Quantum mechanics is a branch of physics which
deals with physical phenomena at nanoscopic
scales where the action is on the order of
the Planck constant. It departs from classical
mechanics primarily at the quantum realm of
atomic and subatomic length scales. Quantum
mechanics provides a mathematical description
of much of the dual particle-like and wave-like
behavior and interactions of energy and matter.
Quantum mechanics provides a substantially
useful framework for many features of the
modern periodic table of elements including
the behavior of atoms during chemical bonding
and has played a significant role in the development
of many modern technologies.
In advanced topics of quantum mechanics, some
of these behaviors are macroscopic and emerge
at only extreme energies or temperatures.
For example, the angular momentum of an electron
bound to an atom or molecule is quantized.
In contrast, the angular momentum of an unbound
electron is not quantized. In the context
of quantum mechanics, the wave–particle
duality of energy and matter and the uncertainty
principle provide a unified view of the behavior
of photons, electrons, and other atomic-scale
objects.
The mathematical formulations of quantum mechanics
are abstract. A mathematical function, the
wavefunction, provides information about the
probability amplitude of position, momentum,
and other physical properties of a particle.
Mathematical manipulations of the wavefunction
usually involve bra–ket notation which requires
an understanding of complex numbers and linear
functionals. The wavefunction formulation
treats the particle as a quantum harmonic
oscillator, and the mathematics is akin to
that describing acoustic resonance. Many of
the results of quantum mechanics are not easily
visualized in terms of classical mechanics.
For instance, in a quantum mechanical model
the lowest energy state of a system, the ground
state, is non-zero as opposed to a more "traditional"
ground state with zero kinetic energy. Instead
of a traditional static, unchanging zero energy
state, quantum mechanics allows for far more
dynamic, chaotic possibilities, according
to John Wheeler.
The earliest versions of quantum mechanics
were formulated in the first decade of the
20th century. About this time, the atomic
theory and the corpuscular theory of light
first came to be widely accepted as scientific
fact; these latter theories can be viewed
as quantum theories of matter and electromagnetic
radiation, respectively. Early quantum theory
was significantly reformulated in the mid-1920s
by Werner Heisenberg, Max Born and Pascual
Jordan,; Louis de Broglie and Erwin Schrödinger;
and Wolfgang Pauli and Satyendra Nath Bose.
Moreover, the Copenhagen interpretation of
Niels Bohr became widely accepted. By 1930,
quantum mechanics had been further unified
and formalized by the work of David Hilbert,
Paul Dirac and John von Neumann with a greater
emphasis placed on measurement in quantum
mechanics, the statistical nature of our knowledge
of reality, and philosophical speculation
about the role of the observer. Quantum mechanics
has since permeated throughout many aspects
of 20th-century physics and other disciplines
including quantum chemistry, quantum electronics,
quantum optics, and quantum information science.
Much 19th-century physics has been re-evaluated
as the "classical limit" of quantum mechanics
and its more advanced developments in terms
of quantum field theory, string theory, and
speculative quantum gravity theories.
The name quantum mechanics derives from the
observation that some physical quantities
can change only in discrete amounts, and not
in a continuous way.
History
Scientific inquiry into the wave nature of
light began in the 17th and 18th centuries
when scientists such as Robert Hooke, Christiaan
Huygens and Leonhard Euler proposed a wave
theory of light based on experimental observations.
In 1803, Thomas Young, an English polymath,
performed the famous double-slit experiment
that he later described in a paper entitled
"On the nature of light and colours". This
experiment played a major role in the general
acceptance of the wave theory of light.
In 1838, with the discovery of cathode rays
by Michael Faraday, these studies were followed
by 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 can be discrete, and
the 1900 quantum hypothesis of Max Planck.
Planck's hypothesis that energy is radiated
and absorbed in discrete "quanta" precisely
matched the observed patterns of black-body
radiation.
In 1896, Wilhelm Wien empirically determined
a distribution law of black-body radiation,
known as Wien's law in his honor. Ludwig Boltzmann
independently arrived at this result by considerations
of Maxwell's equations. However, it was valid
only at high frequencies, and underestimated
the radiance at low frequencies. Later, Max
Planck corrected this model using Boltzmann
statistical interpretation of thermodynamics
and proposed what is now called Planck's law,
which led to the development of quantum mechanics.
