Quantum mechanics (QM; also known as quantum
physics, quantum theory, the wave mechanical
model, or matrix mechanics), including quantum
field theory, is a fundamental theory in physics
which describes nature at the smallest scales
of energy levels of atoms and subatomic particles.Classical
physics, the physics existing before quantum
mechanics, describes nature at ordinary (macroscopic)
scale. Most theories in classical physics
can be derived from quantum mechanics as an
approximation valid at large (macroscopic)
scale.
Quantum mechanics differs from classical physics
in that energy, momentum, angular momentum
and other quantities of a bound system are
restricted to discrete values (quantization);
objects have characteristics of both particles
and waves (wave-particle duality); and there
are limits to the precision with which quantities
can be measured (uncertainty principle).Quantum
mechanics gradually arose from theories to
explain observations which could not be reconciled
with classical physics, such as Max Planck's
solution in 1900 to the black-body radiation
problem, and from the correspondence between
energy and frequency in Albert Einstein's
1905 paper which explained the photoelectric
effect. Early quantum theory was profoundly
re-conceived in the mid-1920s by Erwin Schrödinger,
Werner Heisenberg, Max Born and others. The
modern theory is formulated in various specially
developed mathematical formalisms. In one
of them, a mathematical function, the wave
function, provides information about the probability
amplitude of position, momentum, and other
physical properties of a particle.
Important applications of quantum theory include
quantum chemistry, quantum optics, quantum
computing, superconducting magnets, light-emitting
diodes, and the laser, the transistor and
semiconductors such as the microprocessor,
medical and research imaging such as magnetic
resonance imaging and electron microscopy.
Explanations for many biological and physical
phenomena are rooted in the nature of the
chemical bond, most notably the macro-molecule
DNA.
== 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 titled
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, Michael Faraday discovered cathode
rays. 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 packets) 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, Planck
corrected this model using Boltzmann's statistical
interpretation of thermodynamics and proposed
what is now called Planck's law, which led
to the development of quantum mechanics.
Following Max Planck's solution in 1900 to
the black-body radiation problem (reported
1859), Albert Einstein offered a quantum-based
theory to explain the photoelectric effect
(1905, reported 1887). Around 1900–1910,
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.
Among the first to study quantum phenomena
in nature were Arthur Compton, C. V. Raman,
and Pieter Zeeman, each of whom has a quantum
effect named after him. Robert Andrews Millikan
studied the photoelectric effect experimentally,
and Albert Einstein developed a theory for
it. At the same time, Ernest Rutherford experimentally
discovered the nuclear model of the atom,
for which 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):
E
=
h
ν
{\displaystyle E=h\nu \ }
,
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 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. He
won the 1921 Nobel Prize in Physics for this
work.
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.
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. The Copenhagen
interpretation of Niels Bohr became widely
accepted.
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). In
1926 Erwin Schrödinger suggested a partial
differential equation for the wave functions
of particles like electrons. And when effectively
restricted to a finite region, this equation
allowed only certain modes, corresponding
to discrete quantum states—whose properties
turned out to be exactly the same as implied
by matrix mechanics. 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.It was found that subatomic
particles and electromagnetic waves are neither
simply particle nor wave but have certain
properties of each. This originated the concept
of wave–particle duality.By 1930, quantum
mechanics had been further unified and formalized
by the work of David Hilbert, Paul Dirac and
John von Neumann with greater emphasis on
measurement, the statistical nature of our
knowledge of reality, and philosophical speculation
about the 'observer'. It has since permeated
many disciplines, including quantum chemistry,
quantum electronics, quantum optics, and quantum
information science. Its speculative modern
developments include string theory and quantum
gravity theories. It also provides a useful
framework for many features of the modern
periodic table of elements, and describes
the behaviors of atoms during chemical bonding
and the flow of electrons in computer semiconductors,
and therefore plays a crucial role in many
modern technologies.While quantum mechanics
was constructed to describe the world of the
very small, it is also needed to explain some
macroscopic phenomena such as superconductors,
and superfluids.The word quantum derives from
the Latin, meaning "how great" or "how much".
In quantum mechanics, it refers to a discrete
unit assigned 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 subatomic 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 were solely described by classical mechanics,
electrons would not orbit the nucleus, since
orbiting electrons emit radiation (due to
circular motion) and would quickly 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 chemical
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
quantum entanglement
principle of uncertainty
wave–particle dualityHowever, later, in
October 2018, physicists reported that quantum
behavior can be explained with classical physics
for a single particle, but not for multiple
particles as in quantum entanglement and related
nonlocality phenomena.
