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Quantum Mechanics
Quantum mechanics
Quantum mechanics, also referred to as
quantum physics, is a
physical theory that describes the behavior of
matter at short length scales. It provides a quantitative explanation for
two types of phenomena that
classical mechanics and
classical electrodynamics cannot account for:
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Some observable physical quantities, such as the total
energy of a
blackbody, take on
discrete rather than
continuous values. This phenomenon is called quantization, and
the smallest possible intervals between the discrete values are called
quanta (singular: quantum, from the Latin word for "quantity",
hence the name "quantum mechanics.") The size of the quanta typically varies
from system to system.
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Under certain experimental conditions, microscopic objects
like
atoms and
electrons exhibit
wave-like behavior, such as
interference. Under other conditions, the same species of objects exhibit
particle-like behavior ("particle" meaning an object that can be localized to
a particular region of
space), such as
scattering. This phenomenon is known as
wave-particle duality.
The foundations of quantum mechanics were established during
the first half of the
20th century by the work of
Niels Bohr,
Werner Heisenberg,
Erwin Schrödinger,
Paul Dirac, and others. Some fundamental aspects of the theory are still
being actively studied. Quantum mechanics has also been adopted as the
underlying theory of many fields of physics and
chemistry, including
condensed matter physics,
quantum chemistry, and
particle physics.
Quantum mechanics describes the instantaneous state of a
system with a
wave function that encodes the
probability distribution of all measurable properties, or observables.
Possible observables for a system include
energy,
position,
momentum, and
angular momentum. Quantum mechanics does not assign definite values to the
observables, instead making predictions about their
probability distributions. The wavelike properties of matter are explained
by the interference of wave functions.
Wave functions can change as time progresses. For example, a
particle moving in empty space may be described by a wave function that is a
wave packet centered around some mean
position. As time progresses, the center of the wave packet changes, so that
the particle becomes more likely to be located at a different position. The time
evolution of wave functions is described by the
Schrödinger equation.
Some wave functions describe probability distributions that
are constant in time. Many systems that would be 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 circling the
atomic nucleus, whereas in quantum mechanics it is described by a static,
spherically symmetric probability cloud surrounding the nucleus.
When a measurement is performed on an observable of the
system, the wavefunction turns into one of a set of wavefunctions that are
called
eigenstates of the observable. This process is known as
wavefunction collapse. The relative probabilities of collapsing into each of
the possible eigenstates is described by the instantaneous wavefunction just
before the collapse. Consider the above example of a particle moving in empty
space. If we measure the particle's position, we will obtain a random value
x. In general, it is impossible for us to predict with certainty the value
of x which we will obtain, although it is probable that we will obtain
one that is near the center of the wave packet, where the amplitude of the wave
function is large. After the measurement has been performed, the wavefunction of
the particle collapses into one that is sharply concentrated around the observed
position x.
During the process of wavefunction collapse, the wavefunction
does not obey the Schrödinger equation. The Schrödinger equation 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 eigenstate to which the wavefunction collapses is
probabilistic, not deterministic. The probabilistic nature of quantum mechanics
thus stems from the act of measurement.
One of the consequences of wavefunction collapse is that
certain pairs of observables, such as position and momentum, can never be
simultaneously ascertained to arbitrary precision. This effect is known as
Heisenberg's
uncertainty principle.
In the mathematically rigorous formulation 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 (called the state space.) The exact nature of the
Hilbert space is dependent on the system; for example, the state space for
position and momentum states is the space of
square-integrable functions. 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,
plays a central role.
Each observable is represented by a densely-defined
Hermitian 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. We can therefore 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.
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 (i.e. non-quantum) physics when a system
becomes large, which is known as the classical or correspondence
limit. One may 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 relativistic 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
quantum chromodynamics, which describes the interactions of the subnuclear
particles, the
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.
Semi-classical approximations are techniques that make it
possible to formulate a quantum problem with some physical quantities replaced
by their classical analogues, in an effort to reduce the complexity of the
model. Even within non-relativistic quantum mechanics, a fully microscopic
treatment generally requires large-scale numerical computations. Analytic
quantum solutions that describe the system behavior in terms of known
mathematical functions are available only for a small class of systems, of which
the
harmonic oscillator and the
hydrogen atom are the most important representatives.
Even the
helium atom, containing just one more electron than hydrogen, defies all
attempts at a fully analytic treatment in quantum mechanics. In such a
situation, approximate semi-classical results can provide valuable insights. The
necessary methods rely on a detailed understanding of the corresponding
classical mechanics, allowing in particular for the existence of chaos. The
study of these approximations belongs to the field of
quantum chaos.
Much of modern
technology operates under quantum mechanical principles. Examples include
the
laser, the
electron microscope, and
magnetic resonance imaging. Most of the calculations performed in
computational chemistry rely on quantum mechanics.
Many of the phenomena studied in
condensed matter physics are fully quantum mechanical, and cannot be
satisfactorily modeled using classical physics. This includes the
electronic properties of
solids, such as
superconductivity and
semiconductivity. The study of semiconductors has 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.
Since its inception, the many counter-intuitive results of
quantum mechanics have provoked strong philosophical debate and many
interpretations. See
interpretation of quantum mechanics for more detail.
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 quantum mechanics must be
incomplete, and produced a series of objections to the theory. The most famous
of these was the
EPR paradox.
John Stewart Bell's theoretical solution to the EPR paradox, and its later
experimental verification, disproved a large class of such hidden variable
theories and persuaded the majority of physicists that quantum mechanics is not
an approximation to a nominally classical hidden-variable theory.
The
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 postulates the existence of a non-local, universal
wavefunction (Schrödinger equation) which allows distant particles to interact
instantaneously. It is not popular among physicists largely because it is
considered very inelegant.
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.
In the
1940s, quantum electrodynamics was developed by
Feynman,
Dyson,
Schwinger, and
Tomonaga. It served as a role model for subsequent quantum field theories.
The many worlds interpretation was formulated by
Everett in
1956.
Quantum chromodynamics had a long history, 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.
Recently, there has been much interest in
quantum information.
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I do not like it, and I am sorry I ever had anything to
do with it.
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Erwin Schrödinger, speaking of quantum mechanics
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Those who are not shocked when they first come across
quantum mechanics cannot possibly have understood it.
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Niels Bohr
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God does not play dice with the cosmos.
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Albert Einstein
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Einstein, don't tell God what to do.
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Niels Bohr in response to Einstein
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I think it is safe to say that no one understands
quantum mechanics.
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Richard Feynman
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It's always fun to learn something new about quantum
mechanics.
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Benjamin Schumacher
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If that turns out to be true, I'll quit physics.
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Max von Laue,
Nobel Laureate
1914, of de Broglie's thesis on electrons having wave properties.
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Anyone wanting to discuss a quantum mechanical problem
had better understand and learn to apply quantum mechanics to that problem.
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Willis Lamb,
Nobel Laureate
1955
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