Among the first to study quantum phenomena
in nature were Arthur Compton, C.V. Raman,
Pieter Zeeman, each of whom has a quantum
effect named after him. Robert A. Millikan
studied the Photoelectric effect experimentally
and Albert Einstein developed a theory for
it. At the same time Niels Bohr developed
his theory of the atomic structure which was
later confirmed by the experiments of Henry
Moseley. In 1913, Peter Debye extended Niels
Bohr's theory of atomic structure, introducing
elliptical orbits, a concept also introduced
by Arnold Sommerfeld. This phase is known
as Old quantum theory.
According to Planck, each energy element,
E, is proportional to its frequency, ν:
where h is Planck's 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. In fact, he considered
his quantum hypothesis a mathematical trick
to get the right answer rather than a sizable
discovery. However, in 1905 Albert Einstein
interpreted Planck's quantum hypothesis realistically
and used it to explain the photoelectric effect
in which shining light on certain materials
can eject electrons from the material.
The foundations of quantum mechanics were
established during the first half of the 20th
century by Max Planck, Niels Bohr, Werner
Heisenberg, Louis de Broglie, Arthur Compton,
Albert Einstein, Erwin Schrödinger, Max Born,
John von Neumann, Paul Dirac, Enrico Fermi,
Wolfgang Pauli, Max von Laue, Freeman Dyson,
David Hilbert, Wilhelm Wien, Satyendra Nath
Bose, Arnold Sommerfeld and others. In the
mid-1920s, developments in quantum mechanics
led to its becoming the standard formulation
for atomic physics. In the summer of 1925,
Bohr and Heisenberg published results that
closed the "Old Quantum Theory". Out of deference
to their particle-like behavior in certain
processes and measurements, light quanta came
to be called photons. From Einstein's simple
postulation was born a flurry of debating,
theorizing, and testing. Thus the entire field
of quantum physics emerged, leading to its
wider acceptance at the Fifth Solvay Conference
in 1927.
The other exemplar that led to quantum mechanics
was the study of electromagnetic waves, such
as visible and ultraviolet 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
further developed this idea to show that an
electromagnetic wave such as light could also
be described as a particle with a discrete
quantum of energy that was dependent on its
frequency. Einstein was able to use the photon
theory of light to explain the photoelectric
effect for which he won the 1921 Nobel Prize
in Physics. This led to a theory of unity
between subatomic particles and electromagnetic
waves in which particles and waves are neither
simply particle nor wave but have certain
properties of each. This originated the concept
of wave–particle duality.
While quantum mechanics traditionally described
the world of the very small, it is also needed
to explain certain recently investigated macroscopic
systems such as superconductors, superfluids,
and large organic molecules.
The word quantum derives from the Latin, meaning
"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. The
discovery that particles are discrete packets
of energy with wave-like properties led to
the branch of physics dealing with atomic
and sub-atomic systems which is today called
quantum mechanics. It underlies the mathematical
framework of many fields of physics and chemistry,
including condensed matter physics, solid-state
physics, atomic physics, molecular physics,
computational physics, computational chemistry,
quantum chemistry, particle physics, nuclear
chemistry, and nuclear physics. Some fundamental
aspects of the theory are still actively studied.
Quantum mechanics is essential to understanding
the behavior of systems at atomic length scales
and smaller. If the physical nature of an
atom was solely described by classical mechanics
electrons would not "orbit" the nucleus since
orbiting electrons emit radiation and would
eventually collide with the nucleus due to
this loss of energy. This framework was unable
to explain the stability of atoms. Instead,
electrons remain in an uncertain, non-deterministic,
"smeared", probabilistic, wave–particle
orbital about the nucleus, defying the traditional
assumptions of classical mechanics and electromagnetism.
Quantum mechanics was initially developed
to provide a better explanation and description
of the atom, especially the differences in
the spectra of light emitted by different
isotopes of the same element, as well as subatomic
particles. In short, the quantum-mechanical
atomic model has succeeded spectacularly in
the realm where classical mechanics and electromagnetism
falter.
Broadly speaking, quantum mechanics incorporates
four classes of phenomena for which classical
physics cannot account:
quantization of certain physical properties
wave–particle duality
principle of uncertainty
quantum entanglement.
