Quantum mechanics (QM) is a set of principles
describing the
physical reality at
the atomic level of matter (
molecules and
atoms) and the
subatomic (
electrons,
protons, and even smaller
particles). These descriptions include
the simultaneous wavelike and particlelike behavior of both
matter and radiation ("
wave–particle duality").
Quantum Mechanics is a mathematical description of reality, like
any scientific model. Some of its predictions and implications go
against the "common sense" of how humans see a set of bodies (a
system) behave. This isn't necessarily a failure of QM  it's more
of a reflection of how humans understand space and time on larger
scales (e.g.,
centimetres,
seconds) rather than much smaller. QM says that the
most complete description of a system is its wavefunction, which is
just a number varying between time and place. One can derive things
from the wavefunction, such as the position of a particle, or its
momentum. Yet the wavefunction describes probabilities, and some
physical quantities which
classical
physics would assume are both fully defined together
simultaneously for a system are not simultaneously given definite
values in QM. It is not that the experimental equipment is not
precise enough  the two quantities in question just really aren't
defined at the same time by the Universe. For instance,
location and
velocity just do not exist simultaneously for a
body (this is called the
Heisenberg uncertainty principle — see
its formula in the box to the right).
Certain systems, however, do exhibit quantum
mechanical effects on a
larger scale;
superfluidity (the
frictionless flow of a liquid at temperatures near absolute zero)
is one wellknown example. Quantum theory also provides accurate
descriptions for many previously unexplained phenomena such as
black body radiation and the
stability of
electron orbitals. It
has also given insight into the workings of many different
biological systems, including
smell receptors and
protein structures.
Even so,
classical physics often
can be a good approximation to results otherwise obtained by
quantum physics, typically in circumstances with
large numbers of particles or large quantum numbers. (However, some
open questions remain in the field of
quantum chaos.)
Overview
The word
quantum is
Latin
for "how great" or "how much." In quantum mechanics, it refers to a
discrete unit that quantum theory assigns to certain
physical quantities, such as the
energy of an
atom at rest (see
Figure 1, at right). The discovery that
waves
have discrete energy packets (called
quanta)
that behave in a manner similar to
particle led to the branch of physics
that deals with atomic and subatomic systems which we today call
quantum mechanics. It is the underlying
mathematical framework of many fields of
physics , including
condensed matter physics,
solidstate physics,
atomic physics,
molecular physics,
computational chemistry,
quantum chemistry,
particle physics, and
nuclear physics. The foundations of quantum
mechanics were established during the first half of the twentieth
century by
Werner Heisenberg,
Max Planck,
Louis de Broglie,
Albert Einstein,
Niels
Bohr,
Erwin Schrödinger,
Max Born,
John von
Neumann,
Paul Dirac,
Wolfgang Pauli,
David Hilbert, and
others.
Some fundamental aspects of the theory are still actively
studied.
Quantum mechanics is essential to understand the behavior of
systems at
atomic length scales and smaller.
For example, if
classical
mechanics governed the workings of an atom,
electrons would rapidly travel towards and collide
with the
nucleus, making stable atoms
impossible. However, in the natural world the electrons normally
remain in an uncertain, nondeterministic "smeared" (waveparticle
wave function) orbital path around or "through" the nucleus,
defying
classical
electromagnetism.
Quantum mechanics was initially developed to provide a better
explanation of the atom, especially the
spectra of
light emitted by
different
atomic species. The quantum theory
of the atom was developed as an explanation for the electron's
staying in its
orbital, which could
not be explained by Newton's laws of motion and by
Maxwell's laws of classical
electromagnetism.
In the formalism of quantum mechanics, the state of a system at a
given time is described by a
complex
wave function (sometimes referred to
as orbitals in the case of atomic electrons), and more generally,
elements of a complex
vector space.
