Download 15. Crafting the Quantum.IV

15. Crafting the Quantum: Chap 7, Conclusion.
1.
Unsettled Questions of Atomic Physics.
A. The Zeeman effect and number mysteries.
• 1896: Zeeman observes a splitting of the doublet in the spectrum
of a sodium atom in the presence of a weak magnetic field.
Pieter Zeeman
double spectral lines (doublet)
in absence of magnetic field...
...split into many more lines
in presence of magnetic field.
The Zeeman effect:
A splitting of the line spectra of
an atom in a weak magnetic field.
• 1897: Lorentz derives an explanation based on his electron theory.
- Suppose a spectral line is due to an electron with mass
me and charge e vibrating at frequency ν.
- Then in the presence of a magnetic field H, this frequency can take one of three values: ν or ν ±Δν, where
|H||e |
Δν =
4πme
• So: In the presence of a magnetic field, a
single spectral line ("singlet") of frequency
ν is split into three lines separated by an
amount Δν.
• And: This "triplet structure" is actually
observed for hydrogen and helium atoms!
H. A. Lorentz
singlet in
absence of H...
...splits into triplet
in presence of H.
Distance between
triplet lines = Δν
• But: This triple-splitting for singlets isn't what is observed for doublets or
triplets when a magnetic field is switched on.
• The Anomalous Zeeman Effect = The more complex splitting of spectra lines
in the presence of a magnetic field beyond the Lorentz singlet-to-triplet
splitting.
• 1907: Runge discovers the following pattern:
"The hitherto observed complex splittings of spectral lines in a magnetic field exhibit the following perculiarity: the distances of the components from the centre are integral multiples of just a fraction of the
normal separation, a = |H||e|/4πme. So far the fractions a/2, a/3,
a/4, a/6, a/7, a/11, and a/12 have been definitely observed."
Karl Runge
Runge's Law:
s
|H||e |
Δν = a ⋅ , where s and r are integers and a =
.
r
4πme
• For the "normal" (singlet-to-triplet) Zeeman effect, r = 1 and s = 0 or ±1.
Sommerfeld (1916) On the Zeeman Effect in Hydrogen Spectrum.
• For an electron in an elliptical Bohr orbit, there are two Bohr-Sommerfeld
quantum conditions:
y
∫ p dφ = kh,
φ
∫ p dr = rh,
r
r
k = 1, 2, 3, ...
φ
x
r = 0, 1, 2, 3, ...
• k = "azimuthal" quantum number:
determines shape of orbit.
H
θ
• r = "radial" quantum number:
determines size of orbit.
• n = k + r = "principle" quantum
number: denotes energy of orbit.
• m = "magnetic" quantum number:
determines spatial orientation of orbit with
respect to an external magnetic field H.
x
y
cosθ = m/k
• Frequencies of hydrogen lines in magnetic field are split by
Δν = Δm |H||e|/4πme.
K
angular
momentum
vector
Sommerfeld's (1920) "A Number Mystery in the Theory of the Zeeman Effect".
Combination Principle:
Each spectral line is made up of two terms corresponding to the energies of the intial and final states.
Sublevels ("series")
labeled by k.
• Each energy level is characterized by the
principle quantum number n = k + r.
• Each energy level is divided into sublevels
("series") given by values of k = 1, 2, 3, ...
and labeled by s, p, d, and b.
Selection Principle:
Only transitions with Δk = ±1 or 0
are allowed.
Energy levels
labeled by n.
Each transition
(arrow) produces
a spectral line.
Possible transitions for sodium atom.
• Now: Apply Combination Principle to Runge's Law:
⎛ s
⎛ s r − s r ⎞⎟
s 2 ⎞⎟
⎜
1
2 1⎟
Δν = Δν1 − Δν 2 = ⎜⎜a −a ⎟⎟ = a ⎜⎜⎜ 1 2
⎟⎟ .
⎟
⎟
⎜⎝ r1
⎜
r2 ⎠
⎝ r1r2 ⎟⎠
• So:
"Magneto-Optic Splitting Rule":
Δν (s1r2 − s 2r1 ) s
=
= .
a
r1r2
r
The following then accounts for observed Zeeman effects:
1. For singlet spectral lines, r = 1 for any energy sublevel (s-, p-, d-, or b-).
2. For triplets, r = 1, 2, 3, 4
for s-, p-, d-, and b-sublevels.
3. For doublets, r = 1, 3, 5, 7
for s-, p-, d-, and b-sublevels.
• Example:
- For a triplet spectral line belonging to the p and s series, r = 1 ⋅ 2 = 2.
- For a doublet spectral line belonging to the p and s series, r = 1 ⋅ 3 = 3.