== 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 symbolized as 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, to arbitrary
precision. 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 density,
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 "wave function collapse"
(see, for example, the relative state interpretation).
The basic idea is that when a quantum system
interacts with a measuring apparatus, their
respective wave functions 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 wave function will instantaneously be
an eigenstate (or "generalized" eigenstate)
of that observable. This process is known
as wave function 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 wave function collapsing into each
of the possible eigenstates.
For example, the free particle in the previous
example will usually have a wave function
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 wave function at an initial
time—it makes a definite prediction of what
the wave function will be at any later time.During
a measurement, on the other hand, the change
of the initial wave function into another,
later wave function 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 wave functions 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.
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
wave function 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 dihydrogen cation, 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
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 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 (crucially, that the space
has an inner product) and that observables
of that system are Hermitian operators acting
on vectors in 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 (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
−
e
2
/
(
4
π
ϵ
0
r
)
{\displaystyle \textstyle -e^{2}/(4\pi \epsilon
_{_{0}}r)}
Coulomb potential. This "semi-classical" approach
fails if quantum fluctuations in the electromagnetic
field play an important role, such as in the
emission of photons by charged particles.
Quantum field theories for the strong nuclear
force and the weak nuclear force have 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 as
illustrated by the Einstein–Podolsky–Rosen
(EPR) paradox — an attack on a certain philosophical
interpretation of 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 (i.e. approaching absolute
zero) 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 (consisting 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, often considered "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.
=== Copenhagen interpretation of quantum versus
classical kinematics ===
A big difference between classical and quantum
mechanics is that they use very different
kinematic descriptions.In Niels Bohr's mature
view, quantum mechanical phenomena are required
to be experiments, with complete descriptions
of all the devices for the system, preparative,
intermediary, and finally measuring. The descriptions
are in macroscopic terms, expressed in ordinary
language, supplemented with the concepts of
classical mechanics. The initial condition
and the final condition of the system are
respectively described by values in a configuration
space, for example a position space, or some
equivalent space such as a momentum space.
Quantum mechanics does not admit a completely
precise description, in terms of both position
and momentum, of an initial condition or "state"
(in the classical sense of the word) that
would support a precisely deterministic and
causal prediction of a final condition. In
this sense, advocated by Bohr in his mature
writings, a quantum phenomenon is a process,
a passage from initial to final condition,
not an instantaneous "state" in the classical
sense of that word. Thus there are two kinds
of processes in quantum mechanics: stationary
and transitional. For a stationary process,
the initial and final condition are the same.
For a transition, they are different. Obviously
by definition, if only the initial condition
is given, the process is not determined. Given
its initial condition, prediction of its final
condition is possible, causally but only probabilistically,
because the Schrödinger equation is deterministic
for wave function evolution, but the wave
function describes the system only probabilistically.For
many experiments, it is possible to think
of the initial and final conditions of the
system as being a particle. In some cases
it appears that there are potentially several
spatially distinct pathways or trajectories
by which a particle might pass from initial
to final condition. It is an important feature
of the quantum kinematic description that
it does not permit a unique definite statement
of which of those pathways is actually followed.
Only the initial and final conditions are
definite, and, as stated in the foregoing
paragraph, they are defined only as precisely
as allowed by the configuration space description
or its equivalent. In every case for which
a quantum kinematic description is needed,
there is always a compelling reason for this
restriction of kinematic precision. An example
of such a reason is that for a particle to
be experimentally found in a definite position,
it must be held motionless; for it to be experimentally
found to have a definite momentum, it must
have free motion; these two are logically
incompatible.Classical kinematics does not
primarily demand experimental description
of its phenomena. It allows completely precise
description of an instantaneous state by a
value in phase space, the Cartesian product
of configuration and momentum spaces. This
description simply assumes or imagines a state
as a physically existing entity without concern
about its experimental measurability. Such
a description of an initial condition, together
with Newton's laws of motion, allows a precise
deterministic and causal prediction of a final
condition, with a definite trajectory of passage.
Hamiltonian dynamics can be used for this.
Classical kinematics also allows the description
of a process analogous to the initial and
final condition description used by quantum
mechanics. Lagrangian mechanics applies to
this. For processes that need account to be
taken of actions of a small number of Planck
constants, classical kinematics is not adequate;
quantum mechanics is needed.
