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Pedestrian Notes on Quantum Mechanics Date 2004-7-6 17:03 |
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Pedestrian Notes on Quantum Mechanics
_____________________________________
Haret C. Rosu
Instituto de Fisica de la Universidad de Guanajuato,
Apdo Postal E-143, Leon, Gto, Mexico
Institute of Gravitation and Space Sciences,
Bucharest, Romania
rosu@ifug3.ugto.mx
(received: March 22, 1997)
"Get your facts straight, and then you can distort them as much as you
please."
-- Mark Twain
I present an elementary essay on some issues related to
foundations of nonrelativistic quantum mechanics, which
is written in the spirit of extreme simplicity, making
it an easy-to-read paper. Moreover, one can find a useful
collection of ideas and opinions expressed by many
well-known authors in this vast research field.
I. Indefinables
_______________
Physics is first of all the science of measurement. As Lord
Kelvin put it
I often say that when you can measure what you are speaking
about, and express it in numbers, you know something about it.
According to Kelvin a collection of thoughts cannot advance to the
stage of Science without numbers. Any observable of interest in physics should
be measurable or expressed in terms of measurable quantities. Length and time
are two of the indefinables of classical mechanics, since on an intuitive base
there are no simpler or more fundamental quantities in terms of which length
and time may be expressed. The problem of space-time picture of the physical
world is connected with the rigour of exact description of nature requiring,
say, differential equations, by means of which we are able to gauge the
intuitive space-time scales of any motion. The full number of indefinables in
mechanics is three, as all its quantities could be expressed by only three
indefinables. The third mechanical indefinable is usually the _mass_, but also
_force_ may be chosen [1]. Human beings in their everyday lives are
continuously "measuring" the mechanical indefinables, as well as other
indefinables of physics, e.g., the temperature, by means of their
physiological senses. Of course, it is a very rough "measurement", because it
could be expressed in words, not in numbers. Words and numbers are
complementary units of knowledge. Pure numbers do not tell us much about
Nature unless we assign them some significance. As a good example consider the
number 3.52. Just a (real) number as any other. But now write it as
2 pi
____
gamma
e
For some physicists it has already a meaning. Finally, let us write down the
full chain, i.e.,
Delta_0 2 pi
______ = _____________ = 3.52
T_c gamma
e
It tells us that 3.52 is the BCS value for the ratio between the gap at zero
temperature and the critical temperature for the transition to the
superconductor phase. One gets 3.52 only in BCS theory.
Because of its beauty, I am temted to give a second example which has
been quoted by Noyes [2] from the books of Stillman Drake on Galileo. So, what
about the number 1.1107... Nothing special at first glimpse. But now let us
give a first significate: 1.1107... = pi / (2 sqrt[2]) . Geometrically it is
the ratio between the quarterperimeter of the circle and the side of the
square inscribed into that circle. Geometrical (i.e., spatial) measurements
and thinking were much developed by Old Greeks. But Galileo was the first to
give 1.1107... as the ratio of two times, namely the time `t_p' it takes for a
pendulum of a specific length `l' to swing to the vertical through a small arc
to the time `t_f' it takes a body to fall from rest through a distance equal
to the length of the pendulum. Galileo's measurement was 942/850 = 1.108, but
he was not aware he measured `pi / (2 sqrt[2])' . However he gave a remarkable
formulation of the "law of gravity". The Galilean gravitation states that the
ratio of the pendulum time to the falling time as specified above is the
constant 1.108, "anywhere that bodies fall and pendulums oscillate".
To obtain the number of indefinables (NOI) a community of physicists
should adopt rules of their measurements, especially since NOI is not a fixed
number, and new types of experiments might augment NOI. The rule of classical
indefinables is to choose a durable standard of unit for each of them and to
have a good dividing engine. This has been achieved rather easy for the meter
of length but not so easy in the case of the meter of time (second). In the
latter case the great difficulty has been for a long time the missing of an
accurate dividing machine. Large errors were continuously accumulating over
historical epochs, and people in the fields of Religion and Politics were
forced to apply corrections at some times. Atomic standards (lasers) have been
introduced since early sixties having a natural atomic time-dividing machine (
the atomic frequency). However, even the very precise atomic standards display
statistical results [3], and there are also reported abnormalities during
solar eclipses [4]. At present, interferometers could be used for dividing
purposes too, and computing machines are usually attached to measuring devices
for a more rapid conversion of the physical interactions into real numbers. As
regarding computers, there should be a continuous effort to study their rate
of producing numbers which is not depending only on the used algorithm, and to
have more involved definitions of computable numbers (so-called Turing Problem
[5]).
Establishing measurement rules for indefinables is extremely important
for the conceptual constructions in the realm of physics. It is not at all an
easy job in the case of quantum indefinables. Since the microscopic world is
described by another kind of mechanics, the celebrated quantum mechanics, it
may be that new indefinables come into play. One new indefinable is the
wavefunction, also called the state vector [6], or, more realistically, the
wavepacket. These are concepts essentially of mathematical origin, and hence
*a priori* calculable ones. In a certain sense they correspond to the
fundamental indefinable of euclidean geometry- the point. The geometrical
point has neither dimensions nor any attached units, but we can assign numbers
(coordinates) to it. In this way one may come to the conclusion that quantum
mechanics is more a mathematical theory rather than a physical one. It is a
"wave" mechanics allowing a corpuscular duality. The measurement problem will
be *ab inizio* extremely delicate for quantum mechanics, just because it is a
theory containing unmeasurable (mathematical) indefinables. To the
mathematical indefinables one could always assign generalized probabilistic
meanings depending substantially on the measuring scheme. The mathematical and
psycho/philosophical literature is extremely rich in various axiomatic schemes
in probability theory and its mental implications [7]. Let us mention the
so-called *belief functions* on which Halpern and Fagin have recently
elaborated [8]. These are functions that allocate a number between 0 and 1 to
every subset of a given set of objects. Such functions have been introduced by
Dempster in 1967 [9], and it would be quite interesting to have a quantum
mechanics based on them, e.g., to write down a *em belief* density matrix.
It became common lore to say that the measurement process is a more or
less instantaneous effect inducing a reduction/collapse of the wavepacket,
which is interrupting its quantum unitary evolution as described by the
Schroedinger equation. As far as the meaning of measurement is not quite clear
whenever one is dealing with probabilistic concepts, the whole quantum theory
is subject to severe questions of interpretation and therefore is open to deep
philosophical problems. The axioms of the standard probability theory are not
fixed for ever. It is a fundamental scientific goal to exploit various
modifications of the probabilistic axioms rendering possible new
interpretations of quantum mechanics [10].
II. The concept of massiveness
______________________________
Let us start this section with an excerpt from Glimm [11]
Between quantum length scales (atomic diameters
of about 10^-10 m) and the earth's diameter
(10^6 m) there are about 16 length scales. Most
of technology and much of science occurs in this
range. Between the Planck length and the diameter
of the universe there are 70 length scales. 70, 16,
or even 2 is a very large number. Most theories
become intractable when they require coupling between
even two adjacent length scales. Computational resources
are generally not sufficient to resolve multiple length
scales in 3 dimensional problems and even in many
two-dimensional problems.
At the present time, *technology* is penetrating into the nanometer scale
[12], and even atomic scale [13]. By *technology* one should understand: i)
machine tools (i.e., processing equipment), ii) measuring instruments
(inspection equipment), iii) super-inspection factors (e.g., well-qualified
human beings). We have already machine tools at the nanometer scale [14], and
one can process shapes down there at only one order of magnitude away from the
atomic scale.
