12.12. Crafting the Quantum: Chaps 1-3.Crafting the Quantum: Chaps 1-3.• History of development of theoretical physics: 1890-1926.

• Cast of Characters:

• Key issues:

What is the content of theoretical physics?

What are its methods?

How should it be taught?

Arnold Sommerfeld

Max Planck

Niels Bohr

Albert Einstein

Werner Heisenberg

Wolfgang Pauli

Claim: "...theoretical physics at the turn of the twentieth century cannot beunderstood as a 'distillation' of theory from physics, but rather must be seenas having been actively constructed from multiple and varied parts." (pg. 14)

• Two contrasting approaches:

Claim: Contrary to Kuhn, "...for those working within the context of aphysics of problems, neither crises nor revolutions came to pass in the mid1920s." (pg. 10)

Physics of Principle (Planck, Einstein, Bohr):

Method: principled analysis based on thermodyn & statistics.

Experiment limited to testing of conclusions.

Ex: Planck.

Method: problem-based analysis based on electrodyn & mechanics.

Experiments as constitutive element at multiple states in theproduction of theoretical work.

Physics of Problems (Sommerfeld):

I. Sommerfeld's Physics of Problems.• A blend of mathematics, engineering, and physics.

1. Sommerfeld as Mathematician.• 1891: Thesis in Königsberg. "Arbitrary Functions in Mathematical Physics."

1872: Erlangen Programme: Klein's categorization ofgeometries in terms of their symmetry group.

Compare with Mobius strip:2-dim non-oriented open surface.

Felix Klein

• 1893: Assistant to Felix Klein in Göttingen.

1882: Klein bottle: 2-dim non-oriented closed surface.

2. Sommerfeld as Engineer.• 1900: Appointed Professor of Technical Mechanics at the Rheinisch-

Westphalisch Technische Hochschule (RWTH), Aachen.

• 1870: Aachen Hochschule established.

Mid-19th century: Verien Deutsches Ingenieure(Society of German Engineers).

Legitimation of engineering profession: artisinalengineers vs. polytechnic graduates.

Theory vs. practice at polytechnics: How muchtheory should an engineer study?

• 1880s: Cultural conflicts between traditional universities and technicalinstitutes in Germany.

A defense of the technical sciences and a claim for the relevance oftheory to such sciences.

Seth: Advocates mathematical engineering (mathematical formulationsof engineering subjects).

• Klein's fear: University mathematics will be pushed out ofengineering schools, and then universities, too!

• Sommerfeld's view: 1903 speech on "The Scientific Aims andResults of Modern Technical Mechanics".

Seth: Advocates engineering mathematics (application of mathematicalmodeling in engineering)

Klein contributes theory and simple examples.

Sommerfeld contributes more technicalapplications.

• Joint project: (1897) Über die Theorie des Kreisels(On the Theory of the Gyroscope).

2. Sommerfeld as Physicist.• 1906: Appointed Professor of Theoretical Physics at University of Munich.

The Electromagnetic Worldview

• 1873: Maxwell's A Treatise on Electricity and Magnetism.

Hertzian radiator Hertzian resonator

... cause currentin loop there...

EM waves producedby spark here...

... and sparkin gap here.

• 1888: H. Hertz generates and detects electromagnetic waves.

Heinrich Hertz

• 1897-98: J. J. Thompson measurescharge-to-mass ratio of electron.

• 1901: W. WeinAll mechanics can be reducedto electromagnetic theory.

Characteristics (Seth, pg. 32):

A distaste for, and mistrust of, mechanical modeling, especially asapplied to microscopic phenomena;

a belief that the only physical realities were electromagnetic in nature;

a programmatic commitment toward a "concentration of effort onproblems whose solution promised to secure a universal physics basedsolely on electromagnetic laws and concepts".

• Example of a paradigm?

• Seth (pg. 33): Not quite. Those who used it were selective but not exclusiveor dogmatic.

