Re: Epistemology 201: The Science of Science
From: robert j. kolker (nowhere_at_nowhere.net)
Date: 03/13/05
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Date: Sun, 13 Mar 2005 13:48:19 -0500
Lester Zick wrote:
>
>
> Quantum mechanics is necessary. Quantum postulates are not a
> mechanics.
The mathematics along with its interpretations ARE the theory.
Modern quantum theory identifies quantum states as vectors in a
seperable Hilbert space. Observables are Hermitean operators on the
Hilbert Space. Possible values of the observable are eigenvalues of the
Hermitean operator. In the early stages of the field quantum mechamics
pertained to the wave mechanics derived from the Schroeding Equaton or
the Matrix Mechanics derived from Heisneberg's formulaton. Now-a-days
the terms quantum mechanics and quantum theory are used rather
interchangably since it has been shown that Schroedinger's and
Heisenberg's formulation are equivalent. The theory was generalized to
subsume Special Theory of Relativity by Dirac and to resolve wave
particle duality. The theory has been generalized to describe the
interactions of all particles and fields (other than gravitational
fields). The so-called Standard Model.
Here is some glossary information on how the terms -quantum theory-,
-quantum mechanics- and -quantum field theory- are used:
quantum mechanics
Dictionary
quantum mechanics
n. (used with a sing. or pl. verb)
Quantum theory, especially the quantum theory of the structure and
behavior of atoms and molecules.
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The American Heritage® Dictionary of the English Language, Fourth
Edition Copyright © 2004, 2000 by Houghton Mifflin Company. Published by
Houghton Mifflin Company. All rights reserved.
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Technology
quantum mechanics
The branch of physics developed in the first part of the 20th century
that was highly successful in explaining the behavior of atoms,
molecules and nuclei. Developed between 1900 and 1930 and combined with
the general and special theory of relativity, it revolutionized the
field of physics. The new concepts, which were the particle properties
of radiation, the wave properties of matter, quantization of physical
properties and the idea that one can no longer know exactly where a
single particle such as an electron is at any one time were necessary to
explain all of the new experimental evidence that was available at the
time. For example, quantum mechanics explains the behavior of
semiconductors which are used to make the myriad of devices we use every
day.
Following are the important contributors to the foundation of quantum
mechanics and the principles they uncovered.
Year Researcher Quantum Mechanics Concept
1901 Planck Blackbody radiation
1905 Einstein Photoelectric effect
1913 Bohr Quantum theory of spectra
1922 Compton Scattering of photons
off electrons
1924 Pauli Exclusion principle
1925 de Broglie Matter waves
1926 Schroedinger Wave equation
1927 Heisenberg Uncertainty principle
1927 Davison and
Germer Wave properties of electrons
1927 Born Interpretation of the
wavefunction
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Encyclopedia
quantum field theory, study of the quantum mechanical interaction of
elementary particles and fields. Quantum field theory applied to the
understanding of electromagnetism is called quantum electrodynamics
(QED), and it has proved spectacularly successful in describing the
interaction of light with matter. The calculations, however, are often
complex. They are usually carried out with the aid of Feynman diagrams
(named after American physicist Richard P. Feynman), simple graphs that
represent possible variations of interactions and provide an elegant
shorthand for precise mathematical equations. Quantum field theory
applied to the understanding of the strong interactions between quarks
and between protons, neutrons, and other baryons and mesons is called
quantum chromodynamics (QCD); QCD has a mathematical structure similar
to that of QED.
Columbia University Press
The Columbia Electronic Encyclopedia, Sixth Edition Copyright © 2003,
Columbia University Press. Licensed from Columbia University Press. All
rights reserved. www.cc.columbia.edu/cu/cup/
Directory > General Reference > Encyclopedia > quantum mechanics
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Science
quantum mechanics
The branch of physics that deals with the behavior of matter at the
level of the atom, the nucleus, and the elementary particle. At this
level, energy, mass, momentum, and other quantities do not vary
continuously, as they do in the large-scale world, but come in discrete
units, or quanta. (See Bohr atom and photon.)