Already at the end of 1959 Feynman [15] delivered a remarkable talk on
manipulating and controlling things on a small scale. As a matter of fact,
human beings are closer to atomic scales than, say, galactic scales, and
besides, to fabricate small things is an absolute technological requirement. A
list of reasons why we want to make things small was provided by Pease [16] of
which we cite: it is *fun*, smaller devices work *faster*, smaller devices
consume *less power*, smallness is intrinsically *good*, it is scientifically
*important*. One can add to the list of Pease *small is beautiful* introduced
by Fubini and Molinari [17].
Nonetheless, even within mesoscopic world the measurement problem will
preserve its features. We shall continue to establish correlations between a
"system" observable and an "apparatus" observable, i.e., we shall do our
measurements in the same common manner [18]. The apparatus should be massive
with respect to the system (massive not necessarily meaning of macroscopic
size), and should have a "pointer", which cannot be but in localized quantum
states. According to DeWitt, *massiveness* of the pointer compared to the
measured system is absolutely necessary to get experimental results/outcomes.
However, the concept of *massiveness* is not elaborated by DeWitt (unless to
say that M_ap >> m_s). At the nanometer scale, mesoscopic tips are by now in
much use making possible the measurement of Van der Waals forces between the
tip and the surface under investigation at distances smaller than 100 nm. That
means forces in the range 10^-6 - 10^-7 N are measured either when the tip is
moved or the surface is slowly approaching the tip [19].
Massiveness is related to localization, and may be very helpful in
distinguishing amongst various theories of quantum evolution. The fundamental
goal of this family of theories is to explain in a unifying way microscopic
(quantum) objects and macroscopic ones. Since it is reasonable to think that
massiveness is indeed related to the localization features of the system, it
would be interesting to study in detail the conditions under which various
systems make the transition to massiveness. By this transition which, to this
day, is one of the most unclear in quantum mechanics [20], one should not mean
in a compulsory manner the various semiclassical (h-bar --> 0; or better
h-bar / m --> 0 ; notice K.R.W. Jones [21] who showed that one can also keep
`h-bar' fixed) approximations to quantum mechanics or the transition to
macrophysics N --> infinity. It is merely a transition in the sense of the
Born-Oppenheimer approximation. This approximation, going back to 1927, refers
to molecular wave functions and is essential in the interpretation of
molecular spectra. It is a perturbation expansion in a small parameter defined
as the fourth root of the electron mass divided by the mass of the nuclei,
kappa = (m/M)^(1/4). Denote by `H' a common Hamiltonian for a system of heavy
and light particles and by `H_cM' the hamiltonian of the center of mass
motion. The problem is to find out the manner in which the eigenvalues `W_i',
and eigenfunctions `psi' of H' = H - H_cM depend on `kappa' [22]. The main
hypothesis of Born- Oppenheimer is the existence of an equilibrium position
`X_0' of the heavy particles such that the eigenvalues `W_i' and the scaled
eigenfunctions,
3/2(N-1)
phi (xi ,x) = x psi (x xi + X_o, x) ,
are analytic in `kappa^2' and `kappa', respectively, for fixed `m' in the
neighbourhood of zero.
Very useful would be to consider the transitions to macroscopic world
and to massiveness as problems with multiple time scales, which are pervading
many areas in applied mathematics [23].
Let us end this section with the relationship between *big* and
*small* in quantum mechanics. For this, I shall present excerpts from Dirac's
*Principles* [24]. In the first chapter, "The Principle of Superposition",
Dirac states
So long as *big* and *small* are merely relative concepts,
it is no help to explain the big in terms of the small. It
is therefore necessary to modify classical ideas in such a
way as to give an absolute meaning to size.
On the same page one can read
We may define an object to be big when the disturbance accompanying
our observation of it may be neglected, and small when the disturbance
cannot be neglected. This definition is in close agreement with the
common meanings of big and small....In order to give an absolute
meaning to size, such as is required for any theory of the ultimate
structure of matter, we have to assume that *there is a limit to the
fineness of our powers of observation and the smallness of accompanying
disturbance - a limit which is inherent in the nature of things and can
never be surpassed by improved technique or increased skill on the part
of the observer*.
III. Quantum mechanics and diffusions
______________________________________
Words like _particle_ and _motion_ could be considered also in the
class of undefined (primary) concepts [25]. Electrons, neutrons, neutrinos and
other _'ons_ and _'inos_ could be only names that we accept because of their
intuitive power. All the so-called elementary particles which have been
introduced in the last one hundred years can be considered as particular forms
of propagating (transport) processes and energy carriers [26]. Indeed,
Schroedinger equation is the basic equation of quantum world, but diffusion
equations are, no doubt, more general equations. We say `no doubt' just
because already in 1933, Fuerth [27] showed that Schroedinger equation could
be written as a diffusion equation with an imaginary diffusion coefficient,
D_qm = i h-bar / 2 m . This imaginary diffusivity is vexing and many stopped
the analogy at this point. On the other hand, negative diffusivity is more
natural and one may encounter it in multicomponent systems, implying local
increase in the energy of the system as discussed by Ghez [28]. Let us
consider a one-dimensional Schroedinger equation
d psi
i h-bar _____ = H phi
d t
where H = (p^2 / 2 m) + V and
d
p = -i h-bar ____ .
d x
The diffusive character of such an equation for `psi' is obvious if we take
into account a source term related to the potential energy, and the momentum
playing the role of flux. For historical reasons the diffusion interpretations
( they may come in three classes: in configuration space, in phase space, and
in imaginary time) were not favored during a long lapse of time, though today
mainly because of the impetus provided by *quantum optics*, we became
accustomed with such methods as quantum-jump [29] and quantum-state diffusion
[30] to simulate dissipation processes. Indeed, Schroedinger obtained his wave
mechanics by means of a more intuitive analogy in which he put together the
Hamilton-Jacobi theory, relating geometrical optics and particle dynamics,
with de Broglie's matter waves. One could say that what Schroedinger did was
to randomize a purely classical theory by means of de Broglie hypothesis. It
was a way of randomizing within the classical formalism, but, more generally,
one should be aware of the multitude of randomization procedures [31].
The apparent difficulty of imaginary diffusivity is not essential when
interpreting it in the proper way. The result can well be a more general
theory. The picture of the *World* is that of an infinite number of clusters
in the sense of percolation theory. Classes of clusters could be defined in
terms of their relative diffusivities and fluxes. Some of them are "static"
relative to other more kinetic ones [32]. In this diffuson context, the
imaginary character of the diffusion coefficient for quantum particles is
related to the passage from a parabolic differential equation to a hyperbolic
one [33].
Even a presentation of the one dimensional diffusion equation, first
in the discretized form and then in the continuous limit, on the lines of the
*Primer* of Ghez [28] is very helpful to understand the diffusive aspects of
the Schroedinger equation, and I recommend the reader to look in that book.
The toy model of Ghez is a pedagogical isotropic one-dimensional random work,
in which one consider points on a line with an arbitrary fixed origin. For the
passage to the continuum limit one must introduce a jump distance between the
points and a continuous particle distribution, depending not only on time but
also on the space variable such that to coincide at the discrete sites with
the discrete particle distribution.
There are also papers dealing with the connection between a classical
Markov process of diffusion type and the quantum mechanical form of the
Hamiltonian for a classical charged particle in an electromagnetic field [34].
These two problems are equivalent as far as one is concerned with the
expectation values for the particle energies in the two cases. Consider a
continuity equation of the type d rho / d t = nabla . (rho v) where
v = v_o - D nabla ln(rho).