Mathematical formulations
In the mathematically rigorous formulation
of quantum mechanics developed by Paul Dirac,
David Hilbert, John von Neumann, and Hermann
Weyl the possible states of a quantum mechanical
system are represented by unit vectors. Formally,
these reside in a complex separable Hilbert
space—variously called the "state space"
or the "associated Hilbert space" of the system—that
is well defined up to a complex number of
norm 1. In other words, the possible states
are points in the projective space 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 square-integrable 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 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 attain only
those discrete eigenvalues.
In the formalism of quantum mechanics, the
state of a system at a given time is described
by a complex wave function, also referred
to as state vector in 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 given region
of space, but with their exact positions 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 momentum.
According to one interpretation, as the result
of a measurement the wave function containing
the probability information for a system collapses
from a given initial state to a particular
eigenstate. The possible results of a measurement
are the eigenvalues of the operator representing
the observable—which explains the choice
of Hermitian operators, for which all the
eigenvalues are real. The probability distribution
of an observable in a given state can be found
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 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 Bohr-Einstein 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. Newer interpretations of quantum
mechanics have been formulated that do away
with the concept of "wavefunction collapse".
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.
Generally, quantum mechanics does not assign
definite values. Instead, it makes a prediction
using a probability distribution; that is,
it describes the probability of obtaining
the possible outcomes from measuring an observable.
Often these results are skewed by many causes,
such as dense probability clouds. 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,
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.
In the everyday world, it is natural and intuitive
to think of everything 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's position and momentum or its
energy and time; rather, it provides only
a range of probabilities in which 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.
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 of that
observable. This process is known as wavefunction
collapse, a controversial and much-debated
process that 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
the wavefunction 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 x0. 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 x0, 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.
The time evolution of a quantum state is described
by the Schrödinger equation, in which the
Hamiltonian generates the time evolution.
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, on the other hand, the
change of the initial wavefunction into another,
later wavefunction is not deterministic, it
is unpredictable. A time-evolution simulation
can be seen here.
Wave functions change as time progresses.
The Schrödinger equation describes how wavefunctions
change in time, playing 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.
However, the wave packet will also spread
out as time progresses, which means that the
position becomes more uncertain with time.
This also has the effect of turning a position
eigenstate into a broadened wave packet that
no longer represents a position eigenstate.
Some wave functions produce probability distributions
that are constant, or independent of time—such
as when in a stationary state of constant
energy, time vanishes in 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.
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 "wave-like" behavior of quantum states.
As it turns out, analytic solutions of the
Schrödinger equation are available for only
a very small number of relatively simple 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 does the hydrogen atom—has defied all
attempts at a fully analytic treatment.
There exist several techniques for generating
approximate solutions, however. In the important
method known as perturbation theory, one uses
the analytic result for a simple quantum mechanical
model to generate a result for a more complicated
model that is related to the simpler model
by 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 only
weak deviations from classical behavior. These
deviations can then be computed based on the
classical motion. This approach is particularly
important in the field of quantum chaos.
Mathematically equivalent formulations of
quantum mechanics
There are numerous mathematically equivalent
formulations of quantum mechanics. One of
the oldest and most commonly used formulations
is the "transformation theory" proposed by
Paul Dirac, which unifies and generalizes
the two earliest formulations of quantum mechanics
- matrix mechanics and wave mechanics.
Especially since Werner Heisenberg was awarded
the Nobel Prize in Physics in 1932 for the
creation of quantum mechanics, the role of
Max Born in the development of QM was overlooked
until the 1954 Nobel award. The role is noted
in a 2005 biography of Born, which recounts
his role in the matrix formulation of quantum
mechanics, and the use of probability amplitudes.
Heisenberg himself acknowledges having learned
matrices from Born, as published in a 1940
festschrift honoring Max Planck. In the matrix
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 or discrete. An alternative
formulation of quantum mechanics is Feynman's
path integral formulation, in which a quantum-mechanical
amplitude is considered as a sum over all
possible classical and non-classical paths
between the initial and final states. This
is the quantum-mechanical counterpart of the
action principle in classical mechanics.
Interactions with other scientific theories
The rules of quantum mechanics are fundamental.
They assert that the state space of a system
is a Hilbert space, and that observables of
that system are Hermitian operators acting
on that space—although they do not tell
us which Hilbert space or which operators.