The New Dictionary of Cultural Literacy, Third Edition Edited by E.D.
Hirsch, Jr., Joseph F. Kett, and James Trefil. Copyright © 2002 by
Houghton Mifflin Company. Published by Houghton Mifflin. All rights
reserved.
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WordNet
Note: click on a word meaning below to see its connections and related
words.
The noun quantum field theory has one meaning:
Meaning #1: the branch of quantum physics that is concerned with the
theory of fields; it was motivated by the question of how an atom
radiates light as its electrons jump from excited states
Directory > Language > WordNet > quantum mechanics
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quantum mechanics
Fig. 1: The wavefunctions of an electron in a hydrogen atom possessing
definite energy (increasing downward: n=1,2,3,...) and angular momentum
(increasing across: s, p, d,...). Brighter areas correspond to higher
probability density for a position measurement. The angular momentum and
energy are quantized, and only take on discrete values like those shown.
Enlarge
Fig. 1: The wavefunctions of an electron in a hydrogen atom possessing
definite energy (increasing downward: n=1,2,3,...) and angular momentum
(increasing across: s, p, d,...). Brighter areas correspond to higher
probability density for a position measurement. The angular momentum and
energy are quantized, and only take on discrete values like those shown.
Quantum mechanics is a physical theory which, for very small objects
such as atoms, produces results that are very different and much more
accurate than those of classical mechanics. It is the underlying
framework of many fields of physics and chemistry, including condensed
matter physics, quantum chemistry, and particle physics. It is derived
from a small set of basic principles, and predicts at least three types
of phenomena that classical mechanics and classical electrodynamics
cannot account for: quantization, wave-particle duality, and quantum
entanglement. It also explains the behavior of many physical systems
that contradict classical mechanics, such as the existence of stable
atoms and the fact that the total radiation emitted by a black body is
finite.
The terms quantum physics and quantum theory are often used as synonyms
of quantum mechanics. Some authors refer to "quantum mechanics" in the
restricted sense of non-relativistic quantum mechanics. Quantum
mechanics should however be taken to mean quantum theory in its most
general sense when used in this article.
The foundations of quantum mechanics were established during the first
half of the 20th century by Max Planck, Albert Einstein, Niels Bohr,
Werner Heisenberg, Erwin Schrödinger, Max Born, Paul Dirac, Richard
Feynman and others. Some fundamental aspects of the theory are still
actively studied.
Description of the theory
Wave functions and measurement
There are a number of mathematically equivalent formulations of quantum
mechanics. One of the earliest and easiest to understand is the "wave
mechanics" formulation invented by Erwin Schrödinger, in which the
instantaneous state of a system is described by a "wavefunction" that
encodes the probability distribution of all measurable properties, or
"observables". Examples of observables include energy, position,
momentum, and angular momentum.
Generally, quantum mechanics does not assign definite values to
observables. Instead, it makes predictions about their probability
distributions; that is, the probability of obtaining each of the
possible outcomes from measuring an observable. Naturally, these
probabilities will depend on the wavefunction at the instant of the
measurement. There are, however, certain wavefunctions that are
associated with a definite value of a particular observable. These are
known as "eigenstates" of the observable ("eigen" meaning "own" in German).
A concrete example will be useful here. Let us consider a free particle.
Its wavefunction is a wave, of arbitrary shape, extending over all of
space, and its position and momentum are observables. An eigenstate of
position is a wavefunction that is very large at a particular position
x, and zero everywhere else. If we perform a position measurement on
such a wavefunction, we will obtain the result x with 100% probability.
An eigenstate of momentum, on the other hand, has the form of a plane
wave. It turns out that the wavelength is equal to h/p, where h is
Planck's constant and p is the momentum of the eigenstate.
Usually, a system will not be in an eigenstate of whatever observable we
are interested in. However, if we measure the observable, the
wavefunction will immediately become an eigenstate of that observable.