Such a continuity equation is in fact a Fokker-Planck equation for the
probability density `rho' for the position vector of a particle following a
Markov process of diffusion type with diffusion coefficient `D'. The
expectation value for the energy of the particle
/
<E> = | [ mv_o^2 / 2 +eV ] rho d^3 x
/
can also be written
/
<E>= | [ mv^2 / 2 + D^2 m / 2 ( nabla ln(rho) )^2 +eV ] rho d^3 x,
/
and the connection with the electromagnetic phenomena can be established by
means of the celebrated Helmholtz theorem for a vector (considering the
velocity as a vector, thus no type of spin)
v = alpha nabla phi + beta A ,
where `phi' and `A' are defined in the usual way up to a gauge transformation,
and `alpha' and `beta' are constants which should be chosen in an appropriate
way to achieve the correspondence. There would be interesting to study the
passage ways from microscopic to macroscopic description of electromagnetic
fields [35] in this framework. The traditional one, going back to Lorentz, and
which is applicable to common molecular media, is by averaging the
differential equations for microscopic quantities by integration over some
macroscopic volume. This is the most trivial procedure for going to the
macroscopic approximation. There are other approaches, e.g., the topological
one as discussed by Brusin [36].
IV. Quantum mechanics and localizations
_______________________________________
The collapse postulate of quantum mechanics is one of the most
debatable points in the conceptual base of this theory, being at the same time
the main desideratum for a modified quantum dynamics [37]. The collapse of the
state vector is required by the formal quantum theory of measurement. One must
assure somehow the decoherence of the macroscopic states of the apparatus in
order to have a definite outcome for any experiment involving quantum
particles. We still do not know if this decoherence is dependent on the
particular interaction and hence on the particular type of measurement or is a
universal feature of the transition from microscopic to macroscopic behaviour.
The first hypothesis is called environmental (Zeh-Joos) localization [38]. On
the other hand, the universal localization, also known as spontaneous (or GRW)
localization is due to Ghirardi, Rimini and Weber [39]. It is difficult to
decide between the two models. In our opinion, they are not completely
opposite ideas. The dynamical (environmental) localization may be specific to
the particular experiment, while the spontaneous localization might be thought
of as related to the transition to massiveness, which one would like to see as
universal. In this way having different purposes, the two standpoints are not
contradictory. At the same time GRW localization could be considered only as a
special type of environmental localization at the scale given by the
parameters of the model. The point is that these parameters have been raised
to the level of fundamental constants of Nature by the authors. Anyway, one
must spell out explicit conditions allowing to pass from a regime of
continuous spontaneous (or dynamic) localization to a discontinuous regime
characteristic to the GRW localization. We recall that in the GRW approach the
N-particle wave function of the non-relativistic Schreodinger quantum
mechanics (NRQM) is coupled to a normalized Gaussian jump factor
(-x^2/2a^2)
J_(GRW)(x) = K e^
The frequency of the jumps and the localization constant are considered as two
new fundamental constants of Nature of the following orders of magnitude
nu_(GRW) ~ 10^(-15) / s = 10^(-8) 1/year and a ~ 10^-5 cm.
The spontaneous localization implied by the GRW model might be tested
experimentally by means of mesoscopic phenomena, e.g., by looking for
instabilities of the mesoscopic growing (thread-like, filamentary) patterns.
Recently, Kasumov, Kislov, and Khodos [40] observed the displacements of the
free ends of threads of amorphous hydrocarbons of 200-500 (angstrom) in
width and 0.2-2.0 um in length relative to a fixed reference point on the
screen of a transmission electron microscope. The minimal displacements were
of about 5 (angstrom), and the observations were made in a regime of
stationarity of the threads, i.e., very low density of the beam current (~ 0.1
pA/cm^2). KKK observed random jumps of the free ends of the carbon threads of
10-30 (angstrom) in length and of frequency of ~1 Hz. They discussed
possible reasons to induce vibrations, and came to the conclusion that no
classical external forces could explain the jumps. They attributed them to the
"quantum potential", and to localizations of GRW type, but the range of the
observed parameters do not correspond to that of the GRW ones.
What I would say is that the jumps or mesoscopic fluctuations of the
carbon threads are a kind of mesoscopic Brownian motion which damps in time,
being different from the microscopic quantum fluctuations which never damps
out.
Moreover, if one takes into account the recent work of Sumpter and
Noid [41] the KKK results can be classified as *red herrings*. Sumpter and
Noid assigned the onset of positional instabilities in samples of carbon
nanotubes to nonlinear resonances controlled by their geometry, i.e., the
contour length around the end of the tube and the length of the tube along its
axis. It is quite probable that the same mechanism applies also to
microtubules in biology. For the connection between "quantum jumps" and
nonlinear resonances in classical phase space see Holthaus and Just [42]
I would like to point out that the GRW-type localization corresponds
to a weak coupling limit of Hamiltonian systems with coherent/squeezing
interaction with the environment [43]. Indeed Gaussian localization is
specific to coherent and squeezed states in the configuration representation.
An immediate scope is to generalize this type of localization to relativistic
quantum mechanics (RQM), and to quantum field theories (QFT). In NRQM one is
dealing with spatial probabilities, that is with probabilities associated with
a spatial domain `Delta-X' at a moment of time T. To go to RQM, one must
extend the spatial probability to a spacetime domain as in [44].
V. Nonlinear wavefunction collapse ?
____________________________________
The quantum wavefunction varies in time in a continuous way, following
the deterministic Schroedinger evolution. When an observer wishes to measure a
physical quantity of a quantum system, the wavefunction corresponding to that
physical quantity is exposed to an apparatus specially designed to do this.
The general effect of the apparatus, usually macroscopic with respect to the
physical system, is to induce a discontinuous change of the wave function from
a superposition of states into just one state. This general effect is known in
the quantum formalism as the collapse of the wavefunction. The open question
is to find out the general mechanism of the collapse of the quantum
wavefunction. In the literature one can find many interesting ideas on this
problem. As a quite acceptable interpretation of the collapsing phenomenon we
mention here the old ideas of Schroedinger, who tried to relate the modulus of
the wavefunction to a materialistic and realistic density of electronic
matter, and *not* to probabilities. For a recent discussion of this viewpoint
the reader is referred to a paper of Barut [45]. This model for the modulus of
the wavefunction can be elaborated further by making use of progress due to
Chew [46].
In the following, we would like to comment on some phenomenological
features of physical collapses from other areas of physics in the hope to gain
more insights in the possible physical picture of quantum mechanical collapses
of admittedly fundamental origin. Our standpoint is that the present status of
the wave function reduction phenomenon is too formal, even though one may find
an abundant literature with interesting presentations of the topic [47]. It is
fair to say that we have no generally accepted physical mechanism of the
reduction process for the time being. In the literature, one can find only
extreme descriptions, claiming for a strong nonlinear process in which gravity
[48] and/or quantum gravity [49] is thought to play an important role. On the
other hand, collapsing phenomena, presumably displaying similar patterns can
be encountered in several other fields of physics, in the case when nonlinear
effects are not balancing any more the dispersive spreading of waves
(solitons). Of course, in such cases one is already outside the restricted
regime of linear dissipation implied by standard quantum mechanics. Moreover,
one can avoid thermodynamical arguments against nonlinear variants of
Schroedinger equation [50] by making use of more general entropies [51].
A relevant example of nonlinear collapse is the Langmuir collapse in
plasma physics. Langmuir collapses belong to the class of wave collapses, a
well-defined topic in nonlinear physics [52]. The collapse of Langmuir wave
packets in two or more dimensions was first predicted by Zakharov [53], and it
is observed in the laboratory. It is a strong non-linear collapse occuring in
strong Langmuir turbulence, which consists of many locally coherent wave
packets interacting with a background of long-wavelength incoherent turbulence
[54]. Langmuir collapses are governed by a non-linear Schroedinger equation of
the type i psi_t + 1/2 Delta psi + |psi|^2 psi = 0
which, as it is well-known, allows singularity formation in a finite time t =
t_0, for s d >= 4 , (`d' is the dimension of space). The phenomenology of the
Langmuir turbulence is extremely interesting. Wave packets are observed to
"nucleate" in existing density depressions. The nucleation of new wave-packets
takes place by the trapping of energy from long-wavelength background
turbulence into localized eigenstates of relaxing density wells. Since the
collapse transfers energy to short scales, where there is strong damping, a
process called "burnout" occur in which energy is transfered to the electrons
and the collapse is stopped. In this way the Langmuir field is dissipated, the
density cavity relaxes and can serve as a nucleation site for a new wave
packet. Perhaps an equivalent physical picture as that of the turbulent wave
collapse might be made available with some modifications for wavefunction
collapse in a nonlinear scheme of quantum mechanics (e.g., a dust plasma
model).
