Download 13. Crafting the Quantum.II

13. Crafting the Quantum: Chaps 4-5.
I. Planck's Physics of Principles
• 1889: Accepts Chair in Theoretical Physics at University of Berlin.
• Stresses thermodynamic approach over mechanical:
"The entire new development of
thermodynamics has occurred independent
of the mechanical theory, solely based on
the two main principles of heat theory."
• Ultimate Goal: To provide a mechanical foundation for the 2nd Law of
thermodynamics based on continuum mechanics, as opposed to atomistic
mechanics.
The 2nd Law of Thermodynamics
• Consider different states of a gas:
• Why does the gas prefer to be in the equilibrium macrostate (last one)?
2nd Law: In a thermally isolated system, heat cannot flow from a
cold place to a hot place without work being performed on the system.
• Boltzmann (1897): Attempts to explain the 2nd Law in terms of properties
associated with the mechanical behavior of atoms (gas molecules, say).
Ludwig Boltzmann
The entropy of a state of a system is a measure of how probable
that state is; and the equilibrium state is the most probable
state; and, isolated systems tend to evolve from less-probable
(low-entropy) states to more probable (high-entropy) states.
1. Planck's "Ideal Processes".
• 1887: "On the Principle of Entropy Increase. 3."
At high temperature and low pressure, a (very)
dilute solution is analogous to an ideal gas.
• Prediction (based on thermodynamical reasoning): A dilute solution will
contain molecules of the solvent, molecules of the solute, and in addition,
dissociated components of the the solute.
• Arrhenius (1887) also predicts
(ionic) dissociation but on the basis
of a detailed mechanical model.
Svante Arrhenius
• Arrhenius and supporters: Planck's analogy between a dilute solution and an
ideal gas is unrealistic, hence cannot be used as a basis for prediction.
- Assumes an infinitely dilute solution.
- Does not represent a physically possible state of matter.
• Planck responds (1891):
"In reality, such a process will admittedly often not be
realizable, because in many cases, at high temperatures,
as are necessary here, chemical transformations occur,
and the molecules are thereby altered."
• In other words: "... the inability to produce in the laboratory the processes
assumed by a theory did not, in itself, negate that theory's validity" (Seth,
pg. 109).
"I do not stand back from holding them [ideal
processes] to be a particular triumph of the human
spirit, with the help of which the connection of the
laws of nature can be pursued into areas that are
utterly closed off from direct experiment."
2. Irreversibility and Radiation.
• 1897-1899: 5 papers with the title "On Irreversible Radiation Processes".
• Goal: To demonstrate that the 2nd Law holds absolutely based on
continuous electrodynamical processes.
• In particular: Show that Hertzian
resonators (idealized oscillators) emit
radiation of a qualitatively different form
than the radiation they absorb.
• Oscillating charged mass.
• Can an intrinsic directionality in the
radiation associated with such beasts
be linked with the directionality
associated with entropy increase?
• Hertzian resonator = ideal process! Thus little need be known about the
detailed structure of a resonator in order to describe its equilibrium behavior.
• Boltzmann's critique of Paper #1:
Maxwell's equations are time-reversal invariant.
So: Any directionality Planck hopes to find in his ideal
resonators must be a product of his own assumptions!
• Planck's response:
The time-reversed versions of the resonators are
disallowed due to a singularity at the origin.
• Based on Hertz's original analysis:
"At the origin is shown, in its correct
position and approximately to correct
scale, the arrangement which was used
in our earlier experiments for exciting
the oscillations. The lines of force are
not continued right up to this picture,
for our formulas assume that the
oscillator is infinitely short, and
therefore become inadequate in the
neighborhood of the finite oscillator."
• Seth: Planck appropriates Hertz's experimental approximation into his
theoretical reasoning.
• But: In practice, resonators are not infinitely small.
• 1898: Paper #3. Planck makes following distinctions:
=
(a) Some components are same size
as total intensity; or (b) systematic
gaps between components; or (c) a
limited number of components.
=
(a) Components are small relative to
total intensity; or (b) unsystematic
or no gaps between components.
ordered
radiation
=
Components interfere constructively
due to regularity in phases and thus
contribute to total intensity.
disordered
radiation
=
No regularity in phases of
components, thus constant intensity.
tuned
radiation
untuned
radiation
Idea: Untuned
radiation is not
periodic.
Idea: Disordered
radiation is at
equilibrium.
• Claim: Untuned, disordered processes are irreversible and are realizable by
certain resonator set-ups.
• But: Boltzmann's point is still valid: Maxwell's equations are time-reversal
invariant.
• Planck's response (Paper #4):
natural
radiation
=
Energy distributed irregularity
over components.
"If a resonator at a given time is excited by natural radiation
of variable intensity, the entry of the reverse process is
absolutely excluded for all later times, as long as the exciting
wave maintains the characteristics of natural radiation."
