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12. Crafting the Quantum: Chaps 1-3 12. Crafting the Quantum: Chaps 1-3. • History of development of theoretical physics: 1890-1926. • Cast of Characters: Arnold Sommerfeld Max Planck Niels Bohr Albert Einstein Werner Heisenberg Wolfgang Pauli • Key issues: What is the content of theoretical physics? What are its methods? How should it be taught? Claim: "...theoretical physics at the turn of the twentieth century cannot be understood as a 'distillation' of theory from physics, but rather must be seen as having been actively constructed from multiple and varied parts." (pg. 14) • Two contrasting approaches: Physics of Principle (Planck, Einstein, Bohr): Ex: Planck. Method: principled analysis based on thermodyn & statistics. Experiment limited to testing of conclusions. Physics of Problems (Sommerfeld): Method: problem-based analysis based on electrodyn & mechanics. Experiments as constitutive element at multiple states in the production of theoretical work. Claim: Contrary to Kuhn, "...for those working within the context of a physics of problems, neither crises nor revolutions came to pass in the mid 1920s." (pg. 10) 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." • 1893: Assistant to Felix Klein in Göttingen. 1872: Erlangen Programme: Klein's categorization of geometries in terms of their symmetry group. 1882: Klein bottle: 2-dim non-oriented closed surface. Felix Klein Compare with Mobius strip: 2-dim non-oriented open surface. 2. Sommerfeld as Engineer. • 1900: Appointed Professor of Technical Mechanics at the Rheinisch- Westphalisch Technische Hochschule (RWTH), Aachen. • 1870: Aachen Hochschule established. • 1880s: Cultural conflicts between traditional universities and technical institutes in Germany. Mid-19th century: Verien Deutsches Ingenieure (Society of German Engineers). Legitimation of engineering profession: artisinal engineers vs. polytechnic graduates. Theory vs. practice at polytechnics: How much theory should an engineer study? • Klein's fear: University mathematics will be pushed out of engineering schools, and then universities, too! Seth: Advocates mathematical engineering (mathematical formulations of engineering subjects). • Sommerfeld's view: 1903 speech on "The Scientific Aims and Results of Modern Technical Mechanics". A defense of the technical sciences and a claim for the relevance of theory to such sciences. Seth: Advocates engineering mathematics (application of mathematical modeling in engineering) • Joint project: (1897) Über die Theorie des Kreisels (On the Theory of the Gyroscope). Klein contributes theory and simple examples. Sommerfeld contributes more technical applications. 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. • 1888: H. Hertz generates and detects electromagnetic waves. EM waves produced ... and spark by spark here... in gap here. Heinrich Hertz ... cause current in loop there... Hertzian radiator Hertzian resonator • 1892: H. A. Lorentz's electron theory. • 1897-98: J. J. Thompson measures charge-to-mass ratio of electron. H. A. Lorentz • 1901: W. Wein All mechanics can be reduced to electromagnetic theory. Characteristics (Seth, pg. 32): A distaste for, and mistrust of, mechanical modeling, especially as applied to microscopic phenomena; a belief that the only physical realities were electromagnetic in nature; a programmatic commitment toward a "concentration of effort on problems whose solution promised to secure a universal physics based solely on electromagnetic laws and concepts". • Example of a paradigm? • Seth (pg. 33): Not quite. Those who used it were selective but not exclusive or dogmatic. Sommerfeld's 1907 Lectures on Planck's Radiation Theory • 1906: Planck's Vorlesungen über die Theorie der Wärmestrahlung (The Theory of Heat Radiation). • Goal: Derive a formula for the energy distribution of "black-body radiation". Black-body radiation = thermal radiation emitted by a perfectly absorbing object. • Initial question: How should a black-body in thermal equilibirum be modeled? Story to come: Rayleigh and Jeans use electrodynamic analysis and obtain a formula that is inconsistent with experimental data. Planck obtains a formula that fits the experimental data, and then works backwards to justify it on the basis of thermodynamic and statistical principles (requires assumption of "discontinuity" about the nature of energy). Sommerfeld's electromagnetic worldview orients him favorably towards Rayleigh-Jeans and away from the principled approach of Planck. Rayleigh and Jeans (1900, 1905): • Suppose black-body is a cavity that traps EM waves in the form of standing waves with different modes. • Then: number of modes per unit frequency 8πν 2 = 3 volume of cavity c • Now: In equilibirum, the average energy per mode is kT. Why? The Equipartition Theorem says that a gas in equilibrium has an average energy of kT for each degree of freedom. The higher the frequency, the more modes you can And: R&J consider the trapped EM waves to have normal fit into the cavity.