These can 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 mechanics
when a system moves to higher energies or—equivalently—larger
quantum numbers, i.e. whereas a single particle
exhibits a degree of randomness, in systems
incorporating millions of particles averaging
takes over and, at the high energy limit,
the statistical probability of random behaviour
approaches zero. In other words, classical
mechanics is simply a quantum mechanics of
large systems. This "high energy" limit is
known as the classical or correspondence limit.
One can even start from an established classical
model of a particular system, then attempt
to guess the underlying quantum model that
would give 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. 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 that has been 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
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 also
been developed. The quantum field theory of
the strong nuclear force is called quantum
chromodynamics, and describes the interactions
of subnuclear particles such as quarks and
gluons. The weak nuclear force and the electromagnetic
force were unified, in their quantized forms,
into a single quantum field theory, by the
physicists Abdus Salam, Sheldon Glashow and
Steven Weinberg. These three men shared the
Nobel Prize in Physics in 1979 for this work.
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
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.
Classical mechanics has also been extended
into the complex domain, with complex classical
mechanics exhibiting behaviors similar to
quantum mechanics.
Quantum mechanics and classical physics
Predictions of quantum mechanics have been
verified experimentally to an extremely high
degree of accuracy. According to the correspondence
principle between classical and quantum mechanics,
all objects obey the laws of quantum mechanics,
and classical mechanics is just an approximation
for large systems of objects. The laws of
classical mechanics thus follow from the laws
of quantum mechanics as a statistical average
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.
Quantum coherence is an essential difference
between classical and quantum theories as
illustrated by the Einstein–Podolsky–Rosen
paradox — an attempt to disprove quantum
mechanics by an appeal to local realism. Quantum
interference involves adding together probability
amplitudes, whereas classical "waves" infer
that there is an adding together of intensities.
For microscopic bodies, the extension of the
system is much smaller than the coherence
length, which gives rise to long-range entanglement
and other nonlocal phenomena characteristic
of quantum systems. Quantum coherence is not
typically evident at macroscopic scales, though
an exception to this rule may occur at extremely
low temperatures at which quantum behavior
may manifest itself macroscopically. This
is in accordance with the following observations:
Many macroscopic properties of a classical
system are a direct consequence of the quantum
behavior of its parts. For example, the stability
of bulk matter, the rigidity of solids, and
the mechanical, thermal, chemical, optical
and magnetic properties of matter are all
results of the 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
particles of extremely small size or velocities
approaching the speed of light, the laws of
classical, often considered "Newtonian", physics
remain accurate in predicting the behavior
of the vast majority of "large" objects at
velocities much smaller than the velocity
of light.
Relativity and quantum mechanics
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, they have proven
extremely difficult to incorporate into one
consistent, 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 many of the more "philosophical
consequences and interpretations" of quantum
mechanics, such as the lack of deterministic
causality. He is famously quoted as saying,
in response to this aspect, "My God does not
play with dice". He also had difficulty with
the assertion that a single subatomic particle
can occupy numerous areas of space at one
time. However, he was also the first to notice
some of the apparently exotic consequences
of entanglement, and used them to formulate
the Einstein–Podolsky–Rosen paradox in
the hope of showing that quantum mechanics
had unacceptable implications if taken as
a complete description of physical reality.
This was 1935, but in 1964 it was shown by
John Bell that - although Einstein was correct
in identifying seemingly paradoxical implications
of quantum mechanical nonlocality - these
implications could be experimentally tested.
Alain Aspect's initial experiments in 1982,
and many subsequent experiments since, have
definitively verified quantum entanglement.
According to the paper of J. Bell and the
Copenhagen interpretation—the common interpretation
of quantum mechanics by physicists since 1927
- and contrary to Einstein's ideas, quantum
mechanics was not, at the same time a "realistic"
theory and a "local" theory.
The Einstein–Podolsky–Rosen 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. Quantum
entanglement forms the basis of quantum cryptography,
which is used in high-security commercial
applications in banking and government.
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 particular applications.