This abstract mathematical object allows for the calculation of
probabilities of outcomes of concrete
experiments. For example, it allows one to compute the probability
of finding an electron in a particular region around the nucleus at
a particular time. Contrary to classical mechanics, one can never
make simultaneous predictions of
conjugate variables, such as position
and momentum, with accuracy. For instance, electrons may be
considered to be located somewhere within a region of space, but
with their exact positions being unknown. Contours of constant
probability, often referred to as “clouds,” may be drawn around the
nucleus of an atom to conceptualize where the electron might be
located with the most probability. Heisenberg's
uncertainty principle quantifies the
inability to precisely locate the particle given its
conjugate.
The other
exemplar that led to quantum
mechanics was the study of
electromagnetic waves such as light.
When it was found in 1900 by Max Planck that the energy of waves
could be described as consisting of small packets or quanta,
Albert Einstein exploited this idea
to show that an electromagnetic wave such as light could be
described by a particle called the
photon
with a discrete energy dependent on its frequency. This led to a
theory of unity between
subatomic particles and electromagnetic waves called
wave–particle duality in which
particles and waves were neither one nor the other, but had certain
properties of both. While quantum mechanics describes the world of
the very small, it also is needed to explain certain “
macroscopic quantum systems” such as
superconductors and
superfluids.
Broadly speaking, quantum mechanics incorporates four classes of
phenomena that classical physics cannot account for: (I) the
quantization (discretization)
of
certain physical
quantities, (II)
waveparticle
duality, (III) the
uncertainty
principle, and (IV)
quantum
entanglement. Each of these phenomena is described in detail in
subsequent sections.
History
The
history of quantum mechanics began essentially
with the 1838 discovery of
cathode rays
by
Michael Faraday, the 1859
statement of the
black body
radiation problem by
Gustav
Kirchhoff, the 1877 suggestion by
Ludwig Boltzmann that the energy states of
a physical system could be discrete, and the 1900 quantum
hypothesis by
Max Planck that any energy
is radiated and absorbed in quantities divisible by discrete
‘energy elements’, E, such that each of these energy elements is
proportional to the
frequency ν
with which they each individually radiate
energy, as defined by the following formula:
 E = h \nu = \hbar \omega\,
where
h is
Planck's
Action Constant. Planck insisted that this was simply an aspect
of the processes of absorption and emission of radiation and had
nothing to do with the physical reality of the radiation itself.
However, at that time, this appeared not to explain the
photoelectric effect (1839), i.e. that
shining light on certain materials can function to eject electrons
from the material. In 1905, basing his work on Planck’s quantum
hypothesis,
Albert Einstein
postulated that
light itself consists of
individual quanta. These later came to be called
photons (1926). From Einstein's simple postulation
was born a flurry of debating, theorizing and testing, and thus,
the entire field of quantum physics.
Quantum mechanics and classical physics
Predictions of quantum mechanics have been verified experimentally
to a very high degree of accuracy. Thus, the current logic of
correspondence principle
between classical and quantum mechanics is that all objects obey
laws of quantum mechanics, and classical mechanics is just a
quantum mechanics of large systems (or a statistical quantum
mechanics of a large collection of particles). Laws of classical
mechanics thus follow from laws of quantum mechanics at the limit
of large systems or large
quantum
numbers. However,
chaotic systems
do not have good quantum numbers, and
quantum chaos studies the relationship between
classical and quantum descriptions in these systems.
The main differences between classical and quantum theories have
already been mentioned above in the remarks on the
EinsteinPodolskyRosen paradox. Essentially the
difference boils down to the statement that quantum mechanics is
coherent (addition of
amplitude), whereas classical
theories are
incoherent (addition of
intensities). Thus, such quantities as
coherence
lengths and
coherence times come into play. For
microscopic bodies the extension of the system is certainly much
smaller than the
coherence length;
for macroscopic bodies one expects that it should be the other way
round. An exception to this rule can occur at extremely low
temperatures, when quantum behavior can manifest itself on more
"macroscopic" scales (see
BoseEinstein condensate).
