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 Uυ is
the only independent variable,
dSυ
/ dUυ =
(1/ε)log [(1 + Uυ / ε)] + ( 1/ε)(1+
Uυ
/ ε)/ (1+ Uυ
/ ε)
-
(1/ε)log (Uυ / ε) - (Uυ
/ε)(1/ε)(1/Uυ
/ε)
The
red factors cancel, giving
dSυ
/ dUυ = (1/ε)log
[(1 + Uυ / ε)] + (1/ε)
— (1/ε)log (Uυ / ε) — (1/ε),
where
the red terms now cancel. With C reinserted, and dSυ
/ dUυ replaced with
1/T, the result is
1/T
= C{(1/ε)log [(1 + Uυ / ε)] —
(1/ε)log (Uυ / ε)}
=
(C/ε){(log [(1 + Uυ / ε)]
— log (Uυ / ε)} = (C/ε){(log
[(1 + Uυ / ε) / (Uυ
/ ε)]}
or
ε
/CT = log[(1 + Uυ / ε) / (Uυ
/ ε)].
Taking
exponentials of both sides
exp(ε
/CT ) = (1 + Uυ / ε) / (Uυ
/ ε)
= 1
/ (Uυ /
ε) + (Uυ /
ε) / (Uυ /
ε)
= ε
/ Uυ
+ 1
We
want an expression for the average oscillator energy Uυ
, so some more algebra is required, namely,
ε/Uυ
= exp(ε /CT ) – 1,
or
Uυ
= 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 Uυ for the
average resonator energy. The “correct relationship” he’s
referring to is Planck's "simple relation," uυdυ
= (8πυ2/c3)Uυ
dυ, where Uυ
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
Uυ again instead of Ē (if the Greek letter nu comes out as u, I'll try to fix that later...)
Uυ
=
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
= nrn-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
Uυ
= 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:
Uυ
= 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
Uυ
= 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/ kBT) – 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).