This process is known as wavefunction collapse. If we know the
wavefunction at the instant before the measurement, we will be able to
compute the probability of collapsing into each of the possible
eigenstates. For example, the free particle in our previous example will
usually have a wavefunction that is a wave packet centered around some
mean position x0, neither an eigenstate of position nor of momentum.
When we measure the position of the particle, it is impossible for us to
predict with certainty the result that we will obtain. It is probable,
but not certain, that it will be near x0, where the amplitude of the
wavefunction is large. After we perform the measurement, obtaining some
result x, the wavefunction collapses into a position eigenstate centered
at x.
Wave functions can change as time progresses. An equation known as the
Schrödinger equation describes how wave functions change in time, a role
similar to Newton's second law in classical mechanics. The Schrödinger
equation, applied to our free particle, predicts that the center of a
wave packet will move through space at a constant velocity, like a
classical particle with no forces acting on it. However, the wave packet
will also spread out as time progresses, which means that the position
becomes more uncertain. This also has the effect of turning position
eigenstates into broadened wave packets that are not position eigenstates.
Some wave functions produce probability distributions that are constant
in time. Many systems that are treated dynamically in classical
mechanics are described by such "static" wave functions. For example, an
electron in an unexcited atom is pictured classically as a particle
moving in a circular trajectory around the atomic nucleus, whereas in
quantum mechanics it is described by a static, spherically symmetric
probability cloud surrounding the nucleus (Fig. 1).
The time evolution of wave functions is deterministic in the sense that,
given a wavefunction at an initial time, it makes a definite prediction
of what the wavefunction will be at any later time. During a
measurement, the change of the wavefunction into another one is
probabilistic, not deterministic. The probabilistic nature of quantum
mechanics thus stems from the act of measurement. There are some
interpretations of quantum mechanics that do away with the concept of
"wavefunction collapse" by altering the concept of what constitutes a
"measurement" in quantum mechanics. For further details, see for example
the relative state interpretation.
Quantum mechanical effects
As mentioned in the introduction, there are several classes of phenomena
that appear under quantum mechanics which have no analogue in classical
physics. These are sometimes referred to as "quantum effects".
The first type of quantum effect is the quantization of certain physical
quantities. In the example we have given, of a free particle in empty
space, both the position and the momentum are continuous observables.
However, if we restrict the particle to a region of space (the so-called
"particle in a box" problem), the momentum observable will become
discrete; it will only take on the values h/L, where L is the length of
the box. Such observables are said to be quantized, and they play an
important role in many physical systems. Examples of quantized
observables include angular momentum, the total energy of a bound
system, and the energy contained in an electromagnetic wave of a given
frequency.
Another quantum effect is the uncertainty principle, which is the
phenomenon that consecutive measurements of two or more observables may
possess a fundamental limitation on accuracy. In our free particle
example, it turns out that it is impossible to find a wavefunction that
is an eigenstate of both position and momentum. This implies that
position and momentum can never be simultaneously measured with
arbitrary precision, even in principle: as the precision of the position
measurement improves, the maximum precision of the momentum measurement
decreases, and vice versa. Those variables for which it holds (e.g.
momentum and position, or energy and time) are canonically conjugate
variables in classical physics.
Another quantum effect is the wave-particle duality. It has been shown
that, under certain experimental conditions, microscopic objects like
atoms or electrons exhibit particle-like behavior, such as scattering.
("Particle-like" in the sense of an object that can be localized to a
particular region of space.) Under other conditions, the same type of
objects exhibit wave-like behavior, such as interference. We can observe
only one type of property at a time.
Another quantum effect is quantum entanglement. In some cases, the wave
function of a system composed of many particles cannot be separated into
independent wave functions, one for each particle. In that case, the
particles are said to be entangled. Entangled particles display
remarkable and counter-intuitive properties. For example, a measurement
made on a particle can produce, through the collapse of the total
wavefunction, an instantaneous effect on the other particles with which
it is entangled, even if they are far apart.