=== General 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.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 physical 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 in competition with general relativity,
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"
(or infinitely curved) and not readily amenable
to measurement or probing.
Another popular theory is Loop quantum gravity
(LQG), a theory first proposed by Carlo Rovelli
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.
== 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 Niels Bohr and Werner Heisenberg—remains
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 conjugate nature of evidence obtained
under different experimental situations.
Albert Einstein, himself one of the founders
of quantum theory, did not accept some of
the more philosophical or metaphysical interpretations
of quantum mechanics, such as rejection of
determinism and of causality. He is famously
quoted as saying, in response to this aspect,
"God does not play with dice". He rejected
the concept that the state of a physical system
depends on the experimental arrangement for
its measurement. He held that a state of nature
occurs in its own right, regardless of whether
or how it might be observed. In that view,
he is supported by the currently accepted
definition of a quantum state, which remains
invariant under arbitrary choice of configuration
space for its representation, that is to say,
manner of observation. He also held that underlying
quantum mechanics there should be a theory
that thoroughly and directly expresses the
rule against action at a distance; in other
words, he insisted on the principle of locality.
He considered, but rejected on theoretical
grounds, a particular proposal for hidden
variables to obviate the indeterminism or
acausality of quantum mechanical measurement.
He considered that quantum mechanics was a
currently valid but not a permanently definitive
theory for quantum phenomena. He thought its
future replacement would require profound
conceptual advances, and would not come quickly
or easily. The Bohr-Einstein debates provide
a vibrant critique of the Copenhagen Interpretation
from an epistemological point of view. In
arguing for his views, he produced a series
of objections, 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 theories that rely on
added hidden variables. Experiments have been
performed confirming the accuracy of quantum
mechanics, thereby demonstrating that quantum
mechanics cannot be improved upon by addition
of hidden variables. Alain Aspect's initial
experiments in 1982, and many subsequent experiments
since, have definitively verified quantum
entanglement. By the early 1980s, experiments
had shown that such inequalities were indeed
violated in practice—so that there were
in fact correlations of the kind suggested
by quantum mechanics. At first these just
seemed like isolated esoteric effects, but
by the mid-1990s, they were being codified
in the field of quantum information theory,
and led to constructions with names like quantum
cryptography and quantum teleportation.Entanglement,
as demonstrated in Bell-type experiments,
does not, however, violate causality, since
no transfer of information happens. Quantum
entanglement forms the basis of quantum cryptography,
which is proposed for use in high-security
commercial applications in banking and government.
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 only observe
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 (including
the physicist's memory). In light of these
Bell tests, Cramer (1986) formulated his transactional
interpretation which is unique in providing
a physical explanation for the Born rule.
Relational quantum mechanics appeared in the
late 1990s as the modern derivative of the
Copenhagen Interpretation.
== Applications ==
Quantum mechanics has had enormous success
in explaining many of the features of our
universe. Quantum mechanics is often the only
theory 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 are
joined by covalent bond to form molecules.
The application of quantum mechanics to chemistry
is known as quantum 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.
In many aspects modern technology operates
at a scale where quantum effects are significant.
=== Electronics ===
Many modern electronic devices are designed
using quantum mechanics. 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, computer and
telecommunication devices. Another application
is for making laser diode and light emitting
diode which are a high-efficiency source of
light.
Many electronic devices operate under effect
of quantum tunneling. It even exists in the
simple light switch. The switch would not
work if electrons could not quantum tunnel
through the layer of oxidation on the metal
contact surfaces. Flash memory chips found
in USB drives use quantum tunneling to erase
their memory cells. Some negative differential
resistance devices also utilize quantum tunneling
effect, such as resonant tunneling diode.
Unlike classical diodes, its current is carried
by resonant tunneling through two or more
potential barriers (see right figure). Its
negative resistance behavior can only be understood
with quantum mechanics: As the confined state
moves close to Fermi level, tunnel current
increases. As it moves away, current decreases.
Quantum mechanics is necessary to understanding
and designing such electronic devices.
=== Cryptography ===
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.
An inherent advantage yielded by quantum cryptography
when compared to classical cryptography is
the detection of passive eavesdropping. This
is a natural result of the behavior of quantum
bits; due to the observer effect, if a bit
in a superposition state were to be observed,
the superposition state would collapse into
an eigenstate. Because the intended recipient
was expecting to receive the bit in a superposition
state, the intended recipient would know there
was an attack, because the bit's state would
no longer be in a superposition.