This is in accordance with the following observations:
Many “macroscopic” properties of “classic” systems are direct
consequences of quantum behavior of its parts. For example,
stability of bulk matter (which consists of atoms and
molecules which would quickly collapse under
electric forces alone), rigidity of this matter, mechanical,
thermal, chemical, optical and magnetic properties of this
matter—they are all results of interaction of
electric charges under the rules of quantum
mechanics.
While the seemingly exotic behavior of matter posited by quantum
mechanics and relativity theory become more apparent when dealing
with extremely fastmoving or extremely tiny particles, the laws of
classical “Newtonian” physics still remain accurate in predicting
the behavior of surrounding (“large”) objects—of the order of the
size of large molecules and bigger—at velocities much smaller than
the
velocity of light.
Theory
There are numerous mathematically equivalent formulations of
quantum mechanics. One of the oldest and most commonly used
formulations is the
transformation
theory proposed by Cambridge
theoretical physicist Paul Dirac, which unifies and generalizes the two
earliest formulations of quantum mechanics,
matrix mechanics (invented by
Werner Heisenberg) and
wave mechanics (invented by
Erwin Schrödinger).
In this formulation, the
instantaneous
state of a quantum system encodes the probabilities of its
measurable properties, or "
observables".
Examples of observables include
energy,
position,
momentum, and
angular momentum. Observables can be either
continuous (e.g., the position
of a particle) or
discrete
(e.g., the energy of an electron bound to a hydrogen atom).
Generally, quantum mechanics does not assign definite values to
observables. Instead, it makes predictions using
probability distributions; that is,
the probability of obtaining possible outcomes from measuring an
observable. Oftentimes these results are skewed by many causes,
such as dense
probability clouds
or quantum state nuclear attraction. Naturally, these probabilities
will depend on the quantum state at the "instant" of the
measurement. Hence, uncertainty is involved in the value. There
are, however, certain states that are associated with a definite
value of a particular observable. These are known as "eigenstates"
of the observable ("eigen" can be roughly translated from
German as inherent or as a characteristic).
In the everyday world, it is natural and intuitive to think of
everything (every observable) as being in an eigenstate. Everything
appears to have a definite position, a definite momentum, a
definite energy, and a definite time of occurrence. However,
quantum mechanics does not pinpoint the exact values of a particle
for its position and momentum (since they are
conjugate pairs) or its energy and time
(since they too are conjugate pairs); rather, it only provides a
range of probabilities of where that particle might be given its
momentum and momentum probability. Therefore, it is helpful to use
different words to describe states having
uncertain values and states
having
definite values (eigenstate).
For example, consider a
free particle.
In quantum mechanics, there is
waveparticle duality so the
properties of the particle can be described as the properties of a
wave. Therefore, its
quantum state can
be represented as a
wave of arbitrary shape and
extending over space as a
wave
function. The position and momentum of the particle are
observables. The
Uncertainty Principle states that both
the position and the momentum cannot simultaneously be measured
with full precision at the same time. However, one can measure the
position alone of a moving free particle creating an eigenstate of
position with a wavefunction that is very large (a
Dirac delta) at a particular position
x
and zero everywhere else. If one performs a position measurement on
such a wavefunction, the result
x will be obtained with
100% probability (full certainty). This is called an eigenstate of
position (mathematically more precise: a
generalized position
eigenstate (eigendistribution)). If the
particle is in an eigenstate of position then its momentum is
completely unknown. On the other hand, if the particle is in an
eigenstate of momentum then its position is completely unknown.In
an eigenstate of momentum having a
plane
wave form, it can be shown 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 the observable we are interested
in. However, if one measures the observable, the wavefunction will
instantaneously be an eigenstate (or generalized eigenstate) of
that observable. This process is known as
wavefunction collapse, a debatable
process. It involves expanding the system under study to include
the measurement device. If one knows the corresponding wave
function at the instant before the measurement, one will be able to
compute the probability of collapsing into each of the possible
eigenstates. For example, the free particle in the previous example
will usually have a wavefunction that is a
wave packet centered around some mean position
x_{0}, neither an eigenstate of position nor of
momentum. When one measures the position of the particle, it is
impossible to predict with certainty the result. It is probable,
but not certain, that it will be near
x_{0}, where
the amplitude of the wave function is large. After the measurement
is performed, having obtained some result
x, the wave
function 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 the aforementioned
example of the 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 (which can be thought of as infinitely
sharp wave packets) into broadened wave packets that are no longer
position eigenstates.Some wave functions produce probability
distributions that are constant or independent of time, such as
when in a
stationary
state of constant energy, time drops out of the absolute square
of the wave function. Many systems that are treated dynamically in
classical mechanics are described by such "static" wave functions.