Mathematical formulation
In the mathematically rigorous formulation of quantum mechanics,
developed by Paul Dirac and John von Neumann, the possible states of a
quantum mechanical system are represented by unit vectors (called "state
vectors") residing in a complex separable Hilbert space (variously
called the "state space" or the "associated Hilbert space" of the
system.) The exact nature of this Hilbert space is dependent on the
system; for example, the state space for position and momentum states is
the space of square-integrable functions, while the state space for the
spin of a single electron is just the product of two complex planes. The
time evolution of a quantum state is described by the Schrödinger
equation, in which the Hamiltonian, the operator corresponding to the
total energy of the system, generates time evolution.
Each observable is represented by a densely-defined Hermitian (or
self-adjoint) linear operator acting on the state space. Each eigenstate
of an observable corresponds to an eigenvector of the operator, and the
associated eigenvalue corresponds to the value of the observable in that
eigenstate. If the operator's spectrum is discrete, the observable can
only attain those discrete eigenvalues. During a measurement, the
probability that a system collapses to each eigenstate is given by the
absolute square of the inner product between the eigenstate vector and
the state vector just before the measurement. The possible results of a
measurement are the eigenvalues of the operator - which explains the
choice of Hermitian operators, for which all the eigenvalues are real.
We can find the probability distribution of an observable in a given
state by computing the spectral decomposition of the corresponding
operator. Heisenberg's uncertainty principle is represented by the
statement that the operators corresponding to certain observables do not
commute.
The details of the mathematical formulation are contained in the article
Mathematical formulation of quantum mechanics. See also the discussion
in the article on Quantum logic.
It turns out that analytic solutions of Schrödinger's equation are only
available for a small number of model Hamiltonians, of which the quantum
harmonic oscillator and the hydrogen atom are the most important
representatives. Even the helium atom, which contains just one more
electron than hydrogen, defies all attempts at a fully analytic
treatment. There exist several techniques for generating approximate
solutions. For instance, in the method known as perturbation theory one
uses the analytic results for a simple quantum mechanical model to
generate results for a more complicated model related to the simple
model by, for example, the addition of a weak potential energy. Another
method is the "semi-classical equation of motion" approach, which
applies to systems for which quantum mechanics produces weak deviations
from classical behavior. The deviations can be calculated based on the
classical motion. This approach is important for the field of quantum chaos.
An alternative formulation of quantum mechanics is Feynman's path
integral formulation, in which a quantum-mechanical amplitude is
considered as a sum over histories between initial and final states;
this is the quantum-mechanical counterpart of action principles in
classical mechanics.
Interactions with other scientific theories
The fundamental rules of quantum mechanics are very broad. They state
that the state space of a system is a Hilbert space and the observables
are Hermitian operators acting on that space, but do not tell us which
Hilbert space or which operators. These must be chosen appropriately in
order to obtain a quantitative description of a quantum system. An
important guide for making these choices is the correspondence
principle, which states that the predictions of quantum mechanics reduce
to those of classical physics when a system becomes large. This "large
system" limit is known as the classical or correspondence limit. One can
therefore start from an established classical model of a particular
system, and attempt to guess the underlying quantum model that gives
rise to the classical model in the correspondence limit.
When quantum mechanics was originally formulated, it was applied to
models whose correspondence limit was non-relativistic classical
mechanics. For instance, the well-known model of the quantum harmonic
oscillator uses an explicitly non-relativistic expression for the
kinetic energy of the oscillator, and is thus a quantum version of the
classical harmonic oscillator.
Early attempts to merge quantum mechanics with special relativity
involved the replacement of the Schrödinger equation with a covariant
equation such as the Klein-Gordon equation or the Dirac equation. While
these theories were successful in explaining many experimental results,
they had certain unsatisfactory qualities stemming from their neglect of
the relativistic creation and annihilation of particles. A fully
relativistic quantum theory required the development of quantum field
theory, which applies quantization to a field rather than a fixed set of
particles. The first complete quantum field theory, quantum
electrodynamics, provides a fully quantum description of the
electromagnetic interaction.