However, the lack of a correct theory of quantum
gravity is an important issue in cosmology
and the search by physicists for an elegant
"Theory of Everything". Consequently, resolving
the inconsistencies between both theories
has been a major goal of 20th and 21st century
physics. Many prominent physicists, including
Stephen Hawking, have labored for many years
in the attempt to discover a theory underlying
everything. This TOE would combine not only
the different models of subatomic physics,
but also derive the four fundamental forces
of nature - the strong force, electromagnetism,
the weak force, and gravity - from a single
force or phenomenon. While Stephen Hawking
was initially a believer in the Theory of
Everything, after considering Gödel's Incompleteness
Theorem, he has concluded that one is not
obtainable, and has stated so publicly in
his lecture "Gödel and the End of Physics".
Attempts at a unified field theory
The quest to unify the fundamental forces
through quantum mechanics is still ongoing.
Quantum electrodynamics, which is currently
the most accurately tested physical theory,(blog)
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
1014 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 1019 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. One of those searching for
a coherent TOE is Edward Witten, a theoretical
physicist who formulated the M-theory, which
is an attempt at describing the supersymmetrical
based string theory. M-theory posits that
our apparent 4-dimensional spacetime is, in
reality, actually an 11-dimensional spacetime
containing 10 spatial dimensions and 1 time
dimension, although 7 of the spatial dimensions
are - at lower energies - completely "compactified"
and not readily amenable to measurement or
probing.
Another popular theory is Loop quantum gravity,
a theory that describes the quantum properties
of gravity. It is also a theory of quantum
space and quantum time, because in general
relativity the geometry of spacetime is a
manifestation of gravity. LQG is an attempt
to merge and adapt standard quantum mechanics
and standard general relativity. The main
output of the theory is a physical picture
of space where space is granular. The granularity
is a direct consequence of the quantization.
It has the same nature of the granularity
of the photons in the quantum theory of electromagnetism
or the discrete levels of the energy of the
atoms. But here it is space itself which is
discrete. More precisely, space can be viewed
as an extremely fine fabric or network "woven"
of finite loops. These networks of loops are
called spin networks. The evolution of a spin
network over time, is called a spin foam.
The predicted size of this structure is the
Planck length, which is approximately 1.616×10−35
m. According to theory, there is no meaning
to length shorter than this. Therefore LQG
predicts that not just matter, but also space
itself, has an atomic structure. Loop quantum
Gravity was first proposed by Carlo Rovelli.
Philosophical implications
Since its inception, the many counter-intuitive
aspects and results of quantum mechanics have
provoked strong philosophical debates and
many interpretations. Even fundamental issues,
such as Max Born's basic rules concerning
probability amplitudes and probability distributions
took decades to be appreciated by society
and many leading scientists. Richard Feynman
once said, "I think I can safely say that
nobody understands quantum mechanics." According
to Steven Weinberg, "There is now in my opinion
no entirely satisfactory interpretation of
quantum mechanics."
The Copenhagen interpretation - due largely
to the Danish theoretical physicist Niels
Bohr - remains the quantum mechanical formalism
that is currently most widely accepted amongst
physicists, some 75 years after its enunciation.
According to this interpretation, the probabilistic
nature of quantum mechanics is not a temporary
feature which will eventually be replaced
by a deterministic theory, but instead must
be considered a final renunciation of the
classical idea of "causality". It is also
believed therein that any well-defined application
of the quantum mechanical formalism must always
make reference to the experimental arrangement,
due to the complementarity nature of evidence
obtained under different experimental situations.
Albert Einstein, himself one of the founders
of quantum theory, disliked this loss of determinism
in measurement. Einstein held that there should
be a local hidden variable theory underlying
quantum mechanics and, consequently, that
the present theory was incomplete. He produced
a series of objections to the theory, the
most famous of which has become known as the
Einstein–Podolsky–Rosen paradox. John
Bell showed that this "EPR" paradox led to
experimentally testable differences between
quantum mechanics and local realistic theories.
Experiments have been performed confirming
the accuracy of quantum mechanics, thereby
demonstrating that the physical world cannot
be described by any local realistic theory.
The Bohr-Einstein debates provide a vibrant
critique of the Copenhagen Interpretation
from an epistemological point of view.
