THEORETICAL PHYSICS: QUANTIZATION OF A PENDULUM SYSTEM
ScienceWeek http://scienceweek.com
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

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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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Related Material:

ENTANGLEMENT, DECOHERENCE, AND THE QUANTUM-CLASSICAL BOUNDARY

Quantum mechanical entanglement is a phenomenon that has caught
the imagination of the public as one of the more bizarre
consequences of fundamental physical theory. Entanglement is
unique to quantum mechanics, and involves a relationship (a
"superposition of states") between the possible quantum states of
two entities such that when the possible states of one entity
collapse to a single state as a result of suddenly imposed
boundary conditions, a similar and related collapse occurs in the
possible states of the entangled entity no matter where or how
far away the entangled entity is located. Entanglement arises
from the wave function equation of quantum mechanics, which has
an array of possible function solutions rather than a single
function solution, with each possible solution describing a set
of possible probabilistic quantum states of the physical system
under consideration. Upon fixation of the appropriate boundary
conditions, the array of possible solutions collapses into a
single solution. For many quantum mechanical physical systems,
the fixation of boundary conditions is a theoretical and
fundamental consequence of some interaction of the physical
system with something outside that system, e.g., an interaction
with the measuring device of an observer.

In this context, two entities that are described by the same
array of possible solutions to the wave function equation are
said to be "coherent", and when events decouple these entities,
the consequence is said to be "decoherence". As a physical
phenomenon, entanglement was discussed many years ago, most
particularly following the publication in 1935 of the often
quoted Einstein-Podolsky-Rosen paper (Phys Rev 1935 47:777).
These discussions have been in the form of "gedanken" (thought)
experiments involving two quantum-mechanical entangled entities.
More recently, however, there have been laboratory constructions
of actual quantum mechanical systems exhibiting such entanglement
phenomena, and the reportage of these laboratory arrangements by
the media have engaged the public fancy. Essential here is that
any purely verbal account of quantum mechanical phenomena is
severely limited by the constraint that the properties of quantum
mechanical systems can be precisely described only by the
equations relevant for those systems, and all other descriptions
usually introduce serious ambiguities.

The following points are made by Serge Haroche (Physics Today
1999 July):

1) In quantum mechanics, a particle can be delocalized
(simultaneously occupy various probable positions in space), can
be simultaneously in several energy states, and can even have
several different identities at once. This apparent "weirdness"
behavior is encoded in the wave function of the particle.

2) Recent decades have witnessed a rash of experiments designed
to test whether nature exhibits implausible nonlocality. In such
experiments, the wave function of a pair of particles flying
apart from each other is entangled into a non-separable
superposition of states. The quantum formalism asserts that
detecting one of the particles has an immediate effect on the
other, even if they are very far apart, even far enough apart to
be out of interaction range. The experiments clearly demonstrate
that the state of one particle is always correlated to the result
of the measurement performed on the other particle, and in just
the strange way predicted by quantum mechanics.

3) An important question is: Why and how does quantum weirdness
disappear (decoherence) in large systems? In the last 15 years,
entirely solvable models of decoherence have been presented by
various authors (e.g., Leggett, Joos, Omnes, Zeh, Zurek), these
models based on the distinction in large objects between a few
relevant macroscopic observables (e.g., position or momentum) and
an "environment" described by a huge number of variables, such as
positions and velocities of air molecules, number of black-body
radiation photons, etc. The idea of these models, essentially, is
that the environment is "watching" the path followed by the
system (i.e., interacting with the system), and thus effectively
suppressing interference effects and quantum weirdness, and the
result of this process is that for macroscopic systems only
classical physics obtains.

4) In mesoscopic systems, which are systems between macroscopic
and microscopic dimensions, decoherence may occur slowly enough
to be observed. Until recently, this could only be imagined in a
gedanken experiment, but technological advances have now made
such experiments real, and these experiments have opened this
field to practical investigation.

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