Quantum mechanics (QM – also known as quantum
physics, or quantum theory) is a
branch of physics which deals with physical
phenomena at microscopic 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. It is the non-relativistic limit of
quantum field theory (QFT), a theory
that was developed later that combined quantum
mechanics with relativity.
In advanced topics of quantum mechanics, some
of these behaviors are macroscopic
(see macroscopic quantum phenomena) and emerge
at only extreme (i.e., very low
or very high) energies or temperatures (such
as in the use of superconducting
magnets). The name quantum mechanics derives
from the observation that some
physical quantities can change only in discrete
amounts (Latin quanta), and not
in a continuous (cf. analog) way. For example,
the angular momentum of an
electron bound to an atom or molecule is 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 known as 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 the bra-ket
notation, which requires an understanding
of complex numbers and linear
functionals. The wavefunction treats the object
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, the ground state
in a quantum mechanical model is a non-zero
energy state that is the lowest permitted
energy state of a system, as opposed
to a more "traditional" system that is thought
of as simply being at rest, with
zero kinetic energy. Instead of a traditional
static, unchanging zero 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. At around the same time,
the atomic theory and the
corpuscular theory of light (as updated by
Einstein) 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, who created matrix
mechanics; Louis de Broglie and
Erwin Schrödinger (Wave Mechanics); and Wolfgang
Pauli and Satyendra Nath Bose (statistics
of subatomic particles). 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 branched out into
almost every aspect of 20th century physics
and other disciplines, such as
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.
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" (or "energy elements") 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:
Planck is considered the father of the Quantum
Theory
where h is Planck's constant. Planck (cautiously)
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 sizeable 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 1927 Solvay Conference in Brussels.
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 (1926). 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 (later called the photon) with a
discrete quantum of energy that was
dependent on its frequency. As a matter of
fact, Einstein was able to use the
photon theory of light to explain the photoelectric
effect, for which he won the
Nobel Prize in 1921. 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. Thus
coined the term 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 larger
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 (see
Figure 1). 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 is the underlying
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 classical
mechanics alone governed the
workings of an atom, electrons could not really
"orbit" the nucleus. Since
bodies in circular motion are accelerating,
electrons must emit radiation,
losing energy and eventually colliding with
the nucleus in the process. This
clearly contradicts the existence of stable
atoms. However, in the natural world,
electrons normally remain in an uncertain,
non-deterministic, "smeared",
probabilistic, wave–particle wavefunction
orbital path around (or through) 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:
The quantization of certain physical properties
Wave–particle duality
The uncertainty principle
Quantum entanglement.
Mathematical formulations
See also: Quantum logic
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 (called "state 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 (the
phase factor). 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 (precisely: by a self-adjoint) linear
operator acting on the state
space. Each eigenstate of an observable corresponds
to an eigenvector of the
operator, and the associated eigenvalue corresponds
to the value of the
observable in that eigenstate. If the operator's
spectrum is discrete, the
observable can 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 a certain probability) 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"
(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.
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 ("eigen" can be
translated from German as meaning "inherent"
or "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's position and momentum (since they
are conjugate pairs) or its energy
and time (since they too are conjugate pairs);
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 (eigenstates). Usually,
a system will not be in an eigenstate of the
observable (particle) 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 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 (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 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 (the operator corresponding
to the total energy of the
system) 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 (i.e.,
random). 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
(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 with time. This
also has the effect of turning a position
eigenstate (which can be thought of as
an infinitely sharp wave packet) into a broadened
wave packet that no longer
represents a (definite, certain) position
eigenstate.
Fig. 1: Probability densities corresponding
to the wavefunctions of an electron
in a hydrogen atom possessing definite energy
levels (increasing from the top of
the image to the bottom: n = 1, 2, 3, ...) and
angular momenta (increasing
across from left to right: s, p, d, ...). Brighter
areas correspond to higher
probability density in a position measurement.
Such wavefunctions are directly
comparable to Chladni's figures of acoustic
modes of vibration in classical
physics, and are modes of oscillation as well,
possessing a sharp energy and,
thus, a definite frequency. The angular momentum
and energy are quantized, and
take only discrete values like those shown
(as is the case for resonant
frequencies in acoustics)
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
(Fig. 1) (note, however, that
only the lowest angular momentum states, labeled
s, are spherically symmetric).
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 (for one example) the
addition of a weak potential energy. Another
method is the "semi-classical
equation of motion" approach, which applies
to systems for which quantum
mechanics produces only weak (small) 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 the late 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).
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 (e.g.,
the position of a particle) or discrete (e.g.,
the energy of an electron bound
to a hydrogen atom). 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 histories
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.
List of unsolved problems in physics In the
correspondence limit of quantum
mechanics: Is there a preferred interpretation
of quantum mechanics? How does
the quantum description of reality, which
includes elements such as the "superposition
of states" and "wavefunction collapse", give
rise to the reality we perceive?
When quantum mechanics was originally formulated,
it was applied to models whose
correspondence limit was non-relativistic
classical mechanics. For instance, the
well-known model of the quantum harmonic oscillator
uses an explicitly non-relativistic
expression for the kinetic energy of the oscillator,
and is thus a quantum
version of the classical harmonic oscillator.