For example, a single
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
wavefunction surrounding the nucleus (
Fig. 1). (Note that only the lowest
angular momentum states, labeled
s, are spherically
symmetric).
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 not deterministic, but
rather unpredictable, i.e.,
random. A
timeevolution simulation can be seen here.
[4162]
The
probabilistic nature of quantum
mechanics thus stems from the act of measurement. This is one of
the most difficult aspects of quantum systems to understand. It was
the central topic in the famous
BohrEinstein debates, in which the
two scientists attempted to clarify these fundamental principles by
way of
thought experiments. In
the decades after the formulation of quantum mechanics, the
question of what constitutes a "measurement" has been extensively
studied.
Interpretations of
quantum mechanics have been formulated to do away with the concept
of "wavefunction collapse"; see, for example, the
relative state interpretation.
The basic idea is that when a quantum system interacts with a
measuring apparatus, their respective wavefunctions become
entangled, so that the original quantum
system ceases to exist as an independent entity. For details, see
the article on
measurement in quantum
mechanics.
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) well defined up to a complex number
of norm 1 (the phase factor). In other words, the possible states
are points in the
projectivization
of a Hilbert space, usually called the
complex projective space. 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
squareintegrable
functions, while the state space for the spin of a single proton is
just the product of two complex planes. Each observable is
represented by a maximally
Hermitian
(precisely: by a
selfadjoint)
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.
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.
The
inner product between two state
vectors is a complex number known as a
probability amplitude. During a
measurement, the probability that a system collapses from a given
initial state to a particular eigenstate is given by the square of
the
absolute value of the probability
amplitudes between the initial and final states. 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 Schrödinger equation acts on the entire probability amplitude,
not merely its absolute value. Whereas the absolute value of the
probability amplitude encodes information about probabilities, its
phase encodes information about the
interference between quantum states.
This gives rise to the wavelike behavior of quantum states.
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, the
particle in a box, the
hydrogen molecular ion 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 "semiclassical 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 quantummechanical amplitude
is considered as a sum over histories between initial and final
states; this is the quantummechanical counterpart of
action principles in classical
mechanics.
Interactions with other scientific theories
The fundamental rules of quantum mechanics are very deep. They
assert 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,
or if it even exists. 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 moves to higher energies or
equivalently, larger quantum numbers. In other words, classic
mechanics is simply a quantum mechanics of large systems. This
"high energy" 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 whosecorrespondence limit was
nonrelativistic classical mechanics. For instance, the
wellknown model of the
quantum harmonic oscillator uses
an explicitly nonrelativistic 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
KleinGordon
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 \scriptstyle \frac{e^2}{4 \pi\ \epsilon_0\ } \frac{1}{r}
Coulomb potential. This
"semiclassical" 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, by the physicists
Carl Jamieson,
Sheldon Glashow and
Steven Weinberg.
It has proven difficult to construct quantum models of
gravity, the remaining
fundamental force. Semiclassical
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.
Example
The particle in a 1dimensional potential energy box is the most
simple example where restraints lead to the quantization of energy
levels.The box is defined as zero potential energy inside a certain
interval and infinite everywhere outside that interval. For the
1dimensional case in the x direction, the timeindependent
Schrödinger equation can be written as:
  \frac {\hbar ^2}{2m} \frac {d ^2 \psi}{dx^2} = E \psi.