* modes with two degrees of freedom. • So: Rayleigh-Jeans Law energy per unit frequency 8πν 2 = ρ(ν,T) = kT 3 volume of cavity c • But: Experimental data on black-body radiation indicate this law is wrong! *Diagrams from http://hyperphysics.phy-astr.gsu.edu/hbase/mod6.html#c2 8πν 2 Rayleigh-Jeans Law: kT 3 c experimental data Radiation Intensity Frequency • Planck: The data fit the following distribution: Planck Law energy per unit frequency 8πν 2 hν = ρ(ν,T) = h = constant 3 hν/kT volume of cavity c e −1 hν • And: This requires that the average energy per mode is and ehν/kT −1 not kT. • How can this disregard for the Equipartition Theorem be justified? Planck's Model for Black-Body Radiation (1900) • Suppose the walls of the black-body cavity are composed of "Hertzian resonators" (i.e., little oscillators). 8πν 2 average energy • Then (1899): ρ(ν, T) = × of resonator c3 hν • So: Planck Law entails the average energy of a resonator is . ehν/kT −1 • The average entropy S of a resonator with this energy E is defined by the thermodynamic relation 1/T = ∂S/∂E. • And: This is the same form as the entropy for the system derived by statistical considerations (a la Boltzmann), provided that the following relation hold for the average energy of a resonator: Planck's quantum hypothesis E = n, where = hν and n = 1, 2, 3, ... • What this means: Planck can derive the experimentally correct energy distribution for black-body radiation provided he assumes the energy of the resonators is "quantized" in discrete units of hν. 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 entailed by his electron theory. "I think it is very possible that Planck's formula is only a good approximation." Seth Claim (pg. 37): "In deciding which theory to reject, the impotence of Planck's resonators outweighed the failure of the Rayleigh-Jeans equation to match available experimental results. ...The choice between Planck's and Jean's formulas was, rather, framed as a choice between two distinct methods." • Kuhn (1987): Lorentz' derivation of the Rayleigh-Jeans formula from the electron theory marked the beginning of acceptance of the quantum discontinuity. "We can adopt it [Planck's formula] only by altering profoundly our fundamental conceptions of electromagnetic phenomena." • Seth Claim (pg. 40): "Proponents of the electromagnetic worldview... may not have regarded the choice between continuity and discontinuity as the central issue. Rather, the question that 'came to challenge' them... was whether the electron theory could produce a Planck-like formula. Once it was accepted that this was impossible, discontinuity was adopted quite readily by this group." II. The Pedagogical Economy of the Sommerfeld School. • Cambridge University, 19th Century: culture of examination. Mathematical Tripos. Rise of private tutors: private specialized seminars vs. public survey lectures. Edward Routh Introduction of textbooks with problems. Private tutor 1855-1888 - Klein's (1898) glowing introduction to Routh's Dynamics of a System of Rigid Bodies: "[This book] must be of the deepest significance for every man that does not want to restrict himself to abstract principles, but who would like to comprehend the application of the principles to concrete problems." Karl Pearson describes differences between British and German students in the 1880s: "Every bit of mathematical research is really a 'problem', or can be thrown into the form of one, and in post-Cambridge days in Heidelberg and Berlin I found this power of problem solving gave one advantages in research over German students, who had been taught mathematics in theory, but not by 'problems'." Karl Pearson 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 research articles. (iii) Informal
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