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. 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 of the possible consistent states of the
measured system and the measuring apparatus
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
non-deterministic behavior governed by probabilities,
because we can observe only the universe that
we, as observers, 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, these "parallel universes" will
never be accessible to us. The 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 at the speed
of light 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 originally
measured. Not only is this completely impractical,
but even if one could theoretically do this,
it would have to destroy any evidence that
the original measurement took place; in light
of these Bell tests, Cramer formulated his
transactional interpretation. Relational quantum
mechanics appeared in the late 1990s as the
modern derivative of the Copenhagen Interpretation.
Applications
Quantum mechanics had enormous success in
explaining many of the features of our world.
Quantum mechanics is often the only tool available
that can reveal the individual behaviors of
the subatomic particles that make up all forms
of matter. Quantum mechanics has strongly
influenced string theories, candidates for
a Theory of Everything.
Quantum mechanics is also critically important
for understanding how individual atoms combine
covalently to form 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 also provide quantitative insight into
ionic and covalent bonding processes by explicitly
showing which molecules are energetically
favorable to which others, and the magnitudes
of the energies involved. Furthermore, most
of the calculations performed in modern computational
chemistry rely on quantum mechanics.
A great deal of modern technological inventions
operate 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 parts
of modern electronics systems and devices.
Researchers are currently seeking robust methods
of directly manipulating quantum states. Efforts
are being made to more fully develop quantum
cryptography, which will theoretically 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
information over arbitrary distances.
Quantum tunneling is vital to the operation
of many devices - even in the simple light
switch, as otherwise the electrons in the
electric current could not penetrate the potential
barrier made up of a layer of oxide. Flash
memory chips found in USB drives use quantum
tunneling to erase their memory cells.
While quantum mechanics primarily applies
to the atomic regimes of matter and energy,
some systems exhibit quantum mechanical effects
on a large scale - superfluidity, the frictionless
flow of a liquid at temperatures near absolute
zero, is one well-known example. Quantum theory
also provides accurate descriptions for many
previously unexplained phenomena, such as
black-body radiation and the stability of
the orbitals of electrons in atoms. It has
also given insight into the workings of many
different biological systems, including smell
receptors and protein structures. Recent work
on photosynthesis has provided evidence that
quantum correlations play an essential role
in this basic fundamental process of the plant
kingdom. Even so, classical physics can often
provide good approximations to results otherwise
obtained by quantum physics, typically in
circumstances with large numbers of particles
or large quantum numbers.
Examples
Free particle
For example, consider a free particle. In
quantum mechanics, there is wave–particle
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 complete precision. However,
one can measure the position of a moving free
particle, creating an eigenstate of position
with a wavefunction that is very large at
a particular position x, and zero everywhere
else. If one performs a position measurement
on such a wavefunction, the resultant x will
be obtained with 100% probability. This is
called an eigenstate of position—or, stated
in mathematical terms, a generalized position
eigenstate. 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.
Step potential
The potential in this case is given by:
The solutions are superpositions of left-
and right-moving waves:
where the wave vectors are related to the
energy via
, and
with coefficients A and B determined from
the boundary conditions and by imposing a
continuous derivative on the solution.
Each term of the solution can be interpreted
as an incident, reflected, or transmitted
component of the wave, allowing the calculation
of transmission and reflection coefficients.
Notably, in contrast to classical mechanics,
incident particles with energies greater than
the potential step are partially reflected.
Rectangular potential barrier
This is a model for the quantum tunneling
effect which plays an important role in the
performance of modern technologies such as
flash memory and scanning tunneling microscopy.
Quantum tunneling is central to physical phenomena
involved in superlattices.
Particle in a box
The particle in a one-dimensional potential
energy box is the most mathematically simple
example where restraints lead to the quantization
of energy levels. The box is defined as having
zero potential energy everywhere inside a
certain region, and infinite potential energy
everywhere outside that region. For the one-dimensional
case in the direction, the time-independent
Schrödinger equation may be written
With the differential operator defined by
the previous equation is evocative of the
classic kinetic energy analogue,
with state in this case having energy coincident
with the kinetic energy of the particle.
The general solutions of the Schrödinger
equation for the particle in a box are
or, from Euler's formula,
The infinite potential walls of the box determine
the values of C, D, and k at x = 0 and x = L
where ψ must be zero. Thus, at x = 0,
and D = 0. At x = L,
in which C cannot be zero as this would conflict
with the Born interpretation. Therefore, since
sin(kL) = 0, kL must be an integer multiple
of π,
The quantization of energy levels follows
from this constraint on k, since
Finite potential well
A finite potential well is the generalization
of the infinite potential well problem to
potential wells having finite depth.