=== Quantum computing ===
Another goal is the development of quantum
computers, which are expected to perform certain
computational tasks exponentially faster than
classical computers. Instead of using classical
bits, quantum computers use qubits, which
can be in superpositions of states. Quantum
programmers are able to manipulate the superposition
of qubits in order to solve problems that
classical computing cannot do effectively,
such as searching unsorted databases or integer
factorization. IBM claims that the advent
of quantum computing may progress the fields
of medicine, logistics, financial services,
artificial intelligence and cloud security.Another
active research topic is quantum teleportation,
which deals with techniques to transmit quantum
information over arbitrary distances.
=== Macroscale quantum effects ===
While quantum mechanics primarily applies
to the smaller 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.
So is the closely related phenomenon of superconductivity,
the frictionless flow of an electron gas in
a conducting material (an electric current)
at sufficiently low temperatures. The fractional
quantum Hall effect is a topological ordered
state which corresponds to patterns of long-range
quantum entanglement. States with different
topological orders (or different patterns
of long range entanglements) cannot change
into each other without a phase transition.
=== Quantum theory ===
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 fundamental process of plants
and many other organisms. 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. Since
classical formulas are much simpler and easier
to compute than quantum formulas, classical
approximations are used and preferred when
the system is large enough to render the effects
of quantum mechanics insignificant.
== Examples ==
=== 
Free particle ===
For example, consider a free particle. In
quantum mechanics, a free matter is described
by a wave function. The particle properties
of the matter become apparent when we measure
its position and velocity. The wave properties
of the matter become apparent when we measure
its wave properties like interference. The
wave–particle duality feature is incorporated
in the relations of coordinates and operators
in the formulation of quantum mechanics. Since
the matter is free (not subject to any interactions),
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 wave function 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 wave function,
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.
=== 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 therefore infinite potential
energy everywhere outside that region. For
the one-dimensional case in the
x
{\displaystyle x}
direction, the time-independent Schrödinger
equation may be written
−
ℏ
2
2
m
d
2
ψ
d
x
2
=
E
ψ
.
{\displaystyle -{\frac {\hbar ^{2}}{2m}}{\frac
{d^{2}\psi }{dx^{2}}}=E\psi .}
With the differential operator defined by
p
^
x
=
−
i
ℏ
d
d
x
{\displaystyle {\hat {p}}_{x}=-i\hbar {\frac
{d}{dx}}}
the previous equation is evocative of the
classic kinetic energy analogue,
1
2
m
p
^
x
2
=
E
,
{\displaystyle {\frac {1}{2m}}{\hat {p}}_{x}^{2}=E,}
with state
ψ
{\displaystyle \psi }
in this case having energy
E
{\displaystyle E}
coincident with the kinetic energy of the
particle.
The general solutions of the Schrödinger
equation for the particle in a box are
ψ
(
x
)
=
A
e
i
k
x
+
B
e
−
i
k
x
E
=
ℏ
2
k
2
2
m
{\displaystyle \psi (x)=Ae^{ikx}+Be^{-ikx}\qquad
\qquad E={\frac {\hbar ^{2}k^{2}}{2m}}}
or, from Euler's formula,
ψ
(
x
)
=
C
sin
⁡
k
x
+
D
cos
⁡
k
x
.
{\displaystyle \psi (x)=C\sin kx+D\cos kx.\!}
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,
ψ
(
0
)
=
0
=
C
sin
⁡
0
+
D
cos
⁡
0
=
D
{\displaystyle \psi (0)=0=C\sin 0+D\cos 0=D\!}
and D = 0. At x = L,
ψ
(
L
)
=
0
=
C
sin
⁡
k
L
.
{\displaystyle \psi (L)=0=C\sin kL.\!}
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,
k
=
n
π
L
n
=
1
,
2
,
3
,
…
.
{\displaystyle k={\frac {n\pi }{L}}\qquad
\qquad n=1,2,3,\ldots .}
The 
quantization of energy levels follows from
this constraint on k, since
E
=
ℏ
2
π
2
n
2
2
m
L
2
=
n
2
h
2
8
m
L
2
.
{\displaystyle E={\frac {\hbar ^{2}\pi ^{2}n^{2}}{2mL^{2}}}={\frac
{n^{2}h^{2}}{8mL^{2}}}.}
The ground state energy of the particles is
E1 for n=1.
Energy of particle in the nth state is En
=n2E1, n=2,3,4,.....
