[Paleopsych] SW: Einstein and Quantizing Chaos

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Theoretical Physics: Einstein and Quantizing Chaos

    The following points are made by A. Douglas Stone (Physics Today 2005
    1) At the 11 May 1917 meeting of the German Physical Society, Albert
    Einstein (1879-1955), then a professor at the University of Berlin,
    presented the only research paper of his career that was written on
    the quantization of energy for mechanical systems.[1] The paper
    contained an elegant reformulation of the Bohr-Sommerfeld quantization
    rules of the old quantum theory, a rethinking that extended and
    clarified their meaning. Even more impressive, the paper offered a
    brilliant insight into the limitations of the old quantum theory when
    applied to a mechanical system that is nonintegrable -- or in modern
    terminology, chaotic. Louis de Broglie (1892-1987) cited the paper in
    his historic thesis on the wave properties of matter,[2] as did Erwin
    Schroedinger (1892-1987) in the second of his seminal papers on the
    wave equation for quantum mechanics.[3] But the 1917 work was then
    ignored for more than 25 years until Joseph Keller independently
    discovered the Einstein quantization scheme in the 1950s.[4] Even so,
    the significance of Einstein's contribution was not fully appreciated
    until the early 1970s when theorists, led by Martin Gutzwiller,
    finally addressed the fundamental difficulty of semiclassically
    quantizing nonintegrable Hamiltonians and founded a subfield of
    research now known as quantum chaos.
    2) Even today, Einstein's insight into the failure of the
    Bohr-Sommerfeld approach is unknown to the large majority of
    researchers working in quantum physics. It seems appropriate, in this
    centennial of Einstein's miracle year, to put the achievement of his
    obscure 1917 paper in a modern context and to explain how he
    identified a simple criterion for determining if a dynamical system
    can be quantized by the methods of the old quantum theory.
    3) Einstein's paper was titled "On the Quantum Theorem of Sommerfeld
    and Epstein." In his title, Einstein was acknowledging physicist Paul
    Epstein, who had written a paper relating the Sommerfeld rule to the
    form of the constants of motion. Epstein's name has not survived in
    the context of the Sommerfeld rule, and the quantization condition
    discussed by Einstein is now referred to as either Bohr-Sommerfeld or
    WKB (Wentzel-Kramers-Brillouin) quantization.
    4) Although Einstein's antipathy to certain aspects of modern quantum
    theory is well known, there appears to be a renewed appreciation this
    year of his seminal contributions to quantum physics. With his
    introduction of the photon concept in 1905, his clear identification
    of wave-particle duality in 1909, his founding of the quantum theory
    of radiation in 1917, and his treatment of the Bose gas and its
    condensation in 1925, Einstein laid much of the foundation of the
    theory. He commented to Otto Stern, "I have thought a hundred times as
    much about the quantum problems as I have about general relativity
    theory." We should add to his list of illustrious achievements another
    advance, modest on the scale of his genius, but brilliant by any other
    standard: the first identification of the problem of quantizing
    chaotic motion.[5]
    References (abridged):
    1. A translation of the paper appears in The Collected Papers of
    Albert Einstein , vol. 6, A. Engel, trans., Princeton U. Press,
    Princeton , NJ (1997), p. 434
    2. L. de Broglie, PhD thesis, reprinted in Ann. Found. Louis de
    Broglie 17, 22 (1992)
    3. E. Schroedinger, Ann. Phys. ( Leipzig ) 489, 79 (1926)
    4. J. B. Keller, Ann. Phys. (N.Y.) 4, 180 (1958). An initial version
    of the paper was published by Keller as a research report in 1953. See
    also J. B. Keller, S. I. Rubinow, Ann. Phys. (N.Y.) 9, 24 (1960)
    5. J. B. Keller, SIAM Rev. 27, 485 (1985)
    Physics Today http://www.physicstoday.org
    Related Material:
    The following points are made by Ian Stewart (Nature 2004 430:731):
    1) A central problem in modern physics is to find effective methods
    for quantizing classical dynamical systems -- modifying the classical
    equations to incorporate the effects of quantum mechanics. One of the
    main obstacles is the disparity between the linearity of quantum
    theory and the nonlinearity of classical dynamics. Recently, Cushman
    et al (Phys. Rev. Lett. 2004 93: 024302) analyzed a quantum version of
    the spring pendulum, whose resonant state was first discussed by
    Enrico Fermi (1901-1954), and which is a standard model for the carbon
    dioxide molecule.
    2) Cushman et al demonstrated that when this system is quantized, the
    allowed states, or eigenstates, fail to form a perfect lattice,
    contrary to simpler examples. Instead, the lattice has a defect, a
    point at which the regular lattice structure is destroyed. They
    demonstrated that this defect can be understood in terms of an
    important classical phenomenon known as "monodromy". A
    quantum-mechanical cliche is Schroedinger's cat, whose role is to
    dramatize the superposition of quantum states by being both alive and
    dead. Classical mechanics now introduces a second cat, which
    dramatizes monodromy through its ability always to land on its feet.
    The work affords important new insights into the general problem of
    quantization, as well as being an example of the relation between
    nonlinear dynamics and quantum theory.
    3) The underlying classical model here is the swing-spring, a mass
    suspended from a fixed point by a spring. The spring is free to swing
    like a pendulum in any vertical plane through the fixed point, and it
    can also oscillate along its length by expanding and contracting. The
    Fermi resonance occurs when the spring frequency is twice the swing
    frequency. The same resonance occurs in a simplified model of the two
    main classical vibrational modes of the carbon dioxide molecule, and
    the first mathematical analysis of the swing-spring was inspired by
    this model.
    4) Using a modern technique of analysis known as reduction, which