The finite potential well problem is mathematically
more complicated than the infinite particle-in-a-box
problem as the wavefunction is not pinned
to zero at the walls of the well. Instead,
the wavefunction must satisfy more complicated
mathematical boundary conditions as it is
nonzero in regions outside the well.
Harmonic oscillator
As in the classical case, the potential for
the quantum harmonic oscillator is given by
This problem can either be treated by directly
solving the Schrödinger, which is not trivial,
or by using the more elegant "ladder method"
first proposed by Paul Dirac. The eigenstates
are given by
where Hn are the Hermite polynomials,
and the corresponding energy levels are
.
This is another example illustrating the quantization
of energy for bound states.
See also
Angular momentum diagrams
EPR paradox
Fractional quantum mechanics
List of quantum-mechanical systems with analytical
solutions
Macroscopic quantum phenomena
Phase space formulation
Spherical basis
Notes
References
Further reading
Bernstein, Jeremy. Quantum Leaps. Cambridge,
Massachusetts: Belknap Press of Harvard University
Press. ISBN 978-0-674-03541-6. 
Bohm, David. Quantum Theory. Dover Publications.
ISBN 0-486-65969-0. 
Eisberg, Robert; Resnick, Robert. Quantum
Physics of Atoms, Molecules, Solids, Nuclei,
and Particles. Wiley. ISBN 0-471-87373-X. 
Liboff, Richard L.. Introductory Quantum Mechanics.
Addison-Wesley. ISBN 0-8053-8714-5. 
Merzbacher, Eugen. Quantum Mechanics. Wiley,
John & Sons, Inc. ISBN 0-471-88702-1. 
Sakurai, J. J.. Modern Quantum Mechanics.
Addison Wesley. ISBN 0-201-53929-2. 
Shankar, R.. Principles of Quantum Mechanics.
Springer. ISBN 0-306-44790-8. 
External links
3D animations, applications and research for
basic quantum effects)
Quantum Cook Book by R. Shankar, Open Yale
PHYS 201 material
The Modern Revolution in Physics - an online
textbook.
J. O'Connor and E. F. Robertson: A history
of quantum mechanics.
Introduction to Quantum Theory at Quantiki.
Quantum Physics Made Relatively Simple: three
video lectures by Hans Bethe
H is for h-bar.
Quantum Mechanics Books Collection: Collection
of free books
Course material
Doron Cohen: Lecture notes in Quantum Mechanics.
MIT OpenCourseWare: Chemistry.
MIT OpenCourseWare: Physics. See 8.04
Stanford Continuing Education PHY 25: Quantum
Mechanics by Leonard Susskind, see course
description Fall 2007
5½ Examples in Quantum Mechanics
Imperial College Quantum Mechanics Course.
Spark Notes - Quantum Physics.
Quantum Physics Online : interactive introduction
to quantum mechanics.
Experiments to the foundations of quantum
physics with single photons.
AQME : Advancing Quantum Mechanics for Engineers —
by T.Barzso, D.Vasileska and G.Klimeck online
learning resource with simulation tools on
nanohub
Quantum Mechanics by Martin Plenio
Quantum Mechanics by Richard Fitzpatrick
Online course on Quantum Transport
FAQs
Many-worlds or relative-state interpretation.
Measurement in Quantum mechanics.
Media
PHYS 201: Fundamentals of Physics II by Ramamurti
Shankar, Open Yale Course
Lectures on Quantum Mechanics by Leonard Susskind
Everything you wanted to know about the quantum
world — archive of articles from New Scientist.
Quantum Physics Research from Science Daily
Overbye, Dennis. "Quantum Trickery: Testing
Einstein's Strangest Theory". The New York
Times. Retrieved April 12, 2010. 
Audio: Astronomy Cast Quantum Mechanics —
June 2009. Fraser Cain interviews Pamela L.
Gay.
Philosophy
"Quantum Mechanics" entry by Jenann Ismael
in the Stanford Encyclopedia of Philosophy
"Measurement in Quantum Theory" entry by Henry
Krips in the Stanford Encyclopedia of Philosophy