• Kuhn (1987): This marks a new phase in Planck's reasoning; namely, the acceptance
of aspects of Boltzmann's atomistic mechanical analysis.
- Boltzmann's attempt to ground 2nd Law required assumption of "molecular
disorder" -- molecules in a gas in an initial state are not artificially ordered.
- 1900: Planck explicitly makes an analogy between "natural radiation" and
"molecular disorder".
• Seth (pg. 125): "To understand Planck's original conception of natural radiation no
recourse to Boltzmannian molecular disorder is required; his own papers show the
independent development of ideas that would later be deemed analogous."
3. Planck's Derivation of Planck's Law.
• 1900 lecture:
"The most essential point of the whole calculation" is the
postulate that the energy of N resonators of frequency ν is
made up entirely of "an entirely determinate number of finite
equal parts" the size of which is determined by the "natural
constant" h so that the "energy element"  is equal to hν.
• Planck asks: How many ways W can P = E/ such energy elements be
divided among N resonators?
 |  |  |  ... | 
• There are N − 1 dividers | and P energy elements .
• If all these symbols are distinguishable, then there
are (P + N − 1)! ways of ordering them.
• But the 's are indistinguable and so are the |'s; so
we've over-counted by a factor of (N − 1)!P!.
(N + P −1)! (N + P)N +P
≈
• So: W =
(N −1)!P !
N NPP
Stirling's approximation:
N! ≈ NN, for large N.
• Boltzmann sez: The entropy S of an energy distribution among states with W
possible arrangements of energy is given by S = k logW + const.
• So: S = k log
(
(N +P)N +P
N NPP
)
{
}
= N (1 + E )k log (1 + E ) − E k log ( E ) + const.
where P = NE/, and
E is the average energy
of a resonator.
• Thus: The average entropy Save = S/N of a resonator is given by,
Save = (1 + E )k log (1 + E ) − E k log ( E ) + const.
hν
hν
hν
hν
• And: This obeys the thermodynamic relation dS/dE = 1/T.
• Which entails E =
• Recall:
ρ=
8πν 2
c3
hν
e
hν/kT
×E
−1
.
Planck's (1899) result for the energy
distribution of the resonator system.
• And: This gives the Planck Law: ρ =
8πν 2
hν
c 3 e hν /kT −1
II. The Dynamical and the Statistical: Sommerfeld, Planck, and
the Quantum Hypothesis
• 1911: Solvay Conference in Brussels.
• Planck and Sommerfeld offer different approaches to the quantum hypothesis.
"Sommerfeld's statement leads to a finite element of action,
Planck's statement to a finite element of phase space; both
things seem to be fundamentaly different to me; one is of a
dynamical, the other of a statistical nature." (Paul Langevin)
1. Sommerfeld's (1911) Dynamical Version of the Quantum Hypothesis.
• Implicit motive: Show how electromagnetism and the quantum hypothesis
are compatible.
• Recall: Planck's (1900) quantum hypothesis states E = nhν, n = 1, 2, 3, ...
• Which means: Planck's constant h = (energy) × (time).
"Phrased completely generally, a large quantity of energy
in a shorter time, a smaller in a longer time is taken up
and given out by matter, so that the product of energy
and time, or (closer to the definition) the time intergral of
the energy is determined through the magnitude of h."
• Or:
Quantum Hypothesis (Sommerfeld version)
∫
τ
0
Hdt =
h
2π
H = Hamiltonian function
describing the energy of a system.
• Sommerfeld: "...with every purely molecular process a fixed, universal
amount of action [i.e., (energy) × (time)] is taken up or given out from the
atom."
Characteristics of Sommerfeld's version of quantum hypothesis (Seth, pg. 150):
(a) Rooted in areas of research to which he had been devoted for more than a
decade:
• 1911 Solvay paper: New version of quantum hypothesis used to calculate
formula for ratio of polarized energy to total energy of X-rays.
• Extends earlier Habilitationschrift (1895) on diffraction of X-rays.
(b) Not a deduction from general axioms, nor a definition; but following from
the consideration of specific problems.
(c) Used in conjunction with data in process of constructing mathematical
expressions.
(d) Illuminates the shift in adherence to the electromagnetic worldview:
• "Sommerfeld's fundamental hypothesis, in his eyes, offered no contradiction to
electrodynamics at all" (Seth, pg. 155.)
"...in fact, it supplements this with
regard to the course of such processes
about which electrodynamics, in and
of itself, knows nothing."
• Seth (pg. 155): "The dichotomy, that is, was not... one between 'classical' and
'modern' physics, but one between quantum and electrodynamic theory. And,
unlike the former dichotomy, this was explicitly and necessarily not an
either/or. In Sommerfeld's vision, the quantum and the electromagnetic field
were both required in order to understand the physical world."