The full apparatus of quantum field theory is often unnecessary for
describing electrodynamic systems. A simpler approach, one employed
since the inception of quantum mechanics, is to treat charged particles
as quantum mechanical objects being acted on by a classical
electromagnetic field. For example, the elementary quantum model of the
hydrogen atom describes the electric field of the hydrogen atom using a
classical 1/r Coulomb potential. This "semi-classical" approach fails if
quantum fluctuations in the electromagnetic field play an important
role, such as in the emission of photons by charged particles.
Quantum field theories for the strong nuclear force and the weak nuclear
force have been developed. The quantum field theory of the strong
nuclear force is called quantum chromodynamics, and describes the
interactions of the subnuclear particles: quarks and gluons. The weak
nuclear force and the electromagnetic force were unified, in their
quantized forms, into a single quantum field theory known as electroweak
theory.
It has proven difficult to construct quantum models of gravity, the
remaining fundamental force. Semi-classical approximations are workable,
and have led to predictions such as Hawking radiation. However, the
formulation of a complete theory of quantum gravity is hindered by
apparent incompatibilities between general relativity, the most accurate
theory of gravity currently known, and some of the fundamental
assumptions of quantum theory. The resolution of these incompatibilities
is an area of active research, and theories such as string theory are
among the possible candidates for a future theory of quantum gravity.
Applications of quantum theory
Quantum mechanics has had enormous success in explaining many of the
features of our world. The individual behavior of the microscopic
particles that make up all forms of matter - electrons, protons,
neutrons, and so forth - can often only be satisfactorily described
using quantum mechanics.
Quantum mechanics is important for understanding how individual atoms
combine to form chemicals. The application of quantum mechanics to
chemistry is known as quantum chemistry. Quantum mechanics can provide
quantitative insight into chemical bonding processes by explicitly
showing which molecules are energetically favorable to which others, and
by approximately how much. Most of the calculations performed in
computational chemistry rely on quantum mechanics.
Much of modern technology operates at a scale where quantum effects are
significant. Examples include the laser, the transistor, the electron
microscope, and magnetic resonance imaging. The study of semiconductors
led to the invention of the diode and the transistor, which are
indispensable for modern electronics.
Researchers are currently seeking robust methods of directly
manipulating quantum states. Efforts are being made to develop quantum
cryptography, which will allow guaranteed secure transmission of
information. A more distant goal is the development of quantum
computers, which are expected to perform certain computational tasks
with much greater efficiency than classical computers. Another active
research topic is quantum teleportation, which deals with techniques to
transmit quantum states over arbitrary distances.
Philosophical consequences
Since its inception, the many counter-intuitive results of quantum
mechanics have provoked strong philosophical debate and many
interpretations. Even fundamental issues such as Max Born's basic rules
concerning probability amplitudes and probability distributions took
decades to be appreciated.
Another difficulty with quantum mechanics is that the nature of an
object isn't known, in the sense that an object's position, or the shape
of the spatial distribution of the probability of presence, is only
known by the properties (charge for example) and the environment
(presence of an electric potential).
The Copenhagen interpretation, due largely to Niels Bohr, was the
standard interpretation of quantum mechanics when it was first
formulated. According to it, the probabilistic nature of quantum
mechanics predictions cannot be explained in terms of some other
deterministic theory, and do not simply reflect our limited knowledge.
Quantum mechanics provides probabilistic results because the physical
universe is itself probabilistic rather than deterministic.
Albert Einstein, himself one of the founders of quantum theory, disliked
this loss of determinism in measurement. He held that there should be a
local hidden variable theory underlying quantum mechanics and
consequently the present theory was incomplete. He produced a series of
objections to the theory, the most famous of which has become known as
the EPR paradox. John Bell showed that the EPR paradox led to
experimentally testable differences between quantum mechanics and local
hidden variable theories. Experiments have been taken as confirming that
quantum mechanics is correct and the real world cannot be described in
terms of such hidden variables. "Loopholes" in the experiments, however,
mean the question is still not quite settled.