Early attempts to merge quantum mechanics
with special relativity involved the
replacement of the Schrödinger equation with
a covariant equation such as the
Klein-Gordon equation or the Dirac equation.
While these theories were
successful in explaining many experimental
results, they had certain
unsatisfactory qualities stemming from their
neglect of the relativistic
creation and annihilation of particles. A
fully relativistic quantum theory
required the development of quantum field
theory, which applies quantization to
a field (rather than a fixed set of particles).
The first complete quantum field
theory, quantum electrodynamics, provides
a fully quantum description of the
electromagnetic interaction. The full apparatus
of quantum field theory is often
unnecessary for describing electrodynamic
systems. A simpler approach, one 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 (known as electroweak
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 (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.
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 (or a statistical quantum mechanics
of a large collection of particles).
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, and is illustrated by the Einstein-Podolsky-Rosen
paradox, Einstein's
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 that
are characteristic of quantum systems. Quantum
coherence is not typically
evident at macroscopic scales — although
an exception to this rule can occur at
extremely low temperatures (i.e. approaching
absolute zero), when quantum
behavior can manifest itself on more macroscopic
scales. 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 (which
consists of atoms and molecules which would
quickly collapse under electric
forces alone), 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
Newtonian physics remain accurate in predicting
the behavior of the vast
majority of "large" objects (on the order
of the size of large molecules or
bigger) 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 (at least with regard to their
primary claims), 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 (see Bell inequality) 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" (TOE). 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" (2002).
Attempts at a unified field theory
The quest to unify the fundamental forces
through quantum mechanics is still
ongoing. Quantum electrodynamics (or "quantum
electromagnetism"), which is
currently (in the perturbative regime at least)
the most accurately tested
physical theory, [unreliable source](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
the leading authorities continuing the search
for a coherent TOE is Edward
Witten, a theoretical physicist who formulated
the groundbreaking 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" (or
infinitely curved) and not readily amenable
to measurement or probing.
Another popular theory is Loop quantum gravity
(LQG), 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 (cf. Planck scale
energy). 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 (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 non-deterministic behavior
governed by probabilities, because we can
observe only the universe (i.e., the
consistent state contribution to the aforementioned
superposition) 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 (to include the physicist's
memory!); in light of these Bell tests, Cramer
(1986) 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 (electrons, protons, neutrons, photons,
and others). Quantum mechanics
has strongly influenced string theories, candidates
for a Theory of Everything (see
reductionism).
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 working mechanism of a resonant tunneling
diode device, based on the
phenomenon of quantum tunneling through potential
barriers
A great deal of modern technological inventions
operate 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 (MRI).
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
(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 resultant x will
be obtained with 100% probability (i.e., with
full certainty, or complete
precision). This is called an eigenstate of
position—or, stated in mathematical
terms, 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.
3D confined electron wave functions for each
eigenstate in a Quantum Dot. Here,
rectangular and triangular-shaped quantum
dots are shown. Energy states in
rectangular dots are more ‘s-type’ and
‘p-type’. However, in a triangular dot,
the wave functions are mixed due to confinement
symmetry.
Step potential
Scattering at a finite potential step of height
V0, shown in green. The
amplitudes and direction of left- and right-moving
waves are indicated. Yellow
is the incident wave, blue are reflected and
transmitted waves, red does not
occur. E > V0 for this figure.
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
and the coefficients A and B are 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. In contrast to classical
mechanics, incident particles
with energies higher than the size of the
potential step are still partially
reflected.
Rectangular potential barrier
Rectangular potential barrier
This is a model for the quantum tunneling
effect, which has important
applications to modern devices such as flash
memory and the scanning tunneling
microscope.
Particle in a box
1-dimensional potential energy box (or infinite
potential well)
The particle in a one-dimensional potential
energy box is the most 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 can be written
as:
Writing the differential operator
the previous equation can be seen to be evocative
of the classic kinetic energy
analogue
with as the energy for the state , which in
this case coincides 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 presence of the walls of the box determines
the values of C, D, and k. At
each wall (x = 0 and x = L), ψ = 0. Thus
when x = 0,
and so D = 0. When x = L,
C cannot be zero, since this would conflict
with the Born interpretation.
Therefore, sin kL = 0, and so it must be that
kL is an integer multiple of π.
And additionally,
The quantization of energy levels follows
from this constraint on k, since
Finite potential well
Finite potential well
This is the generalization of the infinite
potential well problem to potential
wells of finite depth.
Harmonic oscillator
Quantum harmonic oscillator
Some trajectories of a harmonic oscillator
(i.e. a ball attached to a spring) in
classical mechanics (A-B) and quantum mechanics
(C-H). In quantum mechanics, the
position of the ball is represented by a wave
(called the wavefunction), with
the real part shown in blue and the imaginary
part shown in red. Some of the
trajectories (such as C,D,E,and F) are standing
waves (or "stationary states").
Each standing-wave frequency is proportional
to a possible energy level of the
oscillator. This "energy quantization" does
not occur in classical physics,
where the oscillator can have any energy.
As in the classical case, the potential for
the quantum harmonic oscillator is
given by:
This problem can be solved either by solving
the Schrödinger equation directly,
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