• 1892: H. A. Lorentz's electron theory.

H. A. Lorentz

Story to come:

Sommerfeld's 1907 Lectures on Planck's Radiation Theory

• 1906: Planck's Vorlesungen über die Theorie der Wärmestrahlung(The Theory of Heat Radiation).

• Initial question: How should a black-body in thermal equilibirum bemodeled?

• Goal: Derive a formula for the energy distribution of "black-body radiation". Black-body radiation = thermal radiation emitted by a perfectly absorbing object.

Rayleigh and Jeans use electrodynamic analysis and obtain a formula that isinconsistent with experimental data.

Planck obtains a formula that fits the experimental data, and then worksbackwards to justify it on the basis of thermodynamic and statisticalprinciples (requires assumption of "discontinuity" about the nature of energy).

Sommerfeld's electromagnetic worldview orients him favorably towardsRayleigh-Jeans and away from the principled approach of Planck.

Rayleigh and Jeans (1900, 1905):

• Now: In equilibirum, the average energy per mode is kT.

• But: Experimental data on black-body radiation indicate this law is wrong!

energy per unit frequency

volume of cavity= ρ(ν,T) =

8πν2

c3kT

Rayleigh-Jeans Law• So:

• Suppose black-body is a cavity that traps EM waves in theform of standing waves with different modes.

*Diagrams from http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html#c2

number of modes per unit frequency

volume of cavity=

8πν2

c3

• Then:

The higher the frequency,the more modes you canfit into the cavity.*

Why? The Equipartition Theorem says that a gas in equilibriumhas an average energy of kT for each degree of freedom.

And: R&J consider the trapped EM waves to have normalmodes with two degrees of freedom.

Rayleigh-Jeans Law:

8πν2

c3kT

experimental data

Frequency

Rad

iation

Inte

nsi

ty

• And: This requires that the average energy per mode is and

not kT.

hνehν/kT −1

• How can this disregard for the Equipartition Theorem be justified?

• Planck: The data fit the following distribution:

energy per unit frequency

volume of cavity= ρ(ν,T) =

8πν2

c3

hνehν/kT −1

Planck Law

h = constant

Planck's Model for Black-Body Radiation (1900)

• The average entropy S of a resonator with this energy E is defined by thethermodynamic relation 1/T = ∂S/∂E.

• Suppose the walls of the black-body cavity are composedof "Hertzian resonators" (i.e., little oscillators).

hνehν/kT −1

.• So: Planck Law entails the average energy of a resonator is

• Then (1899): ρ(ν, T) =

8πν2

c3× average energy

of resonator

• What this means: Planck can derive the experimentally correct energydistribution for black-body radiation provided he assumes the energy of theresonators is "quantized" in discrete units of hν.

• And: This is the same form as the entropy for the system derived bystatistical considerations (a la Boltzmann), provided that the followingrelation hold for the average energy of a resonator:

E = n, where = hν and n = 1, 2, 3, ...

Planck's quantum hypothesis

Sommerfeld's Critique of Planck• Resonator model can't guarantee cavity is in equilibrium.

Interaction between resonators is meant to bring system into equilibrium.

But: Only resonators at same frequency can interact.

• To demonstrate that Planck's formula is the equilibrium energy distribution,it must be shown to maximize the entropy.

But: Planck's formula results from his quantum hypothesis alone.

• Moreover, Lorentz (1908) showed that the Rayleigh-Jeans formula is entailedby his electron theory.

"I think it is very possible that Planck'sformula is only a good approximation."

Seth Claim (pg. 37): "In deciding which theory to reject, the impotence ofPlanck's resonators outweighed the failure of the Rayleigh-Jeans equation tomatch available experimental results. ...The choice between Planck's andJean's formulas was, rather, framed as a choice between two distinct methods."