VI. Remarks on various other topics
____________________________________
A. Friction modifications of quantum mechanics
______________________________________________
Modifications of quantum mechanics may be thought of in terms of
friction terms for the more general situation of open quantum systems. The
problem of the ways of including various types of friction in the quantum
mechanical framework has been an active field for decades. Many authors
considered the dissipation in the form of friction as a means to reconciliate
quantum mechanics and general relativity, and also as able to cast light on
the transition between classical and quantum physics. Even though the
dissipation of energy seems to be more appropriately described in terms of a
density operator approach, there has been always a steady activity towards
understanding friction at the level of wave functions [55].
In this area, the damped harmonic oscillator is considered to be "the
primary textbook example of the quantum theory of irreversible processes", to
quote Milburn and Walls [56].
Some time ago, Ellis, Mavromatos and Nanopoulos [57] studied string
theory models from the frictional point of view. They gave reasons to believe
that the light particles in string theories obey an effective quantum
mechanics modified by the inclusion of a quantum-gravitational friction term,
induced by the couplings of the massive string states. According to these
authors the string frictional term has a formal similarity to simple models of
environmental quantum friction.
Finally, Beciu [58] sketched a proof showing that a friction term for a
cosmological fluid still retaining the symmetries of a perfect fluid at the
level of the stress tensor is equivalent to an inflaton field.
B. Wave-particle dualities
__________________________
Historically speaking, the wave-particle dualities were established
before the advent of the quantum differential equations. We say dualities and
not duality because, not only for historical reasons, one must distinguish the
duality of photons from that of massive particles, say electrons.
The wave-particle duality of light is defined by the Einstein relation
E = h / nu = h c / lambda. This duality of light was used by Einstein to
explain the photoelectric effect, by the Nobel-prize formula for the kinetic
energy of the emitted electrons (1/2) m v^2 =E - E_o where E = h / nu is the
quantized energy of the incident photons and E_o = e / phi is the threshold
energy with `phi' the work function.
The duality of massive particles, on the other hand, was established
by de Broglie two decades after Einstein's duality. The wavelength and the
momentum of an electron (and of any other massive particle) is given by
lambda = h / p .
It is worth noting the fact that the two dualities are related to each
other through the photoelectric effect, hbar^2 k^2 / 2 m = h c / lambda - e phi.
Usually, the textbooks and the literature present the wave-particle
dualities as a logical result of Young slit experiments. As a rule, a more or
less detailed discussion of the complementarity principle is accompaning the
discussion of the Young experiment. Interesting ideas concerning the slit
complementarity and duality have been put forth by Wootters and Zurek [59],
Bartell [60] and Bardou [61]. These authors made attempts to transcend the
rather dogmatic presentation of this fundamental topic. Bartell introduced the
idea of intermediate particle-wave behavior. Most probably, we need
generalizations of the concepts of wave and particle, of their interactions,
and a deep scrutiny of the effects of the type of experiment.
In the last couple of years, the investigation of particle-wave
dualities became a very active one, mainly because of the rapid progress of
some new technologies. Perhaps, one of the most interesting experiments is
that performed by Mizobuchi and Ohtake [62], which is just a repetition of the
old double prism experiment done by Bose as long ago as 1897, however not with
microwaves but, following a suggestion of Ghose, Home, and Agarwal [63], with
single photon states. An *and*-logic for the wave-particle duality at the
single-photon level has been claimed.
An open problem in detecting photons is the precise meaning of the
photon in the detection process. The point is that we are detecting signals,
and these signals depend on the experimental detection schemes. The signals
will give some pulses in the detectors. Thus the full detection process is
governed by some electronic relationships in the signal-pulse-detector system
[64].
Understanding better the manifestations of wave-particle dualities for
light can be highly relevant in photonics and optical computing [65].
At this point, let me quote from the recent paper entitled
"Anti-photon" of W.E. Lamb, Jr. [66]
... there is no such thing as a photon. Only a comedy of errors
and historical accidents led to its popularity among physicists
and optical scientists.
Then, of course, the wave-particle duality for light will be loosing
its physical picture but will gain in mathematical rigor.
C. The problem of the constancy of the Planck constant
_____________________________________________________
It was remarked by Barut [67] that the free electromagnetic field has
no scale. There are only frequencies. Moreover, Planck originally derived his
formula from the properties of the oscillator on the boundary of the
black-body cavity, not from the quantization of the field. The common practice
of quantization of the fields came later. Therefore, we believe that even
today careful experimental checks of the constancy of the Planck constant
should be made, and in fact have been made in some laboratories [68]. Barut
showed that a formulation of quantum mechanics without the fundamental
constants `h-bar', `m' and `e' is possible [67] [69]. It looks like a pure
wave theory in terms of frequencies alone, and it might be used more
profitably in experiments where one measures frequency differences. In this
case, the energy becomes a secondary concept, and different quantum systems
are characterized by an intrinsic proper frequency `omega_o'. On the other
hand, one can consider quantum dynamics with two Planck constants, like did
Diosi [70].
As soon as we depart from the assumption of the constancy of the
Planck constant by merely considering a variable Planck parameter `H', but
nonetheless preserving the constancy of `H / m ' we may consider some kind of
quantization at large scales, planetary or even galactic ones. In fact there
is a quite vast literature on megaquantum effects. We draw attention to the
fact that such effects are related to the interpretation of `H / m' as a
pseudo-Planck constant which is associated to some gravitational systems
(e.g., the Solar System [71], quasars [72]).
A viewpoint to be recorded was put forward by Landsman [73]. He
claimed that only dimensionless combinations of `h-bar' and a parameter
characteristic of the physical system under study are variable in Nature. The
references [71] [72] seem to confirm this idea.
We would like to point briefly on the possible effect of the spatial
scale of the measurement scheme on the numerical value of fundamental
constants. We shall use as an example the fine-structure constant alpha = e^2
/ h-bar c. At the present time, we know a very precise macroscopic phenomenon,
namely the quantum Hall effect, from which the fine-structure constant can be
obtained from the quantized Hall resistance. (I consider quantized Hall
resistance a more precise experiment as compared to that involving the proton
gyromagnetic ratio, proton magnetic moment and Josephson frequency-to-voltage
ratio). The numerical value obtained from the quantum Hall effect is [74]:
1/alpha=137.0359943(127). On the other hand, the standard atomic measurement
(coming from the anomalous magnetic moment of the electron) gives 1/alpha
=137.0359884(79). The two values differ only at the level of 0.1 ppm. The QED
corrections are confined to distances of the order of the Compton wavelength
of the electron, whereas the primary interaction in the quantum Hall effect is
between the electrons in the metal and those circulating in the coils which
produce the magnetic field. The spatial scale in this case is of the order of
a few cm. It would be extremely interesting to relate the very small
differences in the numerical values of the fundamental constants to the
spatial scale of the phenomena used to measure them. Presumably, there might
be correlations between the last different digits of the numerical values of
the fundamental constants and the spatial scale of the measuring device used
to determine that value. At least some self-similar correlations are to be
expected.
