Planck oscillator average energy Part II

See the last part of my previous post for equations leading up to here. Leaving out the constant C for the moment, and using the total rather than the partial derivative, since U_υ  is the only independent variable,

dS_υ / dU_υ  =   (1/ε)log [(1 + U_υ / ε)] + ( 1/ε)(1+ U_υ / ε)/ (1+ U_υ / ε)

                                                           - (1/ε)log (U_υ / ε) - (U_υ /ε)(1/ε)(1/U_υ /ε)

The red factors cancel, giving

dS_υ / dU_υ = (1/ε)log [(1 + U_υ / ε)] + (1/ε) — (1/ε)log (U_υ / ε) — (1/ε),

where the red terms now cancel. With C reinserted, and dS_υ / dU_υ replaced with 1/T, the result is

1/T = C{(1/ε)log [(1 + U_υ / ε)] — (1/ε)log (U_υ / ε)} 

             = (C/ε){(log [(1 + U_υ / ε)] — log (U_υ / ε)} = (C/ε){(log [(1 + U_υ / ε) / (U_υ / ε)]}

or                                        ε /CT = log[(1 + U_υ / ε) / (U_υ / ε)].

Taking exponentials of both sides

exp(ε /CT ) = (1 + U_υ / ε) / (U_υ / ε)

= 1 / (U_υ / ε)  +  (U_υ / ε) / (U_υ / ε)

=  ε / U_υ   +  1

We want an expression for the average oscillator energy U_υ , so some more algebra is required, namely,

ε/U_υ = exp(ε /CT )  –  1,

 or                                                      U_υ = ε /[ exp(ε /CT ) – 1].

This was Planck's new expression for the average energy of a resonator in equilibrium with black-body radiation—new as of 1900. There's a lot more to discuss here: 1. why Planck's constant h and Boltzmann's constant k come out of this calculation,  2. the Rayleigh-Jeans and Wien limits at the low end and high end, respectively, of the black-body frequency spectrum, 3. the Wien displacement law, 4. the "wrong" assumption of indistinguishable energy elements Planck made when he chose the permutation calculation that goes into finding the entropy, and 5. the choice of perfectly reflecting (mirror) walls versus perfectly absorbing (black) walls of the cavity. For mirror walls there is absorption with immediate re-emission, while black walls would absorb and only re-emit after affecting the motions of the atoms of the material, right? Sure! But just in case, check with Born & Wolf or Hecht &Zajac if you want to be further enlightened.

On this last question, Sommerfeld, in his Thermodynamics & Statistical Mechanics book, Section 20, says that when the walls are perfectly reflecting, they cannot come into temperature equilibrium with the radiation, and a “speck of soot” must be introduced into the cavity as a “catalyzer” to achieve equilibrium!  And there's also the question of where the oscillators (resonators) are, exactly. Are they supposed to be in the cavity walls or are they imaginary particles, like soot, floating about in the air in the cavity? Apparently, with black walls, the resonators are in the walls, and with mirror walls, the resonators need to be free-floating in the air of the cavity. In calculations, we sort of assume a vacuum cavity, so that the speed of light = c = γυ can be used in the equations rather than c/n, where n is the index of refraction of the (heated) air in the cavity. For air (at 300 K) n ≈ 1.0003, so we won’t worry about that right now.

Right now we will look at Einstein's 1907 calculation using geometric series to find Planck's expression for the average energy of a simple harmonic oscillator system. Einstein says, "To arrive at Planck's theory of black-body radiation ... one assumes Maxwell's theory of electricity yields the correct relationship between radiation density and Ē.”  Yes, Einstein uses Ē instead of U_υ for the average resonator energy. The “correct relationship” he’s referring to is Planck's "simple relation," u_υdυ = (8πυ²/c³)U_υ dυ, where U_υ is average resonator energy, as discussed in the last part of my Walking further with Planck post.

“On the other hand,” Einstein continues, “one abandons equation (4),

Ē =  ʃE exp [(-N/RT)E]dE / ʃ exp [(-N/RT)E]dE  = RT/N = kT,

i.e., one assumes that it is the application of the molecular-kinetic theory which causes a conflict with experience. ... this stipulation involves the assumption that the energy of the elementary structure under consideration assumes only values infinitesimally close to 0, ε, 2ε, etc."