2. Planck's (1911) Statistical Version of the Quantum Hypothesis.
• Phase space = Space of all possible states of
a physical system.
• Liouville's Theorem: Areas of equal size in
phase space are equiprobable.
• Planck: Let dqdp be an "elementary area
of probability" (i.e., the area of a very
small region of phase space).
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Each point represents a possible
state, labeled by particular values
of position q and momentum p.
- If dqdp is allowed to be infinitely small (if phase space is a continuum), then
Rayleigh-Jeans law results from phase space analysis of black-body radiation.
- If dqdp is constrained to be no smaller than h, then Planck law can be derived.
• So:
Quantum Hypothesis (Planck version)
∫ dqdp = h
Difference between 1900 and 1911 version:
"Above all things it is to be emphasized that, at
least in my opinion, the quantum hypothesis is no
energy hypothesis, but is an action hypothesis."
Planck on Sommerfeld's version:
"Until the definitive statement of such a
dynamical law is achieved, it appears to
me safer to restrict oneself to a statistical
formulation of the quantum hypothesis..."
3. The Bohr-Sommerfeld Quantization Conditions.
• Bohr's (1913) atomic model ("horrid assumptions"):
−
Postulate I
−
Only "stationary" orbits with
radius a are allowed, determined
by a quantum condition on
angular momentum:
+
stable orbits
Neils Bohr
no orbits allowed
in between
pθ = ma2ω = nh/2π.
Postulate II
Energy transfers
occur only when
electrons jump/fall
between stable orbits
via absorption or
emission of radiation.
−
+
• Electron can only jump to
higher allowed orbit by
absorbing requisite amount
of energy E2 − E1.
• As electron falls back,
emits pulse of light with
frequency ν = (E2 − E1)/h.
• Violates classical electromagnetism, which entails accelerating charges emit
radiation and lose energy!
• But: Accurately predicts positions of spectral lines for simple elements:
low pressure tube
Atomic spectra:
- Consist of discrete lines
prism
- Only few colors
(frequencies) present
vapors of a
single element
high voltage
electricity
• Each line corresponds to a drop of an excited electron
from a higher energy orbit to a lower energy orbit.
+
• So: Observed spectral lines correspond to the preferred orbits in the given
atom.
• Can now read off Bohr's model for the hydrodgen atom from the hydrogen
spectrum:
−
+
H-atom at rest...
...excite it with
electric current.
low freq
Ritz-Parson
Series
Balmer
Series
high freq
Lyman
Series
+
• 1915: Sommerfeld's "On the Theory of the Balmer Series".
• Adopts Bohr's model.
- But: It contradicts electromagnetic theory!
- However: Spectrographic evidence convinces Sommerfeld.
• Adopts Planck's statistical version of the quantum hypothesis.
- But: What about Sommerfeld's aversion to "principled"
approach?
- "Sommerfeld had vaunted his [dynamical version] on the
grounds that it complemented, but did not contradict,
electromagnetic theory. Accepting Bohr's model meant that
this could no longer be true." (Seth, pg. xx.)
- And: No mention of thermodynamics. Acceptance of Planck's
statistical version based on pragmatic and not methodological
grounds. (Seth, pg. xx.)
• Bohr: Only certain electron orbits in an atom are allowed: those with
quantized angular momentum pθ = nh/2π.
• Sommerfeld's Generalization: Only certain orbits in the phase space of a
(periodic) physical system are allowed: those with quantized enclosed area.
Quantum Hypothesis (Bohr-Sommerfeld version)
∫ p dq − ∫ p
n
dq = h
n−1
or
∫ p dq = nh
n
p
pn
pn−1
q
•"Orbits" in phase space of a
periodic system characterized by
momentum p and position q.
• Each point on orbit represents a
possible state of the system with
a given constant energy.
• The orbits are closed = periodic
systems return to their initial
states after one "period".
B-S Quantum Hypothesis:
• Area between pn and pn−1 orbits = h; or,
• Area enclosed by pn orbit = nh.
Example: Planck's Hertzian resonator. (q = position; p = momentum)
k = spring constant
m = mass
1 2
p2
Energy E = kq +
2
2m
Frequency ν =
1
k/m
2π
• Motion of one period at constant E is given by an ellipse in phase space:
p
q = −a
p=0
q=0
p=b
•
⇒
q=a
p=0
(0, b)
(a, 0)
(−a, 0)
•
q
(0, −b)
•
⇒
q=0
p = −b
•
All points on the ellipse
represent states of an
oscillating resonator
with constant energy E.
E(0,b) = b 2 /2m or b = 2mE
E(a, 0) = 12 ka 2 or a = 2E/k
• B-S quantum hypothesis: Ellipse area = nh = πab = E
2π
m/k
= E/ν.
• So: E = nhν, which reproduces Planck's original quantum hypothesis.