The general solutions are:
 \psi(x) = A e^{ikx} + B e ^{ikx} \qquad\qquad E =
\frac{\hbar^2 k^2}{2m}
or, from
Euler's formula,
 \psi(x) = C \sin kx + D \cos kx.\!
The presence of the walls of the box determines the values of
C,
D, and
k. At each wall ( and ), .
Thus when ,
 \psi(0) = 0 = C\sin 0 + D\cos 0 = D\!
and so . When ,
 \psi(L) = 0 = C\sin kL.\!
C cannot be zero, since this would conflict with the Born
interpretation. Therefore , and so it must be that
kL is
an integer multiple of π. Therefore,
 k = \frac{n\pi}{L}\qquad\qquad n=1,2,3,\ldots.
The quantization of energy levels follows from this constraint on
k, since
 E = \frac{\hbar^2 \pi^2 n^2}{2mL^2} =
\frac{n^2h^2}{8mL^2}.
Attempts at a unified field theory
As of 2009 the quest for unifying the
fundamental forces through quantum
mechanics is still ongoing.
Quantum electrodynamics (or "quantum
electromagnetism"), which is currently the most accurately tested
physical theory, has been successfully merged with the weak nuclear
force into the
electroweak force
and work is currently being done to merge the electroweak and
strong force into the
electrostrong
force. Current predictions state that at around 10
^{14}
GeV the three aforementioned forces are fused into a single unified
field, Beyond this "grand unification", it is speculated that it
may be possible to merge gravity with the other three gauge
symmetries, expected to occur at roughly 10
^{19} GeV.
However and while special relativity is parsimoniously
incorporated into quantum electrodynamics the expanded
general relativity, currently the best
theory describing the gravitation force, has not been fully
incorporated into quantum theory.
Relativity and quantum mechanics
 Main articles: Quantum
gravity and Theory of
everything
Even with the defining postulates of both Einstein's theory of
general relativity and quantum theory being indisputably supported
by rigorous and repeated
empirical
evidence and while they do not directly contradict each other
theoretically (at least with regard to primary claims), they are
resistant to being incorporated within one cohesive model.
Einstein himself is well known for rejecting some of the claims of
quantum mechanics. While clearly contributing to the field, he did
not accept the more philosophical consequences and interpretations
of quantum mechanics, such as the lack of deterministic
causality and the assertion that a single
subatomic particle can occupy numerous areas of space at one time.
He also was the first to notice some of the apparently exotic
consequences of
entanglement
and used them to formulate the
EinsteinPodolskyRosen
paradox, in the hope of showing that quantum mechanics had
unacceptable implications. This was 1935, but in 1964 it was shown
by John Bell (see
Bell inequality)
that Einstein's assumption was correct, but had to be completed by
hidden variables and thus based on wrong philosophical
assumptions. According to the paper of J. Bell and the
Copenhagen interpretation (the
common interpretation of quantum mechanics by physicists for
decades), and contrary to Einstein's ideas, quantum mechanics
was
 neither a "realistic" theory (since quantum measurements do not
state preexisting properties, but rather they
prepare properties)
 nor a local
theory (essentially not, because the state vector \scriptstyle
\psi\rangle determines simultaneously the probability amplitudes at all sites,
\psi\rangle\to\psi(\mathbf r), \forall \mathbf r).
The EinsteinPodolskyRosen paradox shows in any case that there
exist experiments by which one can measure the state of one
particle and instantaneously change the state of its entangled
partner, although the two particles can be an arbitrary distance
apart; however, this effect does not violate
causality, since no transfer of information
happens. These experiments are the basis of some of the most
topical applications of the theory,
quantum cryptography, which has been on
the market since 2004 and works well, although at small distances
of typically \scriptstyle \le 1000 km.