Einstein doesn't bother to show the actual equation (4) calculation, but I guess it was a routine calculation by that time. One way it can be done is by separately finding the average potential energy and the average kinetic energy of a one-dimensional harmonic oscillator. Then the E in the integrals in the expression above is, respectively, a constant times x-squared or a different constant times p-squared. The result of the integration in each case is kT/2.  Try it!  Adding the potential and kinetic contributions together gives kT.  This is the classical result, which Rayleigh (with a numerical correction made later by Jeans) calculated in 1900.

Page 216 of the Einstein paper shows this result. Page 217 shows the series representation that gives Planck's result. Einstein was apparently the first to obtain Planck's result using the infinite series calculation. Nowadays this is how most introductory textbooks do it, although they use discrete sums instead of the integral notation used by Einstein.

I will show the calculation as done by Koichi Shimoda on page 70 of Introduction to Laser Physics.  We now have E = nhυ, and use U_υ again instead of Ē (if the Greek letter nu comes out as u, I'll try to fix that later...)

U_υ =  Σ nhυ·exp(-nhυ/kT) / Σ exp(-nhυ/kT)

where the sums are over n = 0 to n = ∞. This is Boltzmann’s method for calculating a statistical average, but with discrete sums instead of integrals because we have discrete or quantized energy levels.  Shimoda lets exp(-hυ/kT) = r in order to make it clear that the denominator is the geometric series

Σ exp(-nhυ/kT)  =  Σ rⁿ  = 1/(1-r) =1/[1- exp(-hυ/kT)].

And he writes the numerator as 

Σ nhυ·exp(-nhυ/kT) = hυ Σn rⁿ

then use the derivative of rⁿ

drⁿ/dr  =  nr^n-1

and compensates for the n-1 exponent by using an extra factor of r out front when writing the sum:

Σn rⁿ  = rΣ drⁿ/dr  = r (d/dr )Σ rⁿ ,

since a sum of derivatives is the derivative of the sum.  Now we put in 1/(1-r) for the geometric series sum, and do the derivative

r (d /dr) (1-r)^-1 =  r(-1)(-1)(1-r)^-2  =  r/(1-r)².

Putting the closed-form expressions for numerator and denominator into the equation for average energy gives

U_υ = hυ[r/(1-r)²] / [1/(1-r)]  = hυr(1-r) / (1-r)² = hυr / (1-r),

and using the trick of pulling out a factor of r in the denominator gives

hυr / (1-r)  = hυr / [r( 1/r – 1 )]  =  hυ ( 1/r – 1 )^-1.

Now 1/r is replaced by exp(hυ/kT), and the Planck expression for average energy is the result:

U_υ  =  hυ / [exp(hυ/kT) – 1]

One overall final point about the average energy:  It is also given by the expression we used in calculating the entropy of an oscillator.  Remember that?  It’s just the arithmetic mean, E/N, where E is the total energy of N oscillators.  Planck specified that the energy E is partitioned among P elements, each with energy ε, so the average energy is

U_υ  =  E/N = Pε/N.

This simple average must equal the thermodynamic expression Planck found for average energy:

Pε/N  = ε /[ exp(ε /CT ) – 1],

or                                                       P/N   =  1/[ exp(ε /CT ) – 1].

This is sort of self-explanatory, but not quite. Planck has already said that these N resonators have a common frequency υ. (Whatever happened to the other N’, N’’, … resonators with frequencies υ’, υ’’, … and energies E’, E’’, … ? I don’t know!  I’m hoping to get back to that. November 2017:  Planck does discuss the other resonators, see my 31 Oct 2017 post.)  If we go ahead and use ε = hυ, then P/N could be called the occupation number associated with the frequency υ.   In modern terminology, it is the Planck thermal excitation function,

<n>  =  1/[ exp(hυ/ k_BT) – 1],

giving the  mean  number of photons excited in the field mode at temperature T.

To be discussed later:  Why did Planck choose a direct proportionality between energy and frequency? At what point does the quantization of energy occur? (The proportionality between energy and frequency is not in itself quantization.)  How did Planck find the constants h and k? And the biggest question, how can all this be explained—that a continuous oscillator can have discrete energy levels—or as Einstein put it, what is the "mechanism of energy transfer"?  In classical physics the mechanism is acceleration, caused by a net force acting on an electric charge. So far, there is no mechanism in quantum theory to explain how charges radiate, in spite of the potential energy appearing in the Schrödinger equation. There are only acausal calculational tools, such as transition probabilities and the use of time-dependent perturbation theory, as in the Feynman diagrammatic perturbation approach (well, okay, only for unbound energy states, so we’ll just come back to all that later).