D. Quantum mechanics and cosmology
__________________________________
The previous subsection already introduced us into the much more
ambitious program of describing the universe as a whole in quantum mechanical
terms. The difficult problem of interpretation is not so much with respect to
considering the Hilbert space of the Universe. It is related to the
fundamental fact that there can be no *a priori* division into *observer* and
*observed*. In other words, there is no Feynman's "rest of the Universe".
A generalization of the Copenhagen interpretation such as to be
applied to cosmology was first provided by Everett [75] in 1957. His theory of
"many worlds" has been replaced at the present time by theories of "many
histories" (time-ordered sequences of projection operators [76]), but the
essential ideas remained those of 1957. As a matter of fact, what Everett has
done may be entailed in the process of probabilistic modeling, i.e., the
organization of the space of wave function(s) as a probability space [77].
Everett showed how in his interpretation it is possible to consider
the observer as part of the system (the universe) and how its fundamental
activities- measuring, recording, and calculating probabilities- could be
described by quantum mechanics. As incomplete points in Everett's
interpretation, which has been much clarified subsequently, one should mention
the origin of the classical domain we see all around us, and a more detailed
explanation of the process of "branching" that replaces the notion of
measurement.
The main concept that has emerged in this area is that of decoherence
functionals, and the main debated topic is that of connecting this concept to
the probability interpretation. Recently, Isham and collaborators presented a
classification of the decoherence functionals based on a histories analog of
Gleason's theorem [78]. To be noticed are the "negotiations" on the border
between quantum and classical in the decoherence framework published in
*Physics Today* of April 1993.
E. Quantum jumps
________________
The interesting topic of quantum jumps [79] has to do with the rare
but strong fluctuations that may show up in any stochastic process, be it
classical or quantum. The mathematical theory of large deviation estimations
has been already elaborated in considerable extent [80]. All quantum
mechanical equations have solutions to which probability representations may
be given [81]. The mathematical problem is to find out probability measures of
Poisson processes with jump trajectories, which are similar to the Feynman-Kac
transformation of probability measures for processes with continuous
trajectories. For relativistic equations we have usually Poisson probability
representations, whereas for nonrelativistic equations diffusions in imaginary
time have been worked out, but also Poisson representations are possible. One
can establish the scale at which the transition from the covariant hyperbolic
Dirac dynamics to the non-covariant parabolic dynamics of the Schroedinger
equation occurs [82].
As a further argument that quantum jumps, i.e., "discontinuities in
time of the wavefunction" in the terminology of Zeh [83], are related to rare
fluctuations of stochastic nature, we remark that they are observed even in
single quantum systems [84].
F. Analogies to quantum mechanics
_________________________________
Thinking by analogy is considered to be a clear indication of superior
reasoning and of human intelligence [85]. In physics there are a vast amount
of analogies of much help in the progress of many different branches of this
science. Many analogies are not complete and it is precisely this point to
induce into error all those residing too much on this beautiful aspect of
human thinking. One should keep in mind the danger of extrapolating the
analogies beyond their natural limits, which should be carefully estimated,
and also the risk of using them in the wrong way.
Coming to quantum mechanics, we would like to recall two quite
attractive analogies. The first one is the electric network discussed by Cowan
[86] long ago. The Cowan networks have the distribution of the electric energy
density in three dimensional space similar to that of probability density
waves corresponding to a spinless particle in any potential field.
The second analogy has been recently discussed by J.L. Rosner [87] who
showed that the so-called Smith Chart method used for antenna impedance
matching corresponds in quantum mechanics to a simple conformal transformation
of the logarithmic derivative of the wavefunction. The Smith Chart is a
convenient graphical representation for analysing transmission lines [88], and
clearly may help understanding from a different point of view the tunneling
processes.
G. Human brain and quantum computers/brains
___________________________________________
The flux of literature tells us that *quantum computers* are at good
moments of the *gate* phase and of exploratory discussions of various physical
setups from the quantum computational standpoint. This exciting topic has been
started about two decades ago (though one can think of Szilard, von Neumann,
and Brillouin as well) and might turn into a really major general discipline.
Apparently the functioning of the human brain is not based on quantum
effects. The membrane voltages of the neurons do not imply the Planck
constant, and the important physical processes are essentially the mesoscopic
transport ones. A great advantage of the human brain is a quite flexible
microtubule architecture due to a remarkable phenomenon, the so-called
*dynamic instability* [89]. The origin of this important phenomenon is
debatable, and after having read the note of Sumter and Noid (J. Chem. Phys.
of April 22, 1995) I think that a nonlinear resonance mechanism should be
considered as a good proposal. Many brain mysteries are hidden in the
microtubule assembly characterizing any individual biological brain, and there
is much unexplored physics.
The mesoscopic functioning of the human brain does not imply that an
almost quantum (e.g., nanoscopic) brain cannot be fabricated. For example,
Josephson junctions may be the component units of such a brain since the
relationship between the applied voltage and the emitted frequency involves
Planck's constant.
In his paper "Is quantum mechanics useful ?" [90], Professor Landauer
remarked that *technologies differ in their explicit utilization of quantum
mechanical behaviour*. The important technological task in considering quantum
computers is to print the bit on as small a material structure as physically
possible in order to diminish the energy dissipation in the copying process,
and to substantially reduce the switching time from one bit to another.
Actually, the real technological effort is evolving at the intricate nanometer
scale, which clearly will be essential for the general human progress. The
emphasis on the devices is this time both to understand what they measure and
mostly to estimate their computing capabilities. As mentioned by Feynman [91]
the present transistor systems dissipate 10^10 kT. He considered bits written
"ridiculously", as he said, on a single atom. At present we know this is not
ridiculous since we already are talking about atomic transistors [91].
Acknowledgements
_________________
I am grateful to Prof. Gian Carlo Ghirardi for encouraging me to
participate to the scientific activity on foundations of quantum theory in
Trieste along the summer and fall of 1992, when a first substantial draft of
this paper has been written.
This work was partially supported by the CONACyT Project 4868-E9406.
============================================
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___________
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______________________________
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______________________
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Elementary Particle Physics, Kasimierz, Poland, May 25-29, (1992)
[33] G.B. Nagy, O.E. Ortiz, and O.A. Reula, "The behavior of hyperbolic heat
eqs.'solutions near their parabolic limits", J. Math. Phys. 35, 4334 (1994)
[34] R. Eugene Collins, "Quantum mechanics as a classical diffusion process",
Found. Phys. Lett. 5, 63 (1992). See also, A. Posilicano and S. Ugolini,
"Convergence of Nelson diffusions with time-dependent electromagnetic
potentials", J. Math. Phys. 34, 5028 (1993)
[35] I. Imai, "On the definition of macroscopic electromagnetic quantities",
J. Phys. Soc. Japan 60, 4100 (1991)
[36] I.Ya. Brusin, "A topological approach to the determination of macroscopic
field vectors", Usp. Fiz. Nauk 151, 143 (1987) [Sov. Phys. Usp. 30, 60
(1987)]
IV. Spontaneous GRW localization
_________________________________
[37] A. Shimony, "Desiderata for a modified quantum dynamics", Philosophy of
Science Association 2, 49 (1990). In this paper, Professor Shimony points on
the opinion of Bell, expressed at a workshop at Amherst College in June 1990,
that the stochastic modifications of the standard deterministic quantum
dynamics is the most important new idea in the field of foundations of quantum
mechanics during his professional lifetime. See also, J.S. Bell, "Speakable
and Unspeakable in Quantum Mechanics" (University Press, Cambridge, UK, 1987)
[38] E. Joos and H.D. Zeh, "The emergence of classical properties through
interaction with the environment", Z. Phys. B 59, 223 (1985)
[39] G. C. Ghirardi, A. Rimini, and T. Weber, "Unified dynamics for
microscopic and macroscopic systems", Phys. Rev. D 34, 470 (1986); G.C.