    exploits the rotational symmetry of a system, Cushman et al
    demonstrated that this particular resonance has a curious implication,
    which manifests itself physically as a switching phenomenon. Start
    with the spring oscillating vertically but in a slightly unstable
    state. The vertical "spring mode" motion quickly becomes a "swing
    mode" oscillation, just like a clock pendulum swinging in some
    vertical plane. However, this swing state is transient and the system
    returns once more to its spring mode, then back to a swing mode, and
    so on indefinitely. The surprise is that the successive planes in
    which it swings are different at each stage. Moreover, the angle
    through which the swing plane turns, from one occurrence to the next,
    depends sensitively on the amplitude of the original spring mode.
    5) The apparent paradox here is that the initial state has zero
    angular momentum -- the net spin about the vertical axis is zero. Yet
    the swing state rotates from one instance to the next. Analogously, a
    falling cat that starts upside down has no angular momentum about its
    own longitudinal axis, yet it can invert itself, apparently spinning
    about that axis. The resolution of the paradox, for a cat, is that the
    animal changes its shape by moving its paws and tail in a particular
    way. At each stage of the motion, angular momentum remains zero and is
    thus conserved, but the overall effect of the shape changes is to
    invert the cat. The final upright state also has zero angular
    momentum, so there is no contradiction of conservation. This effect is
    known as the "geometric phase", or monodromy, and is important in many
    areas of physics and mathematics.
    Nature http://www.nature.com/nature
    Related Material:
    The following points are made by Miles Blencowe (Science 2004 304:56):
    1) In the macroscopic world of everyday experience, the motions of
    familiar objects such as dust particles, bumblebees, baseballs,
    airplanes, and planets are accurately described by Newton's laws.
    According to these classical laws, the trajectories of the objects can
    in principle be measured to arbitrary accuracy; any uncertainty in
    their motion is due to the imprecision of the measuring device. In
    contrast, in the microscopic world of atomic and subatomic particles
    such as the hydrogen atom and the electron, the probabilistic laws of
    quantum physics hold sway. Heisenberg's uncertainty principle limits
    the precision of simultaneous measurements of the position and
    velocity of a particle. And there is the superposition principle,
    which allows a particle to be simultaneously in two places. This
    latter principle is responsible for the interference pattern produced
    on a detection screen by a beam of particles that have passed through
    a sufficiently narrow-ruled grating. Such interference patterns have
    been observed even for beams of molecules with mass over 1000 times
    that of a hydrogen atom (1).
    2) Ever since the laws of quantum mechanics were first established
    early last century, physicists and philosophers have been occupied
    with the problem of how the macroscopic classical world emerges from
    the microscopic quantum world (2). Is there an actual boundary between
    the two, where some as yet undiscovered fundamental physical law
    governs the transition from quantum to classical behavior as the
    system size and/or energy scale increases? Or is classical physics
    just an approximation to quantum physics, even at macroscopic scales,
    so that if we were to try hard enough in our experiments, quantum
    behavior would be observed in the motion of macroscopic mechanical
    3) LaHaye et al (3) have described an experiment whose goal is to test
    Heisenberg's uncertainty principle on a vibrating mechanical beam that
    is about a hundredth of a millimeter long. While such a beam is tiny
    by everyday standards, it is equivalent in mass to about 10^(12)
    hydrogen atoms, certainly belonging well outside the traditional,
    microscopic quantum domain. The work of LaHaye et al comes hot on the
    heels of a recent related experiment(4). While neither experiment has
    quite reached the necessary sensitivity to test the uncertainty
    principle, they come much closer than all previous efforts.
    4) Under normal conditions, a mechanical beam will undergo classical
    thermal Brownian motion, vibrating in a random way as it is buffeted
    by the air molecules as well as the fluctuating defects in the beam.
    As the beam is cooled and the surrounding air is expelled, the thermal
    Brownian motion will decrease in amplitude, until just the irreducible
    quantum zero-point fluctuations of the beam in its lowest energy state
    remain. This zero-point motion is a consequence of the uncertainty
    principle that prevents the beam from being in a state of absolute
    rest. The temperature below which the beam must be cooled in order to
    freeze out the Brownian motion is related to the beam's resonant
    frequency. The frequency of the beam used by LaHaye et al(3) is about
    20 million cycles per second (20 MHz), and the lowest temperature to
    which they manage to cool the beam is about 60 millikelvins (mK). This
    is not quite cold enough, however; a 20-MHz beam must be cooled to
    about 1 mK in order for the zero-point motion to be comparable to the
    Brownian motion. On the other hand, a smaller beam with a much higher
    frequency of about 1 billion hertz (1 gigahertz, or GHz) was recently
    demonstrated (5). Such a beam would only need to be cooled to about 50
    mK for the quantum zero-point and classical Brownian motions to be
    comparable in amplitude, close to the lowest temperature that LaHaye
    et al(3) achieve in their experiment.
    References (abridged):
    1. L. Hackermueller et al., Phys. Rev. Lett. 91, 90408 (2003)
    2. A. J. Leggett, J. Phys. Condens. Matter 14, R415 (2002)
    3. M. D. LaHaye et al., Science 304, 74 (2004)
    4. R. G. Knobel, A. N. Cleland, Nature 424, 291 (2003)
    5. X. M. H. Huang et al., Nature 421, 496 (2003)
    Science http://www.sciencemag.org

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