Particle in a box with boundary condition
V(x)=0 -a/2<x<+a/2
In this condition the general solution will
be same, there will a little change to the
final result, since the boundary conditions
are changed
ψ
(
x
)
=
C
sin
⁡
k
x
+
D
cos
⁡
k
x
.
{\displaystyle \psi (x)=C\sin kx+D\cos kx.\!}
At x=0, the wave function is not actually
zero at all value of n.
Clearly, from the wave function variation
graph we have,
At n=1,3,4,...... the wave function follows
a cosine curve with x=0 as origin
At n=2,4,6,...... the wave function follows
a sine curve with x=0 as origin
From this observation we can conclude that
the wave function is alternatively sine and
cosine.
So in this case the resultant wave equation
is
ψn(x) = Acos(knx) n=1,3,5,.............
= Bsin(knx) n=2,4,6,.............
=== 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 wave function is not pinned
to zero at the walls of the well. Instead,
the wave function must satisfy more complicated
mathematical boundary conditions as it is
nonzero in regions outside the well.
=== 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.
=== Harmonic oscillator ===
As in the classical case, the potential for
the quantum harmonic oscillator is given by
V
(
x
)
=
1
2
m
ω
2
x
2
.
{\displaystyle V(x)={\frac {1}{2}}m\omega
^{2}x^{2}.}
This problem can either be treated by directly
solving the Schrödinger equation, which is
not trivial, or by using the more elegant
"ladder method" first proposed by Paul Dirac.
The eigenstates are given by
ψ
n
(
x
)
=
1
2
n
n
!
⋅
(
m
ω
π
ℏ
)
1
/
4
⋅
e
−
m
ω
x
2
2
ℏ
⋅
H
n
(
m
ω
ℏ
x
)
,
{\displaystyle \psi _{n}(x)={\sqrt {\frac
{1}{2^{n}\,n!}}}\cdot \left({\frac {m\omega
}{\pi \hbar }}\right)^{1/4}\cdot e^{-{\frac
{m\omega x^{2}}{2\hbar }}}\cdot H_{n}\left({\sqrt
{\frac {m\omega }{\hbar }}}x\right),\qquad
}
n
=
0
,
1
,
2
,
…
.
{\displaystyle n=0,1,2,\ldots .}
where Hn are the Hermite polynomials
H
n
(
x
)
=
(
−
1
)
n
e
x
2
d
n
d
x
n
(
e
−
x
2
)
,
{\displaystyle H_{n}(x)=(-1)^{n}e^{x^{2}}{\frac
{d^{n}}{dx^{n}}}\left(e^{-x^{2}}\right),}
and the corresponding energy levels are
E
n
=
ℏ
ω
(
n
+
1
2
)
.
{\displaystyle E_{n}=\hbar \omega \left(n+{1
\over 2}\right).}
This is another example illustrating the quantification
of energy for bound states.
=== Step potential ===
The potential in this case is given by:
V
(
x
)
=
{
0
,
x
<
0
,
V
0
,
x
≥
0.
{\displaystyle V(x)={\begin{cases}0,&x<0,\\V_{0},&x\geq
0.\end{cases}}}
The solutions are superpositions of left-
and right-moving waves:
ψ
1
(
x
)
=
1
k
1
(
A
→
e
i
k
1
x
+
A
←
e
−
i
k
1
x
)
x
<
0
{\displaystyle \psi _{1}(x)={\frac {1}{\sqrt
{k_{1}}}}\left(A_{\rightarrow }e^{ik_{1}x}+A_{\leftarrow
}e^{-ik_{1}x}\right)\qquad x<0}
and
ψ
2
(
x
)
=
1
k
2
(
B
→
e
i
k
2
x
+
B
←
e
−
i
k
2
x
)
x
>
0
{\displaystyle \psi _{2}(x)={\frac {1}{\sqrt
{k_{2}}}}\left(B_{\rightarrow }e^{ik_{2}x}+B_{\leftarrow
}e^{-ik_{2}x}\right)\qquad x>0}
,with coefficients A and B determined from
the boundary conditions and by imposing a
continuous derivative on the solution, and
where the wave vectors are related to the
energy via
k
1
=
2
m
E
/
ℏ
2
{\displaystyle k_{1}={\sqrt {2mE/\hbar ^{2}}}}
and
k
2
=
2
m
(
E
−
V
0
)
/
ℏ
2
{\displaystyle k_{2}={\sqrt {2m(E-V_{0})/\hbar
^{2}}}}
.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.
== See also ==
== Notes