See the Bohr-Einstein debates
The Everett many-worlds interpretation, formulated in 1956, holds that
all the possibilities described by quantum theory simultaneously occur
in a "multiverse" composed of mostly independent parallel universes.
While the multiverse is deterministic, we perceive non-deterministic
behavior governed by probabilities because we can observe only the
universe we inhabit.
The Bohm interpretation, formulated by David Bohm, postulates the
existence of a non-local, universal wavefunction (Schrödinger equation)
which allows distant particles to interact instantaneously. Based on
this interpretation, Bohm has speculated that the ultimate nature of
physical reality is not a collection of separate objects (as it appears
to us), but rather an undivided whole that is in perpetual dynamic flux.
However, the Bohm interpretation is not popular among physicists,
largely because it is considered very inelegant.
History
In 1900, Max Planck introduced the idea that energy is quantized, in
order to derive a formula for the observed frequency dependence of the
energy emitted by a black body. In 1905, Einstein explained the
photoelectric effect by postulating that light energy comes in quanta
called photons. In 1913, Bohr explained the spectral lines of the
hydrogen atom, again by using quantization. In 1924, Louis de Broglie
put forward his theory of matter waves.
These theories, though successful, were strictly phenomenological: there
was no rigorous justification for quantization. They are collectively
known as the old quantum theory.
The phrase "quantum physics" was first used in Johnston's Planck's
Universe in Light of Modern Physics.
Modern quantum mechanics was born in 1925, when Heisenberg developed
matrix mechanics and Schrödinger invented wave mechanics and the
Schrödinger equation. Schrödinger subsequently showed that the two
approaches were equivalent.
Heisenberg formulated his uncertainty principle in 1927, and the
Copenhagen interpretation took shape at about the same time. In 1927,
Paul Dirac unified quantum mechanics with special relativity. He also
pioneered the use of operator theory, including the influential bra-ket
notation. In 1932, John von Neumann formulated the rigorous mathematical
basis for quantum mechanics as operator theory.
The field of quantum chemistry was pioneered by Walter Heitler and Fritz
London, who published a study of the covalent bond of the hydrogen
molecule in 1927. Quantum chemistry was subsequently developed by a
large number of workers, including the American chemist Linus Pauling.
Beginning in 1927, attempts were made to apply quantum mechanics to
fields rather than single particles, resulting in what are known as
quantum field theories. Early workers in this area included Dirac,
Pauli, Weisskopf, and Jordan. This area of research culminated in the
formulation of quantum electrodynamics by Feynman, Dyson, Schwinger, and
Tomonaga during the 1940s. Quantum electrodynamics is a quantum theory
of electrons, positrons, and the electromagnetic field, and served as a
role model for subsequent quantum field theories.
The many worlds interpretation was formulated by Everett in 1956.
The theory of quantum chromodynamics was formulated beginning in the
early 1960s. The theory as we know it today was formulated by Polizter,
Gross and Wilzcek in 1975. Building on pioneering work by Schwinger,
Higgs, Goldstone and others, Glashow, Weinberg and Salam independently
showed how the weak nuclear force and quantum electrodynamics could be
merged into a single electroweak force.
Founding experiments
* Thomas Young's double-slit experiment proving the wave nature of
light (c1805)
* Henri Becquerel discovers radioactivity (1896)
* Joseph John Thomson's cathode ray tube experiments (discovers the
electron and its negative charge) (1897)
* The study of Black body radiation between 1850 and 1900.
* Robert Millikan's oil-drop experiment, which suggests that
electric charge occurs as quanta (whole units), (1909)
* Ernest Rutherford's gold foil experiment disproved the plum
pudding model of the atom which suggests that the positive charge and
mass of the atom are almost uniformly distributed. (1911)
* Otto Stern and Walter Gerlach conduct the Stern-Gerlach
experiment, which demonstrates particle spin (1920)
* Clyde L. Cowan and Frederick Reines confirm the existence of the
neutrino in the neutrino experiment (1955)
Many more experiments have been done of course, but this is the short
list given in the article.
Bob Kolker
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