Ghirardi et al., "Spontaneous localization of a system of identical
particles", Nuovo Cim. B 102, 383 (1988); T. Weber, "QMSL revisited",
*ibid.* 106, 1111 (1991). For a comparison of JZ and GRW (or QMSL) models
see M.R. Gallis and G.N. Fleming, "Comparison of quantum open-system models
with localization", Phys. Rev. A 43, 5778 (1991). For some progress toward
generalization of QMSL to relativistic domain see N. Gisin, "Stochastic
quantum dynamics and relativity", Helv. Phys. Acta 62, 363 (1989)
[40] A. Yu. Kasumov, N.A. Kislov, and I.I. Khodos, "Vibration of cantilever of
a supersmall mass", unpublished (1992); V.V. Aristov, N.A. Kislov, and I.I.
Khodos, "Direct electron beam-induced formation of nanometer carbon structures
in STEM", Inst. Phys. Conf. Ser. No. 117: Section 10 (Paper presented at
Microsc. Semicond. Mater. Conf., Oxford, 25-28 March 1991)
[41] B.G. Sumpter and D.W. Noid, "The onset of instability in nanostructures:
The role of nonlinear resonance", J. Chem. Phys. 102, 6619 (1995)
[42] M. Holthaus and B. Just, "Generalized `pi' pulses", Phys. Rev. A 49,
1950 (1994)
[43] L. Accardi and L.Y. Gang, "Squeezing noises as weak coupling limit of
Hamiltonian systems", Rep. Math. Phys. 29, 227 (1991)
[44] N. Yamada and S. Takagi, "Spacetime probabilities in nonrelativistic
quantum mechanics", Prog. Theor. Phys. 87, 77 (1992)
V. Nonlinear wavefunction collapse ?
____________________________________
[45] A.O. Barut, "Schroedinger's interpretation of `psi' as a continuous
charge distribution", Ann. d. Physik 45, 31 (1988)
[46] G.F. Chew, "Quantum-electrodynamic model of isolated macroscopic systems
with a recordable connection to surroundings", Phys. Rev. A 45, 4312 (1992);
"A QED model for concentrating condensed matter", LBL-31385 (1991); see also
F. Doveil et al., "Localization of Langmuir waves in a fluctuating plasma",
Phys. Rev. Lett. 69, 2074 (1992)
[47] P. J. Bussey, "When does the wavefunction collapse ?", Phys. Lett. A
106, 407 (1984); "Wavefunction collapse and the optical theorem", *ibid.* A
118, 377 (1986); A.M. Jayannavar, "Comment on some theories of state
reduction", *ibid.* A 167, 433 (1992). This author shows that the modified
master equation for the density matrix in theories like those of GRW and Diosi
is the same as for a particle subjected to a classical white noise potential.
R. Balian, "On the principles of quantum mechanics and the reduction of the
wave packet", Am. J. Phys. 57, 1019 (1989); P. Pearle, "On the time it takes
a state vector to reduce", J. Stat. Phys. 41, 719 (1985); "Might God toss
coins?", Found. Phys. 12, 249 (1982). W.H. Zurek, "Pointer basis of quantum
apparatus: Into what mixture does the wave packet collapse?", Phys. Rev. D
24, 1516 (1981); "Environment-induced superselection rules", *ibid*. 36,
1862 (1982); W.G. Unruh and W.H. Zurek, "Reduction of a wave packet in quantum
Brownian motion", *ibid*. 40, 1071 (1981); H.P. Stapp, "Noise-induced
reduction of wave packets and faster than light influences", *ibid*. A 46,
6860 (1992); GC Ghirardi, R. Grassi, and P. Pearle, "Comment on "Explicit
collapse and superluminal signals", Phys. Lett. A 166, 435 (1992); GC.
Ghirardi et al., "Parameter dependence and outcome dependence in dynamical
models for statevector reduction", in Trieste Workshop on Fundamentals 1992,
to be published. D.Z. Albert, "On the collapse of the wave function", in
"Sixty-two Years of Uncertainty: Historical, Philosophical, and Physical
Inquires into the Foundations of Quantum Mechanics", A.I. Miller (ed.). New
York: Plenum Press, pp 153-65; L. L. Bonilla and F. Guinea, "Collapse of the
wave packet and chaos in a model with classical and quantum degree of
freedom", Phys. Rev. A 45, 7718 (1992); J. Finkelstein, "Covariant collapse
of the state vector and realism", Found. Phys. Lett. 5, 383 (1992); M.
Namiki and S. Pascazio, "Quantum theory of measurement: Wave-function
collapse, decoherence and related interference phenomena", preprint 1992,
review along the lines of Many-Hilbert-Space approach of Machida and Namiki in
Proc. 2nd Int. Symp. on Found. of QM, p. 355, eds M. Namiki et al. (Phys. Soc.
of Japan, Tokyo, 1987); T. Kobayashi, "Macroscopic and mesoscopic changes of
entropies in detectors and collapse in quantum measurement processes",
UTHEP-235, Tsukuba, Ibaraki (1992); Kobayashi distinguishes between quantum
collapse and pure statistical collapse that could be discriminated by means of
mesoscopic phenomena.
[48] J. Unturbe and J.L. Saenchez-Gomez, "On the role of gravitation in the
possible breakdown of the quantum superposition principle for macroscopic
systems", Nuovo Cim. B 107, 211 (1992); R. Penrose, "Gravity and state
vector reduction", in "Quantum Concepts in Space and Time", R. Penrose and C.
Isham (eds.). Oxford: Clarendon Press, pp. 129-46; L. Diosi, Phys. Rev. A
40, 1165 (1989); GC. Ghirardi, R. Grassi, and A. Rimini,
"Continuous-spontaneous-reduction model involving gravity", *ibid*. 42, 1057
(1990); J.L. Rosales and J.L. Saenchez-Gomez, "Non-linear Schroedinger
equation coming from the action of the particle's gravitational field on the
quantum potential", Phys. Lett. A 166, 111 (1992)
[49] J. Ellis, S. Mohanty, and D.V. Nanopoulos, "Quantum gravity and the
collapse of the wavefunction", Phys. Lett. B 221, 113 (1989); M. Damjanovic,
"Is the collapse a phase transition?", *ibid*. A 134, 77 (1988); K.
Urbanowski, "Reduction process, multiple measurement and decay", Europhys.
Lett. 18, 291-295 (1992); M. Jibu, T. Misawa, and K. Yasue, "Measurement and
reduction of wavefunction in stochastic mechanics", Phys. Lett. A 150, 59
(1990); Y. Toyozawa, "Theory of measurement- A note on conceptual foundation
of quantum mechanics", Prog. Theor. Phys. 87, 293 (1992); J. Daboul,
"Increase of average energy by the process of measurement", Europhys. Lett.
18, 189-94 (1992); F. Benatti, "Entropy divergence in the GRW model", Phys.
Lett. A 132, 13 (1988)
[50] A. Peres, "Nonlinear variants of Schroedinger's equation violate the
second law of thermodynamics", Phys. Rev. Lett. 63, 1114 (1989); Concerning
nonlinear quantum mechanics, already de Broglie wrote book-lengthly upon it in
"Tentative d'interpretation causale et non-lineaire de la mecanique
ondulatoire", (Gauthier-Villars, Paris, 1956) and "Nonlinear Wavemechanics",
(Elsevier, Amsterdam, 1960). Some well-known papers on nonlinear quantum
mechanics are as follows: B. Mielnik, "Generalized quantum mechanics", Commun.
Math. Phys. 37, 221 (1974); R. Haag and U. Bannier, "Comments on Mielnik's
generalized (non linear) quantum mechanics", *ibid*. 60, 1 (1978); T.W.B.
