consider a circle. We can make any rotation at all and still have the circle: we don't have to rotate about a specific angle as we do in \(\displaystyle D_8\). The collection of these continuous symmetry operations is called a Lie group.

For example, the Poincare group has the symmetry \(\displaystyle \mathbb{R} ^{1, 3} \times O(1, 3)\). (The 1, 3 in the real numbers is 3 three spatial translations and one time translation and the 1, 3 of the O is a rotation about three real axes and one time axis. (Clearly the time components of the group are not entirely physical, but they are useful in the group.)

You can find other groups in this link. Go to the colored circle on the right side and click on "Table of Lie groups."

I don't think I made it in 100 words.

-Dan

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