Gravity is negligible in many areas of particle physics, so that
unification between general relativity and quantum mechanics is not
an urgent issue in those applications. However, the lack of a
correct theory of
quantum gravity is
an important issue in
cosmology and
physicists' search for an elegant "
theory of everything". Thus, resolving
the inconsistencies between both theories has been a major goal of
twentieth and twentyfirstcentury physics. Many prominent
physicists, including
Stephen
Hawking, have labored in the attempt to discover a theory
underlying
everything, combining not only different models
of subatomic physics, but also deriving the universe's four forces
—the
strong force,
electromagnetism,
weak force, and
gravity— from a single force or phenomenon. One of
the leading minds in this field is
Edward
Witten, a theoretical physicist who formulated the
groundbreaking
Mtheory, which is an
attempt at describing the supersymmetrical based
string theory.
Applications
Quantum mechanics has had enormous success in explaining many of
the features of our world. The individual behaviour of the
subatomic particles that make up all forms of
matter—
electrons,
protons,
neutrons,
photons and others—can often only be satisfactorily
described using quantum mechanics. Quantum mechanics has strongly
influenced
string theory, a candidate
for a
theory of everything (see
reductionism) and the
multiverse hypothesis. It is also related to
statistical mechanics.
Quantum mechanics is important for understanding how individual
atoms combine covalently to form chemicals or molecules. The
application of quantum mechanics to
chemistry is known as
quantum chemistry. (Relativistic) quantum
mechanics can in principle mathematically describe most of
chemistry. Quantum mechanics can provide quantitative insight into
ionic and
covalent 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 (and thus the
microchip), 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
exponentially faster than classical
computers. Another active research topic is
quantum teleportation, which
deals with techniques to transmit quantum states over arbitrary
distances.
In many devices, even the simple
light
switch,
quantum tunneling is
vital, as otherwise the electrons in the
electric current could not penetrate the
potential barrier made up, in the case of the light switch, of a
layer of oxide.
Flash memory chips
found in
USB drives also use quantum
tunneling to erase their memory cells.
Philosophical consequences
Since its inception, the many
counterintuitive 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.
The
Copenhagen
interpretation, due largely to the Danish theoretical physicist
Niels Bohr, is the interpretation of
quantum mechanics most widely accepted amongst physicists.
According to it, the probabilistic nature of quantum mechanics
predictions cannot be explained in terms of some other
deterministic theory, and does 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 (this dislike is the source of his famous quote,
"God does not play dice with the universe."). Einstein held that
there should be a
local
hidden variable theory underlying quantum mechanics and that,
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 realistic
theories.
Experiments have
been performed confirming the accuracy of quantum mechanics, thus
demonstrating that the physical world cannot be described by local
realistic theories. The
BohrEinstein debates provide a
vibrant critique of the Copenhagen Interpretation from an
epistemological point of view.
The
Everett
manyworlds interpretation, formulated in 1956, holds that all
the possibilities described by quantum theory simultaneously occur
in a "
multiverse" composed of
mostly independent parallel universes. This is not accomplished by
introducing some new axiom to quantum mechanics, but on the
contrary by
removing the axiom of the collapse of the wave
packet: All the possible consistent states of the measured system
and the measuring apparatus (including the observer) are present in
a
real physical (not just formally mathematical, as in
other interpretations)
quantum
superposition. (Such a superposition of consistent state
combinations of different systems is called an
entangled state.) While the multiverse is
deterministic, we perceive nondeterministic behavior governed by
probabilities, because we can observe only the universe, i.e. the
consistent state contribution to the mentioned superposition, we
inhabit. Everett's interpretation is perfectly consistent with
John Bell's experiments and makes
them intuitively understandable. However, according to the theory
of
quantum decoherence, the
parallel universes will never be accessible to us. This
inaccessibility can be understood as follows: once a measurement is
done, the measured system becomes
entangled with both the physicist who measured
it and a huge number of other particles, some of which are
photons flying away towards the other end of the
universe; in order to prove that the wave function did not collapse
one would have to bring all these particles back and measure them
again, together with the system that was measured originally. This
is completely impractical, but even if one could theoretically do
this, it would destroy any evidence that the original measurement
took place (including the physicist's memory).