Kibble, "Relativistic models of nonlinear quantum mechanics", *ibid*. 64, 73
(1978). Kibble expresses the viewpoint that the linearity of quantum mechanics
as manifested through the superposition principle is anomalous, in all other
known cases being only an approximation. J. Ellis et al., "Search for
violations of quantum mechanics", Nucl. Phys. B 241, 381 (1984) T. Kato, "On
nonlinear Schroedinger equations", Ann. Inst. H.P. Phys. Theor. 46, 113
(1987) H.-D. Doebner and G.A. Goldin, "On a general nonlinear Schroedinger
equation admitting diffusion currents", Phys. Lett. A 162, 397 (1992)
Detailed theoretical analysis of the possible nonlinear corrections to quantum
mechanics (a viewpoint different from NLSE) can be found in: S. Weinberg,
"Precision tests of quantum mechanics", Phys. Rev. Lett. 62, 485 (1989);
Ann. Phys. 194, 336 (1989). For some experiments to test the linearity of
quantum mechanics see, T.E. Chupp and R.J. Hoare, "Coherence in freely
precessing ^21 Ne and a test of linearity of quantum mechanics", Phys. Rev.
Lett. 64, 2261 (1990); R.L. Walsworth et al., "Test of the linearity of
quantum mechanics in an atomic system with a hydrogen maser", *ibid*. 64,
2599 (1990)
[51] A.M. Mariz, "On the irreversible nature of the Tsallis and Renyi
entropies", Phys. Lett. A 165, 409 (1992); See also M. Czachor, "Elements of
nonlinear quantum mechanics (ll): Triple bracket generalization of quantum
mechanics", hep-th/9406174
[52] V.E. Zakharov Ed., "Wave Collapses", Proc. Int. Workshop on Wave Collapse
Physics, Novosibirsk, USSR, 20-27 March 1988, Physica D 52, (1991); S.
Dyachenko et al., "Optical turbulence: weak turbulence, condensates and
collapsing filaments in the nonlinear Schroedinger equation", Physica D 57,
96 (1992)
[53] V.E. Zakharov, Zh. Eksp. Teor. Fiz. 62 (1972) 1745 [Sov. Phys. JETP
35, 908 (1972)]. For a good introduction to NLSE for Langmuir waves see, B.D.
Fried and Y.H. Ichikawa, "On the nonlinear Schroedinger equation for Langmuir
waves", J. Phys. Soc. Japan 34, 1073 (1973)
[54] P.A. Robinson and D.L. Newman, "Two-component model of strong Langmuir
turbulence: scalings, spectra, and statistics of Langmuir turbulence", Phys.
Fluids B 2, 2999 (1990); "Density fluctuations in strong Langmuir
turbulence: scalings, spectra, and statistics", *ibid*. B 2, 3016 (1990);
P.A. Robinson, "Transit time damping and the arrest of wave collapse", *ibid*.
B 3, 545 (1991); G.D. Doolen et al., "Nucleation of cavitons in strong
Langmuir turbulence", Phys. Rev. Lett. 54, 804 (1985). Various other types
of nonlinear collapses can be encountered in the literature. See, e.g., V.
Perez-Munuzuri et al., "Collapse of wave fronts in reaction-diffusion
systems", Phys. Lett. A 168, 133 (1992); J.T. Manassah, "Collapse of the
two-dimensional spatial soliton in a parabolic-index material", Optics Lett.
17, 1259-61 (1992)
VI. Various other topics
__________________________
VI.A: Friction modifications of qm
___________________________________
[55] J. Dalibard et al., "Wave-function approach to dissipative processes in
quantum optics", Phys. Rev. Lett. 68, 580 (1992); A.P. Polychronakos and R.
Tzani, "Schroedinger equation for particle with friction", Phys. Lett. B
302, 255-260 (1993)
[56] G.J. Milburn and D.F. Walls, "Quantum solutions of the damped harmonic
oscillator", Am. J. Phys. 51, 1134 (1983); E.G. Harris, "Quantum theory of
the damped harmonic oscillator", Phys. Rev. A 42, 3685 (1990); E. Celeghini,
M. Rasetti, and G. Vitiello, "Quantum dissipation", Ann. Phys. 215, 156
(1992); A.M. Kowalski, A. Plastino, and A.N. Proto, "Information-theoretic
outlook of the quantum dissipation problem", IC/92/214 (August 1992); B.L. Hu,
J.P. Paz, and Y. Zhang, "Quantum Brownian motion in a general environment: I.
Exact master equation with non local dissipation and colored noise", Phys.
Rev. D 45, 2843 (1992). Quantum Brownian motion (QBM) is considered by many
people to be the main paradigm of quantum open systems. I. Oppenheim and V.
Romero-Rochin, "Quantum Brownian motion", Physica A 147, 184 (1987); S. Roy,
"Relativistic Brownian motion and the space-time approach to quantum
mechanics", J. Math. Phys. 21, 71 (1980); V.R. Chechetkin, V.S. Lutovinov,
"Quantum motion of particles in random fields and quantum dissipation:
Schroedinger equation with Gaussian fluctuating potentials", J. Phys. A 20,
4757 (1987)
[57] J. Ellis, N.E. Mavromatos, D.V. Nanopoulos, "String theory modifies
quantum mechanics", Phys. Lett. B 293, 37 (1992)
[58] M.I. Beciu, "Frinflation", Europhys. Lett. 27, 71 (1994)
VI.B: WP dualities
__________________
[59] W.K. Wootters and W.H. Zurek, "Complementarity in the double-slit
experiment: Quantum nonseparability and a quantitative statement of Bohr's
principle", Phys. Rev. D 19, 473 (1979)
[60] L.S. Bartell, "Complementarity in the double-slit experiment: On simple
realizable systems for observing intermediate particle-wave behavior", *ibid*.
D 21, 1698 (1980)
[61] F. Bardou, "Transition between particle behavior and wave behavior", Am.
J. Phys. 59, 458 (1991)
[62] Y. Mizobuchi and Y. Ohtake, "An "experiment to throw more light on
light"", Phys. Lett. A 168, 1 (1992)
[63] P. Ghose, D. Home, and G.S. Agarwal, "An experiment to throw more light
on light", *ibid*. 153, 403 (1991)
[64] L. Kannenberg, "Quantum formalism via signal analysis", Found. Physics
19, 367 (1989); D. Hajela, "On faster than Nyquist signaling: further
estimations on the minimum distance", SIAM J. Appl. Math. 52, 900 (1992);
J.D. Cresser, "Theory of electron detection and photon-photoelectron
correlations in two-photon ionization", J. Opt. Soc. Am. B 6, 1492 (1989);
B.R. Mollow, "Quantum theory of field attenuation", Phys. Rev. 168, 1896
(1968); V.P. Bykov, V.I. Tatarskii, "Causality violation in the Glauber theory
of photodetection", Phys. Lett. A 136, 77 (1989)
[65] H. John Caulfield and J. Shamir, "Wave particle duality considerations in
optical computing", Appl. Opt. 28, 2184 (1989); "Wave-particle duality
processors: characteristics, requirements, and applications", J. Opt. Soc. Am.
A 7, 1314 (1990); I.M. Ross, "Telecommunications in the era of photonics",
Solid State Technology, (April 1992), pp. 36-43, presented before the IEE
Michael Faraday Bicentennial Conf. in London, England, Sept. 25, 1991; C.M.
Caves, "Quantum limits on bosonic communication rates", Rev. Mod. Phys. 66,
481 (1994); P.M. Harman, "Maxwell's other demons", Nature 356, 753 (1992)
[66] W.E. Lamb, Jr., "Anti-photon", Appl. Phys. B 60, 77 (1995)
VI.C: Constancy of the Planck constant
______________________________________
[67] A.O. Barut, "Formulation of wave mechanics without the Planck constant
h-bar", Phys. Lett. A 171, 1 (1992). h-physics, that is physics related to
the Planck constant is a kind of atomism which is not the most general one.