Sommerfeld's (1920) "General Spectroscopic Laws...".
• Some transitions allowed by
Selection Rule Δk = ±1 or 0
are not observed:
Dotted arrows are allowed but not observed.
• Suppose each line is
assigned an "inner" quantum
number j, and suppose it
satisfies the same selection
rule as k: Δj = ±1 or 0.
j-values
• What is the physical significance of j?
"Of its geometric significance we are quite as
ignorant as we are of those differences in the orbits
which underlie the multiplicity of the series terms."
k-values
"The actual cause for the doublets and triplets and
therefore also the cause of the anomolous Zeeman effect
is still unclear to me. Only this much is certain, that in
all whole-numbered relationships, quanta are involved."
• On model-building vs. empirical rules:
"Thus it is, that at the moment we are at a loss with the modellmässigen meaning of the line multiplicities of the non-hydrogenic
elements... All the more valuable are all the lawful regularities
[Gesetzmässigkeiten] that present themselves empirically..."
• On the "Magneto-Optical Splitting Rule":
"It represents for the time being, as I have noted elsewhere, a
'number mystery'... Only so much appears to be certain: that the
integral harmony of our Runge numbers has its final cause in the
action of hidden quantum numbers and quantum relations."
B. Modellmässig vs Gesetzmässig Reasoning
• "...[I]gnorance about causes was traded for a functionalist understanding of
regularities within phenomena... Sommerfeld gave up the search for
modellmässig foundations in order to develop a praxis -- or craft -- involving
'half-empirical' Gesetzmässigkeiten." (Seth, pg. 212.)
Example: The Rise and Fall of Sommerfeld's (1918) "Ellipsenverein" model.
• Initial problem: How can Bohr's model explain X-ray emission and
absorption, which do not occur at the same frequency.
• Sommerfeld (1915): Suppose, instead of feeling the nucelar charge Ze, an
electron in a given orbit feels a reduced charge (Z − l)e due to screening by
lower electrons.
• But: The screening constant l should be an integer and calculations derived
using Bohr's model entail it isn't!
• (1918): Sommerfeld's ellipsenverein model:
- Describes n electrons on n identical ellipses,
with each ellipse spaced at an angle of 360/n.
- At any moment, each electron stands at the
corner of a regular n-sided polygon.
- Collective motion takes the form of a single
circular orbit expanding and contracting
around the nucleus.
- Entails non-integer screening constant l!
"For my feeling, the artful interlocking of the n electronic paths in our 'Ellipsenverein' is nothing unnatural; I see much more a sign therein for the high
harmony of motion that must rule within the atom."
• But: Essential characteristic of the ellipsenverien: the orbits don't all belong
to the same atom!
"One must always keep in view the fact that the
processes..., even if they occur in the very same
element, must take place in different atoms."
• 1921: Bohr declares this cannot be right under the correspondence principle:
All orbitals, of whatever shape, that correspond to a given
element, must be found within a single atom of that element.
• 1924: Sommerfeld concedes:
"In the present state of theory, it seems to me to be most
secure to put the question of model-based meaning in the
background and to first bring the empirical relations to
their simplest arithmetical and geometrical form."
• Former student Pauli on 4th edition of Sommerfeld's Atombau:
"I found it particularly beautiful in the presentation of the complex structure that you have left all modellmässig considerations
to one side. The model idea now finds itself in a difficult,
fundamental crisis... One now has the impression with all models,
that we speak there a language that is not sufficiently adequate
for the simplicity and beauty of the quantum world."
Wolfgang Pauli
Sommerfeld's method as a craft:
• "...A craft, in that, working directly with the data one extrapolates and
interpolates, drawing conclusions not from model-based deductions, but from
arithmetic and graphical approximations, drawing on special experience to
strike a balance between different sets of the always-insufficient information
from spectroscopic data." (Seth, pg. 225.)
2.
From the Old World of Waves to the New World of Quanta.
Contrast between Bohr and Sommerfeld:
• "While Bohr sought an analogy and hence some form of connection between
[electromagnetism and the quantum theory],...Sommerfeld strove...to keep
them apart." (Seth, pg. 226)
Example: Sommerfeld's Spherical Wave Theory
• Recall: In classical electromagnetism, a moving electron emits radiation
through constant coupling to the electromagnetic field (i.e., "aether").
• In Bohr's model, an electron moving in a stationary orbit does not couple to
the "aether"; only during a transition between stationary orbits does an
electron emit radiation.
• This requires an "abstract mode of expression":
"We must speak not of an electron but of a solution to Maxwell's
equations, which is determined by conditions of coupling in the
process of emission between the atom and the ether. The more
abstract mode of expression, to which we are forced, is inevitable
if we wish to follow out logically the view of the quantum theory."