See also
Notes
 See the Davisson–Germer
experiment, which showed the wavelike character of the
electron.
 See Einstein's photoelectric effect, for which he
gained the Nobel prize in physics.

http://discovermagazine.com/2009/feb/13isquantummechanicscontrollingyourthoughts/article_view?b_start:int=1&C
 http://www.merriamwebster.com/dictionary/quantum
 http://mooni.fccj.org/~ethall/quantum/quant.htm
 Compare the list of conferences presented here.
 http://www.oocities.com/mik_malm/quantmech.html
 http://www.statemaster.com/encyclopedia/Quantummechanics
 , Chapter 1, p. 52
 http://www.aip.org/history/heisenberg/p08a.htm
 http://www.crystalinks.com/quantumechanics.html
 J. Mehra and H. Rechenberg, The historical development of
quantum theory, SpringerVerlag, 1982.
 e.g. T.S. Kuhn, Blackbody theory and the
quantum discontinuity 18941912, Clarendon Press, Oxford,
1978.
 A. Einstein, Über einen die Erzeugung und Verwandlung des
Lichtes betreffenden heuristischen Gesichtspunkt (On a heuristic
point of view concerning the production and transformation of
light), Annalen der Physik 17
(1905) 132148 (reprinted in The collected papers of Albert
Einstein, John Stachel, editor, Princeton University Press,
1989, Vol. 2, pp. 149166, in German; see also Einstein's early
work on the quantum hypothesis, ibid. pp. 134148).

http://www.scribd.com/doc/5998949/Quantummechanicscourseiwhatisquantummechanics

http://philsciarchive.pitt.edu/archive/00002328/01/handbook.pdf

http://academic.brooklyn.cuny.edu/physics/sobel/Nucphys/atomprop.html

http://assets.cambridge.org/97805218/29526/excerpt/9780521829526_excerpt.pdf

http://www.spaceandmotion.com/physicsquantummechanicswernerheisenberg.htm
 Especially since Werner Heisenberg was awarded the
Nobel Prize in Physics in 1932 for
the creation of quantum mechanics, the role of Max Born has been obfuscated. A
2005 biography of Born details his role as the creator of the
matrix formulation of quantum mechanics. This was recognized in a
paper by Heisenberg, in 1940, honoring Max Planck. See: Nancy Thorndike Greenspan,
"The End of the Certain World: The Life and Science of Max Born"
(Basic Books, 2005), pp. 124  128, and 285  286.
 http://thwww.if.uj.edu.pl/acta/vol19/pdf/v19p0683.pdf

http://ocw.usu.edu/physics/classicalmechanics/pdf_lectures/06.pdf
 probability clouds are approximate, but better than
the Bohr model,
whereby electron location is given by a probability
function, the wave function eigenvalue, such that the probability is the
squared modulus of the complex amplitude

http://www.actapress.com/PaperInfo.aspx?PaperID=25988&reason=500
 , Chapter , p.
 http://www.dict.cc/germanenglish/eigen.html
 , Chapter 6, p. 79
 http://books.google.com/books?id=tKmEkwke_UC
 http://www.phy.olemiss.edu/~luca/Topics/qm/collapse.html
 , Chapter 8, p. 215

http://farside.ph.utexas.edu/teaching/qmech/lectures/node28.html
 , Chapter 2, p. 36

http://physics.ukzn.ac.za/~petruccione/Phys120/Wave%20Functions%20and%20the%20Schr%F6dinger%20Equation.pdf

http://www.reddit.com/r/philosophy/comments/8p2qv/determinism_and_naive_realism/
 , Chapter 8, p. 215
 P.A.M. Dirac, The Principles of Quantum Mechanics,
Clarendon Press, Oxford, 1930.
