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14. Crafting the Quantum: Chaps 6-7, Conclusion. I. Prinzipienfuchser and Virtuosen

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14. Crafting the Quantum: Chaps 6-7, Conclusion. I. Prinzipienfuchser and Virtuosen 14. Crafting the Quantum: Chaps 6-7, Conclusion. I. Prinzipienfuchser and Virtuosen. 1. Einstein on Principles. • 1925: Einstein letter to Paul Ehrenfest. "There exist Prinzipienfuchser and Virtuosi. We belong, all three of us [Ehrenfest, Einstein, Bohr] to the first category and have (at least certainly we two) little virtuosic ability." • "Prinzipienfuchser" = those obsessed by principles (in an artistic, as opposed to technical, way). • "Virtuosi" = those possessed of great technical skill (that, perhaps, lacks genuine insight). Sommerfeld Born • 1919 article in Nature: Constructive theories vs. theories of principle. "Constructive theories attempt to build up a picture of the more complex phenomena out of the materials of a relatively simple scheme from which they start out." • Example: kinetic theory of gases. "[Theories of priniciple] ...employ the analytic, not the synthetic method. The elements which form their basis and starting point are not hypothetically constructed but empirically discovered ones, general characteristics of natural processes, principles that give rise to mathema- tically formulated criteria which separate processes or the theoretical representations of them have to satisfy." • Examples: thermodynamics, special relativity, general relativity. • Eddington (1918) on general relativity as a theory of principle: "The nearest parallel to it is found in the applications of the second law of thermodynamics, in which remarkable conclusions are deduced from a single principle without any inquiry into the mechanism of the phenomena; similarly, if the principle of equivalence is accepted, it is possible to sride over the difficulties due to ignorance of the nature of Sir Arthur Eddington gravitation and arrive directly at physical results." • Product of 19th century Cambridge pedagogical system: problem solving based on mechanical modeling. • But: Uniquely sympathetic to Einstein's principled approach (Warwick 2003). • Later influence: Paul Dirac's (1930) Principles of Quantum Mechanics: "The growth of the use of transformation theory as applied to first relativity and later to the quantum theory, is the essence of the new method in theoretical physics. Further progress lies in the direction of making our equations invariant under wider and still wider applications." Paul Dirac (Eddington student) 2. Ehrenfest's (1913) Adiabatic Hypothesis. In an infinitely slow, reversible transformation, allowed quantum motions are converted into other allowed quantum motions. Paul Ehrenfest • What this means: If: You know the "allowed motions" (i.e., "adiabatic invariants") of a simple quantum system. And: If you transform the simple system into a more complex system by an "adiabatic" transformation (i.e., an infinitely slow, reversible transformation). Then: The "allowed motions" of the simple system remain the "allowed motions" of the more complex system! • A way of constructing quantum descriptions of complex systems from the quantum descriptions of simple systems. Example: • Let the simple quantum system be a Hertzian (( )) resonator oscillating in an electric field. • Then the following quantity is an "adiabatic invariant" (remains constant under an adiabatic transformation): T P T is the time average of the kinetic energy over period P. = ∫ dt ⋅T ω 0 ω is the frequency of the motion. • T is half the total energy, so Planck's (original) Quantum Hypothesis entails: T 1 = nh ω 2 • Now: Suppose we adiabatically decrease the electric field. • Result: The original harmonic oscillation of the resonator is adiabatically transformed into a uniform rotation. Initial allowed oscillatory motions (ellipses) Final allowed rotational motions • Any adiabatic invariant associated with ⎛ ⎞ ⎛ ⎞ ⎜T ⎟ ⎜T ⎟ 1 the initial oscillatory motions remains ⎜ ⎟ = ⎜ ⎟ = nh ⎝⎜ω ⎠⎟ ⎝⎜ω ⎠⎟ 2 the same for the final rotational motions: initial final The final rotatational frequency is ω = q/4π. The final kinetic energy is T = pq/2. • So: The final momentum is p = ±nh/4π. Moral: You can calculate properties for a complex quantum system from properties of a more simple quantum system if the two are connected by an adiabatic transformation and possess adiabatic invariants. 