• During a transition, what produces radiation is not a classical electron, but a
"solution to Maxwell's equations".
• Quantum theory describes production of radiation (during transitions);
classical theory describes propagation of radiation (after it has been emitted).
• In particular: The emitted radiation takes the form of a classical spherical
electromagnetic wave.
• Recall: Bohr's relation between frequency νn',n'' of emitted radiation and
energy difference En' − En'' of transition:
hνn',n'' = (En' − En'')
describes frequency of
classical spherical EM wave
describes quantum
states of atom
"The aether demands its hνn',n'',
the atom furnishes it by giving up
an amount of energy En' − En''."
• Development of spherical wave theory: 1919-1922 in first 3 editions of
Atomic Structure and Spectral Lines.
• But: Abandoned in 4th edition.
• Why? Compton's (1922) "A Quantum Theory of the Scattering
of X-rays by Light Elements".
The "Compton Effect"
Arthur Compton
• Describes scattering of light by an electron.
• Suppose: Planck's quantum hypothesis applies to light (Einstein 1905).
• And: A quantum of light (a photon) with
frequency ν and energy E has momentum
p = E/c = hν/c.
• Can now model the scattering of light by
an electron as an elastic collision between
a photon and an electron in which
momentum and energy are conserved...
• Conservation of momentum requires: pi = pf + pe
• So:
pe ⋅ pe = (pi − pf) ⋅ (pi − pf)
• Or:
pe 2 = pi 2 + p f 2 − pi p f cos θ
(∗)
pi = hνi/c
• Conservation of energy requires:
hνi + mec 2 = hν f + (mec 2 )2 + (pec)2
• Or:
2
(
pe = hνi /c − hν f /c + me
)
2
− m 2ec 2
• Now: Subtract (∗∗) from (∗):
λf − λi =
Wavelength of
photon λ = c/ν
h
(1 − cos θ)
mec
The "Compton wavelength" of an electron
pf = hνf/c
(∗∗)
• Upshot: Radiation behaves like
discrete particles (photons) with
momenta and energy!
• Additional implication: Matter
(electrons) has associated
wavelength!
"Whereas earlier, I had sought to maintain the wave
theory for pure propagation phenomena for as long
as possible, I have been pushed ever more to the
ground of the extreme quantum theory of light."
• Major blow to Sommerfeld: Spherical wave theory was his alternative to
Bohr's "magic wand" (correspondence principle).
• But: Still critical of Bohr's modellmässig approach...
"The difficulties that emerge ever more clearly
in atomic physics appear to me to arise less
from an exaggerated application of the quantum
theory and much more from a perhaps exaggerated belief in the reality of concepts of models."
3.
The Pauli Exclusion Principle.
• 1921: Landé's improved formula for Zeeman splitting (in terms of energy
differences between spectral lines):
ΔE = mgωh
• m = integer, and ω = 2πν, where ν = |H||e|/4πme.
• g = "g-factor", calculated by comparison with experimental data:
singlets:
g =1
doublets:
2j
g=
,
2k −1
triplets:
⎧
⎪
1 + 1/k,
⎪
⎪
⎪
g =⎪
⎨1 −1/k + 1/(k −1),
⎪
⎪
⎪
1 −1/(k −1),
⎪
⎪
⎩
k = azimuthal quantum number
j = inner quantum number
j = k or k −1
j =k
j = k −1
j = k −2
Bravo, you are able
to work miracles!"
• Heisenberg's Rumpf model:
H
R
K
• Outermost electron with angular
momentum k encoded in vector K.
• Core ("Rumpf") of inner electrons
with angular momentum r encoded in
vector R.
Werner
Heisenberg
• Zeeman effect due to coupling between K
and R in presence of magnetic field H.
• Splitting factor mg corresponds to sum of projections of K and R onto H.
3 1 R2 − K 2
• Landé derives g = + ⋅ 2
2 2 J − 14
R = r/2, K = k − 1/2,
J = j for even multiplets and J = j − 1/2
for odd multiplets (j is total ang mo).
• A modellmässig explanation of the g-factor that fits experimental data!
• But: Assumes R contributes twice as much to splitting factor as K:
mg = |K|cos(K, H) + 2|R|cos(R, H)
• And: Classical electron theory (Larmor's theorem) entails factor of 2 should
not be present!
• 1924. Pauli applies himself to the problem:
"...it is much too difficult for me and I wish that I were a film comedian or something similar and had never heard anything of physics!"
Wolfgang Pauli
• Turning point: Pauli calculates relativistic correction for Rumpf model.