 J. von Neumann, Mathematische Grundlagen der
Quantenmechanik, Springer, Berlin, 1932 (English translation:
Mathematical Foundations of Quantum Mechanics, Princeton
University Press, 1955).
 Derivation of particle in a box, chemistry.tidalswan.com
 Life on the lattice: The most accurate theory we
have.
 "There is as yet no logically consistent and complete
relativistic quantum field theory.", p. 4. — V. B. Berestetskii,
E. M.
Lifshitz, L P Pitaevskii (1971). J. B. Sykes, J. S. Bell
(translators). Relativistic Quantum Theory 4, part
I. Course of Theoretical Physics (Landau and
Lifshitz) ISBN 0 08 016025 5
 http://books.google.com/books?id=vdXU6SD4_UYC

http://en.wikibooks.org/wiki/Computational_chemistry/Applications_of_molecular_quantum_mechanics
 http://plato.stanford.edu/entries/qmactiondistance/
 http://plato.stanford.edu/entries/qmeverett/
 http://wwwphysics.lbl.gov/~stapp/PTRS.pdf
References
The following titles, all by working physicists, attempt to
communicate quantum theory to lay people, using a minimum of
technical apparatus.
 Chester, Marvin (1987) Primer of Quantum Mechanics.
John Wiley. ISBN 0486428788
 Richard Feynman, 1985.
QED: The
Strange Theory of Light and Matter, Princeton University
Press. ISBN 0691083886. Four elementary lectures on quantum electrodynamics and quantum field theory, yet containing
many insights for the expert.
 Ghirardi, GianCarlo, 2004. Sneaking a Look at God's
Cards, Gerald Malsbary, trans. Princeton Univ. Press. The most
technical of the works cited here. Passages using algebra, trigonometry,
and braket notation can be passed
over on a first reading.
 N. David Mermin, 1990, “Spooky actions at a
distance: mysteries of the QT” in his Boojums all the way
through. Cambridge Univ. Press: 11076.
 Victor Stenger, 2000.
Timeless Reality: Symmetry, Simplicity, and Multiple
Universes. Buffalo NY: Prometheus Books. Chpts. 58. Includes
cosmological and philosophical considerations.
More technical:
 Bryce DeWitt, R. Neill
Graham, eds., 1973. The ManyWorlds Interpretation of Quantum
Mechanics, Princeton Series in Physics, Princeton
University Press. ISBN 069108131X
 The beginning chapters make up a very clear and comprehensible
introduction.
 Hugh Everett, 1957, "Relative State
Formulation of Quantum Mechanics," Reviews of Modern
Physics 29: 45462.
 A standard undergraduate text.
 Max Jammer, 1966. The Conceptual
Development of Quantum Mechanics. McGraw Hill.
 Hagen Kleinert, 2004. Path
Integrals in Quantum Mechanics, Statistics, Polymer Physics, and
Financial Markets, 3rd ed. Singapore: World Scientific.
Draft of 4th edition.
 Gunther Ludwig, 1968. Wave Mechanics. London: Pergamon
Press. ISBN 0082032041
 George Mackey (2004). The
mathematical foundations of quantum mechanics. Dover
Publications. ISBN 0486435172.
 Albert Messiah, 1966. Quantum Mechanics (Vol. I),
English translation from French by G. M. Temmer. North Holland,
John Wiley & Sons. Cf. chpt. IV, section III.
 Scerri, Eric R., 2006. The Periodic Table: Its Story and Its
Significance. Oxford University Press. Considers the extent to
which chemistry and the periodic system have been reduced to
quantum mechanics. ISBN 0195305736
 Hermann Weyl, 1950. The Theory
of Groups and Quantum Mechanics, Dover Publications.
 D. Greenberger, K. Hentschel, F. Weinert, eds., 2009.
Compendium of quantum physics, Concepts, experiments, history
and philosophy, SpringerVerlag, Berlin, Heidelberg.
Further reading
External links
 General
 Course material
 FAQs
 Media
 Philosophy