3. Bohr on Principles. • Recall: Problematic aspects of Bohr's 1913 model of the atom. Specification of stationary states doesn't obey classical mechanics! Motion within a stationary state obeys classical mechanics, but not classical electrodynamics! Transitions between stationary states don't obey classical mechanics or classical electrodynamics! Principle of Mechanical Transformability (1918): Ordinary mechanical laws prevail in infinitely slow (adiabatic) transformations of stationary states. • Based on Ehrenfest's Adiabatic Hypothesis. "[Ehrenfest's] principle allows us to overcome a fundamental difficulty. In fact we have assumed that the direct transition between two such [stationary] states cannot be described by ordinary mechanics, while on the other hand we possess no means of defining an energy difference be- tween two states if there exists no possibility for a continuous mechanical connection between them. It is clear, however, that such a connection is afforded by Ehrenfest's principle which allows us to transform mech- anically the stationary states of a given system into those of another..." • The Principle of Mechanical Transformability provided "...a justification for the application of specific arguments from mechanics to what were, by definition, non-mechanical systems." (Seth, pg. 196.) Correspondence Principle (1920): Individual quantum transitions between stationary states correspond to components of the classical frequency spectrum. • "...a means of applying arguments from classical electrodynamics to systems defined by their failure to accord with Maxwell's equations" (Seth pg. 196). • What does it mean? What is the nature of the "correspondence"? Classical Picture: • The path x(t) of an electron undergoing periodic motion with orbital frequency ω is given by a Fourier series: ∞ x(t) = c cos(τωt) ∑ τ τ=1 = c cos(ωt)+c cos(2ωt)+c cos(3ωt)+ 1 2 3 each term is called a "harmonic" • The radiation emitted by an accelerating electron is given by all the frequences in the harmonics of its motion: ω, 2ω, 3ω, ... Bohr's Picture: n' • Radiation is not emitted by an electron in a stationary state, but by an electron n'' transitioning between stationary states: • For a transition between states with energies En' and En'', the emitted radiation has a single frequency given by νn',n'' = (En' − En'')/h. • Three versions of the Correspondence Principle: Correspondence Principle (Frequency Version): νn',n'' = τω, for large values of n, where n' − n'' = τ. For large n, the frequency of emitted radiation for a transition between the n' and n'' states is given by the frequency of the τth harmonic of the classical motion. Correspondence Principle (Intensity Version): 2 Pn',n'' ∝ |cτ| , for large values of n, where n' − n'' = τ, and Pn',n'' is the probability of a transition between the n' and n'' states. For large n, the probability of a transition between the n' and n'' states is 2 proportional to the intensity |cτ| of the τth harmonic of the classical motion. Correspondence Principle (Selection Rule Version): A transition from n' to n'' is allowed if and only if there exists a τth harmonic between the n' and n'' states, where n' − n'' = τ. "Although the process of radiation cannot be described on the basis of the ordinary theory of electrodynamics, according to which the nature of the radiation emitted by an atom is directly related to the harmonic components occurring in the motion of the system, there is found, nevertheless, to exist a far-reaching correspondence between the various types of possible tran- sitions between the stationary states on the one hand, and the various harmonic components of the motion on the other hand." "Bohr has discovered in his principle of correspondence a magic wand (which he himself calls a formal principle), which allows us immediately to make use of the results of the classical wave theory in the quantum theory." (1919) "The magic of the correspondence principle has proved itself generally through the selection rules of the quantum num- bers, in the series and band spectra… Nonetheless I cannot view it as ultimately satisfying on account of its mixing of quantum-theoretical and classical viewpoints.." (1924) • "...a very peculiar idea of a principle" (Seth, pg. 198). "variable and groping...[but in a good non-rigid way]" (1921) Ehrenfest "I always liked [it] just because it gave that kind of lack of rigidity, that flexibility in the picture, which could lead to real mathematical schemes." (1963) Elder Heisenberg Three different views on principles (Seth, pg. 200): • Planck: Principles have the lasting validity of a holy commandment in a world of absolute disunity. • Einstein: Principles are fixed truths of Being in a world of absolute unity. • Bohr: Principles are both approximate and necessarily changeable in a world of dynamic change..
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