- Rumpf electrons move at relativistic speeds.
- So: Splitting of spectral lines should depend on a relativistic factor
that increases as the number of Rumpf electrons increases.
- And: This correction factor should be observable in different elements.
- But: Empirical data indicates no such correction.
• Pauli's conclusion: Rumpf can't be responsible for Zeeman effect!
• Zeeman effect must be due solely to outermost electron.
"The doublet structure of the alkali spectrum, as well as the violation
of Larmor's theorem comes about through a peculiar, classically nondescribable kind of Zweideutigkeit [ambiguity, doubled signification]
of the quantum-theoretical characteristics of the light-electron."
• 1925: "On the Connection of the Closing of Electron Groups in the Atom to
the Complex-Structure of Spectra."
• Electron states in an atom are uniquely characterized by 4 quantum numbers:
principle n, azimuthal k, and two magnetic numbers m1, m2.
• These states obey an "Exclusion Principle":
"There can never be two or more equivalent electrons in an
atom for which, in strong fields, the values of all quantum
numbers... coincide. If an electron is to be found in an atom
for which these quantum numbers (in an external field)
possess determinate values, then this state is 'occupied'."
• Seth: "The rule abandons, as Pauli would emphasize to Bohr, any talk of
orbits and provides instead a formal quantum rule connecting the number of
terms into which a single spectral line could split with the periodic structure
of the table of elements." (pg. 256.)
• A victory of quantum Gesetzmässigkeiten over model-based analysis.
• Explanation of fourth quantum number in terms of spin: Kronig, Goudsmit
and Ulenbeck.
- m2 encodes two degrees of freedom of electron in outermost orbit.
- Suppose these degrees of freedom are associated with the angular
momentum of a spinning electron (in either a clockwise or counter-clockwise
direction about a given axis).
• Pauli is resistant.
- Relativistic considerations: spinning electrons and relativity.
- Sommerfeld's influence: Gesetzmässigkeiten trumps modellmässig reasoning.
Contemporary description of electron states in atoms:
• Electron state characterized by four properties (n, , m, ms), energy n, orbital
angular momentum , z-component of orbital angular momentum m, and
spin ms.
• n = 1, 2, ...;  = 0, 1, 2, ... (n − 1); m = −, ... 0, ..., ; ms = −, +.
• Exclusion Principle: No two electrons can be in the same state (i.e., no two
electrons can have all the same values of these four properties).
n: 1
2
3
4
energy shells
: 0
0 1
012
0123
K shell (n = 1)
L shell (n = 2)
M shell (n = 3)
N shell (n = 4)
etc.
Z
Element
1
H hydrogen
1
2
He helium
2
3
Li lithium
2
1
4
Be beryllium
2
2
5
B boron
2
2 1
6
C carbon
2
2 2
7
N nitrogen
2
2 3
8
O oxygen
2
2 4
9
F fluorine
2
2 5
2
2 6
10 Ne neon
orbitals
s orbital ( = 0)
p orbital ( = 1)
d orbital ( = 2)
f orbital ( = 3)
etc.
Example: The 3 electrons in a
lithium atom are characterized by
(1, 0, 0, +),
(1, 0, 0, −),
(2, 0, 0, +).
4.
Reconsidering Kuhnian Revolutions.
• "Sommerfeld and the members of his school did not register a sense of crisis, a
sense of the existence of paradoxical and insurmountable anomalies, or a
sense of the occurrence of a revolution." (Seth, pg. 266.)
"The new development does not signify a revolution, but a
joyful advancement of what was already in existence, with
many fundamental clarifications and sharpenings." (1929)
"My memory is that everyone was so excited about the possibilities of solving problems, answering questions, the mechanism provided by the new quantum mechanics, that there
was little discussion of those details of interpretation." (1963)
Elder Pauli
"No, there was no great puzzle and I think this is the
greatest characteristic of the Sommerfeld group; we
were not made aware of great puzzles. We were given
the impression that here was a wonderful tool." (1964)
Hans Bethe
• "Scientific revolutions, in other words, are revolutions of conceptual
foundations, not puzzle-solving techniques. Most simply: Science sees
revolutions of principles, not of problems." (Seth, pg. 268.)
Critique of Kuhn:
• Kuhn's analysis focuses on individuals and/or communities.
• "Mesoscopic level" analysis focus on small groups within communities.
• Claim: To understand scientific change, both types of analysis are important.
• "The result is a scientific community made up, in the majority, of those who
solve problems (and eschew the pursuit of revolutions) and a much smaller
group whose focus on principles and foundations means that the only change
that counts is a revolutionary and fundamental one."