[Paleopsych] SW: Einstein and Quantizing Chaos
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Theoretical Physics: Einstein and Quantizing Chaos
http://scienceweek.com/2005/sw050902-6.htm
The following points are made by A. Douglas Stone (Physics Today 2005
August):
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:
THEORETICAL PHYSICS: QUANTIZATION OF A PENDULUM SYSTEM
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:
QUANTUM PHYSICS: ON NANOMECHANICAL QUANTUM LIMITS
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
objects?
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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