The charge atomism, i.e., the existence of localized, charged particles with
different masses does not follow from the Planck physics. See the discussion
of R. Roempe and H.-J. Treder, "Demokrit-Planck", Ann. Physik 45, 37 (1988).
Of relevance here is also D.F. Bartlett and W.F. Edwards, "Invariance of
charge to Lorentz transformation", Phys. Lett. A 151, 259 (1990); H.D.
Weymann, "Finite speed of propagation in heat conduction, diffusion, and
viscous shear motion", Am. J. Phys. 35, 488 (1965). In the last paper, it
is shown that the finite speed of propagation is due to the atomistic
structure of matter.
[68] E. Fishbach, G.L Greene, and R.J. Hughes, "New test of QM: Is Planck's
constant unique ?", Phys. Rev. Lett. 66, 256 (1991); M. Baublitz Jr.,
"Limitations on the possibility of distinct Planck constants", Nuovo Cimento B
110, 121 (1995)
[69] P.S. Wesson, "Constants and cosmology: the nature and origin of
fundamental constants in astrophysics and particle physics", Space Sci. Rev.
59, 407 (1992). According to Wesson, there are no constants which truly
deserve to be called fundamental, and therefore the aim of physics ought to be
to write down laws in which no constants appear.
L.B. Okun, "Fundamental constants of physics", TPI-MINN-91/22-T [ITEP-41-91]
(1991).
[70] L. Diosi, "Quantum dynamics with two Planck constants and the
semiclassical limit", quant-ph/9503023
[71] J.P. Bagby, "A comparison of the Titius-Bode rule with the Bohr atomic
orbitals", Spec. Sci. Technol. 2, 173 (1979); R. Wayte, "Quantization in
stable gravitational systems", Moon and Planets 26, 11 (1982); R. Louise, "A
postulate leading to the Titius Bode law", *ibid*. 26, 93 (1982); "Loi de
Titius Bode et formalism ondulatoire", *ibid*. 26, 389 (1982); "Quantum
formalism in gravitational quantitative application to the Titius-Bode Law",
*ibid*. 27, 59 (1982); A. Buta, "Toward an atomic model of the planetary
system", Rev. Roum. Phys. 27, 321 (1982)
[72] W.G. Tifft, Ap. J. 179, 29 (1973); 181, 305 (1973); W.J. Cocke, "A
theoretical framework for quantized redshifts and uncertainty in cosmology",
Astrophys. Lett. 23, 239 (1983); W.G. Tifft and W. J. Cocke, "Global
redshift quantization", Ap. J. 287, 492 (1984); R. Louise, "Steady wave
model of spiral galaxies and its application in cosmology", Astrophys. Sp.
Sci. 86, 505 (1982); M. DerSarkissian, "Does wave-particle duality apply to
galaxies ?", Lett. Nuovo Cim. 40, 390 (1984); "New consequences of cosmic
QM", *ibid*. 43, 274 (1985); "Possible evidence for gravitational Bohr
orbits in double galaxies", *ibid*. 44, 629 (1985); H. Arp, "Additional
members of the local group of galaxies and quantized redshifts within the two
nearest groups", J. Astrophys. Astr. 8, 241 (1987); D.M. Greenberger,
"Quantization in the Large", Found. Physics 13, (1983); R. Muradian, "Regge
in the sky: Origin of the cosmic rotation", preprint IC/94/143 (1994) and
references therein; R.L. Oldershaw, "Discrete self-similarity between
period-radius relations for variable stars and Rydberg atoms", Spec. Sci.
Tech. 14, 193 (1991)
[73] N.P. Landsman, "Definitions of algebra of observables and the classical
limit of quantum mechanics", Rev. Math. Phys. 5, 775 (1993)
[74] E.R. Cohen and B.N. Taylor, "The 1986 adjustment of the fundamental
physical constants", Rev. Mod. Phys. 59, 1121 (1987); "The fundamental
physical constants", Physics Today (August 1994)
VI.D: Quantum mechanics and cosmology
_____________________________________
[75] H. Everett III, ""Relative state" formulation of qm", Rev. Mod. Phys.
29, 454 (1957)
[76] M. Gell-Mann and J.B. Hartle, "Quantum mechanics in the light of quantum
cosmology", Proc. 3rd Int. Symp. Foundations of Quantum mechanics, Tokyo, pp.
321-343 (1989)
[77] S. Guiasu, "Probability space of wave functions", Phys. Rev. A 36, 1971
(1987)
[78] C.J. Isham, N. Linden, and S. Schreckenberg, "The classification of
decoherence functionals: an analogue of Gleason's theorem", J. Math. Phys.
35, 6360 (1994) %[gr-qc/9406015]
VI.E: Quantum jumps
___________________
[79] T. Erber et al., "Resonance fluorescence and quantum jumps in single
atoms: Testing the randomness of quantum mechanics", Ann. Phys. 190, 254
(1989); R.G. Hulet et al., "Precise test of quantum jump theory", Phys. Rev. A
37, 4544 (1988); D.T. Pegg and P.L. Knight, "Interrupted fluorescence,
quantum jumps, and wave-function collapse", *ibid*. 37, 4303 (1988); M.S.
Kim and P.L. Knight, "Quantum-jump telegraph noise and macroscopic intensity
fluctuations", *ibid*. 36, 5265 (1987)
[80] J.D. Deuschel and D.W. Stroock, "Large deviations", Pure and Applied
Mathematics, vol 137, Academic Press (1987); D.W. Dawson and J. Gartner, "Long
time fluctuations of weakly interacting diffusions", Stochastics 20, 247
(1987); J. Lynch and J. Sethuraman, "Large deviations for processes with
independent increments", Ann. Probab. 15, 610 (1987); M. Ovidiu Vlad and
K.W. Kehr, "A new stochastic description of random processes with memory: the
overall jump rate as a random variable", Phys. Lett. A 158, 149 (1991)
[81] A.A. Konstantinov, V.P. Maslov and A.M. Chebotarev, "Probability
representations of solutions of the Cauchy problem for quantum mechanical
equations", Usp. Mat. Nauk 45:6, 3 (1990) [ Russian Math. Surveys 45:6, 1
(1990)]
[82] S. Succi and R. Benzi, "Lattice Boltzmann equation for quantum
mechanics", ROM2F/92/30 (1992) %[check Phys. Lett.]
[83] H.D. Zeh, "There are no quantum jumps, nor are there particles !", Phys.
Lett. A 172, 189 (1993)
[84] T. Erber and S.J. Putterman, "Quantum jumps in a single atom; prolonged
darkness in the fluorescence of a resonantly driven cascade", Phys. Lett. A
141, 43 (1989); J.C. Bergquist et al., "Observation of quantum jumps in a
single atom", Phys. Rev. Lett. 57, 1699 (1986)
VI.F: Analogies to qm
_____________________
[85] Yi Bo and Xu Jia-Fu, "Analogy model and analogy correspondence", Science
in China A 35, 374 (1992)
[86] E.W. Cowan, "Electric analog network for quantum mechanics", Am. J.
Phys. 34, 1122 (1966)
[87] J.L. Rosner, "The Smith chart and quantum mechanics", Am. J. Phys. 61,
310 (1993)
[88] W.N. Caron, "Antenna Impedance Matching", (American Radio League
Publications, Newington, CT, 1989) VI.G: Human brain and quantum
computers/brains
[89] T. Mitchison and M. Kirschner, "Microtubule assembly nucleated by
isolated centrosomes" and "Dynamic instability of microtubule growth", Nature
312, 232, 237 (1984)
[90] R. Landauer, "Is qm useful ?", preprint, (1994)
[91] R. P. Feynman, "Quantum mechanical computers", Optics News 11, 11,
(Febr. 1985)
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