• Seth Claim (pg. 40): "Proponents of the electromagnetic worldview... maynot have regarded the choice between continuity and discontinuity as thecentral issue. Rather, the question that 'came to challenge' them... waswhether the electron theory could produce a Planck-like formula. Once it wasaccepted that this was impossible, discontinuity was adopted quite readily bythis group."

• Kuhn (1987): Lorentz' derivation of the Rayleigh-Jeans formula from theelectron theory marked the beginning of acceptance of the quantumdiscontinuity.

"We can adopt it [Planck's formula] only byaltering profoundly our fundamentalconceptions of electromagnetic phenomena."

II. The Pedagogical Economy of the Sommerfeld School.• Cambridge University, 19th Century: culture of examination.

Mathematical Tripos.

- Klein's (1898) glowing introduction to Routh's Dynamics of a Systemof Rigid Bodies:

"[This book] must be of the deepest significance forevery man that does not want to restrict himself toabstract principles, but who would like to comprehendthe application of the principles to concrete problems."

Karl Pearson describes differences between British and German studentsin the 1880s:

Karl Pearson

"Every bit of mathematical research is really a 'problem', or canbe thrown into the form of one, and in post-Cambridge days inHeidelberg and Berlin I found this power of problem solvinggave one advantages in research over German students, whohad been taught mathematics in theory, but not by 'problems'."

Edward RouthPrivate tutor1855-1888

Rise of private tutors: private specialized seminars vs. publicsurvey lectures.

Introduction of textbooks with problems.

Sommerfeld's Teaching Methods (1906-1926, Univ. Munich)

(i) Public lectures (open to all students).

(ii) Seminars (by approval only). Review and critique of current researcharticles.

(iii) Informal contact with students:

"seminary" (waiting room outside office)

ski-trips, other social contact outside university

• Importance of (ii) and (iii):

- Tacit knowledge of how to solve particular types of problems: "tricks"learned only via direct contact and practice (Seth, pg. 58).

- Tacit knowledge of how to read and write research articles (Seth, pg. 59).

But: What exactly is "tacit knowledge"?

- Knowledge that cannot be formalized or verbalized?

- Knowledge that is not "explicit"?

• Contrast (iii) with other major centers of physics instruction in Germany:

Bohr's Copenhagen group. Less structured: not asmuch contact.

Niels Bohr

Born's Berlin group. More structured: no analogueof "seminary"; thus not as much contact. Max Born

• Result:

Sommerfeld students/postdocs include 6 Nobel prize winners(Heisenberg, Pauli, Debye, Bethe, Pauling, Rabi).

"What I particularly wonder about you isthat you have stamped so great a number ofyoung talents as if they came out of theground... You must have a gift for activatingand engaging the talents of your listeners."

Einstein on Sommerfeld (1922):

Born on Sommerfeld (1928): 10 professorships in theoretical physicsin universities in which German is spoken are Sommerfeld graduates.

Kuhn and Foucault on the Role of Pedagogy

• Recall Kuhn:

• The central point of instruction is to solveproblems.

• Such instruction serves to indoctrinate studentsinto the current paradigm.

• Role of textbooks is essential to such instruction.

• But: Can problem-solving skill be aquired just through textbooks?

• Warwick (2003): Cambridge graduates acquired tacit knowledge of problem-solving via interactions with private tutors.

• In particular: Problem-solving skills could not be learned solely throughtextbooks (Cambridge textbooks were incomprehensible to students at otherschools).

• Foucault on pedagogy:

• Training involves no essential distinction betweentheory and practice.

• The more relevent distinction is between "initiatorytime" (culture of apprenticeship) and "disciplinarytime" (culture of examination).

• Seth: "Whereas the Cambridge undergraduate system was a near-perfectinstantiation of Foucauldian disciline in action, Sommerfeld's remained amixture of the newer forms of training, particularly in its focus on problemsolving in both pedagogy and research, while nonetheless retaining elementsof older forms of apprenticeship." (pg. 66.)

• Sommerfeld's seminars = the transition between the disciplinary time oflectures, and the initiatory time of original research.