mechanics (QM; also known as quantum physics or quantum
theory), 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.
(the physics existing before quantum mechanics) is a
set of fundamental theories which describes nature at ordinary
(macroscopic) scale. Most theories in classical physics can be derived
from quantum mechanics as an approximation valid at large
mechanics differs from classical
physics in that: energy, momentum and other quantities of a system may
be 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
known (uncertainty principle).[note 1]
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,
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
3 Mathematically equivalent formulations of quantum mechanics
4 Interactions with other scientific theories
Quantum mechanics and classical physics
Copenhagen interpretation of quantum versus classical kinematics
General relativity and quantum mechanics
4.4 Attempts at a unified field theory
6.4 Macroscale quantum effects
7.1 Free particle
Particle in a box
7.3 Finite potential well
7.4 Rectangular potential barrier
7.5 Harmonic oscillator
7.6 Step potential
8 See also
11 Further reading
12 External links
displaystyle hat H psi _ n (t)rangle =ihbar frac partial
partial t psi _ n (t)rangle
displaystyle frac 1 c ^ 2 frac partial ^ 2 phi _ n
partial t ^ 2 - nabla ^ 2 phi _ n + left( frac mc hbar
right) ^ 2 phi _ n =0
Manifold dynamics: Schrödinger and Klein–Gordon equations
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Main article: History of quantum mechanics
Scientific inquiry into the wave nature of light began in the 17th and
18th centuries, when scientists such as Robert Hooke, Christiaan
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.
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.
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
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
Max 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 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
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. The Copenhagen
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.[citation
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.[better source needed]
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 eventually collide with the nucleus due to
this loss of energy. This framework was unable to explain the
stability of atoms. Instead, electrons remain in an uncertain,
non-deterministic, smeared, probabilistic wave–particle orbital
about the nucleus, defying the traditional assumptions of classical
mechanics and electromagnetism.
Quantum mechanics was initially developed to provide a better
explanation and description of the atom, especially the differences in
the spectra of light emitted by different isotopes of the same
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
principle of uncertainty
Mathematical formulation of quantum mechanics
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
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, 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
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
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
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
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
Probability densities corresponding to the wave functions 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 wave functions 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 wave function surrounding the nucleus (Fig. 1) (note,
however, that only the lowest angular momentum states, labeled s, are
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).
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.
Unsolved problem 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 "wave function collapse", give rise to the reality we
(more unsolved problems in physics)
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
displaystyle scriptstyle -e^ 2 /(4pi 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
Classical mechanics has also been extended into the complex domain,
with complex classical mechanics exhibiting behaviors similar to
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
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
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
Attempts at a unified field theory
Main article: Grand unified theory
The quest to unify the fundamental forces through quantum mechanics is
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
Loop quantum gravity (LQG), a theory first
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.
Main article: Interpretations of quantum mechanics
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
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
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.
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
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.
Relational quantum mechanics
Relational quantum mechanics appeared in the late 1990s as the modern
derivative of the Copenhagen Interpretation.
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
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
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
In many aspects modern technology operates at a scale where quantum
effects are significant.
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.
A working mechanism of a resonant tunneling diode device, based on the
phenomenon of quantum tunneling through potential barriers. (Left:
band diagram; Center: transmission coefficient; Right: current-voltage
characteristics) As shown in the band diagram(left), although there
are two barriers, electrons still tunnel through via the confined
states between two barriers(center), conducting current.
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,
Quantum mechanics is necessary to understanding and
designing such electronic 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.
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
A more distant 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
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
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
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.
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
Planck's constant and p is the momentum of the eigenstate.
Particle in a box
1-dimensional potential energy box (or infinite potential well)
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
direction, the time-independent
Schrödinger equation may be
displaystyle - frac hbar ^ 2 2m frac d^ 2 psi dx^ 2
With the differential operator defined by
displaystyle hat p _ x =-ihbar frac d dx
the previous equation is evocative of the classic kinetic energy
displaystyle frac 1 2m hat p _ x ^ 2 =E,
in this case having energy
coincident with the kinetic energy of the particle.
The general solutions of the
Schrödinger equation for the particle in
a box are
displaystyle psi (x)=Ae^ ikx +Be^ -ikx qquad qquad E= frac hbar
^ 2 k^ 2 2m
or, from Euler's formula,
displaystyle psi (x)=Csin kx+Dcos 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,
displaystyle psi (0)=0=Csin 0+Dcos 0=D!
and D = 0. At x = L,
displaystyle psi (L)=0=Csin 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,
displaystyle k= frac npi L qquad qquad n=1,2,3,ldots .
The quantization of energy levels follows from this constraint on k,
displaystyle E= frac hbar ^ 2 pi ^ 2 n^ 2 2mL^ 2 = frac n^
2 h^ 2 8mL^ 2 .
Finite potential well
Main article: 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
Main article: 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.
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 wave function), 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
displaystyle V(x)= frac 1 2 momega ^ 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
displaystyle psi _ n (x)= sqrt frac 1 2^ n ,n! cdot left(
frac momega pi hbar right)^ 1/4 cdot e^ - frac momega x^ 2
2hbar cdot H_ n left( sqrt frac momega hbar xright),qquad
displaystyle n=0,1,2,ldots .
where Hn are the Hermite polynomials
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
displaystyle E_ n =hbar omega left(n+ 1 over 2 right).
This is another example illustrating the quantification of energy for
Main article: Solution of
Schrödinger equation for a 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:
displaystyle V(x)= begin cases 0,&x<0,\V_ 0 ,&xgeq
The solutions are superpositions of left- and right-moving waves:
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
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
displaystyle k_ 1 = sqrt 2mE/hbar ^ 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.
Angular momentum diagrams (quantum mechanics)
Fractional quantum mechanics
List of quantum-mechanical systems with analytical solutions
Macroscopic quantum phenomena
Phase space formulation
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International Institute of Intellectual Co-operation, Paris, 1939, pp.
11–30, reprinted in Niels Bohr, Collected Works, volume 7 (1933 –
1958) edited by J. Kalckar, Elsevier, Amsterdam,
ISBN 0-444-89892-1, pp. 303–322. "The essential lesson of the
analysis of measurements in quantum theory is thus the emphasis on the
necessity, in the account of the phenomena, of taking the whole
experimental arrangement into consideration, in complete conformity
with the fact that all unambiguous interpretation of the quantum
mechanical formalism involves the fixation of the external conditions,
defining the initial state of the atomic system and the character of
the possible predictions as regards subsequent observable properties
of that system. Any measurement in quantum theory can in fact only
refer either to a fixation of the initial state or to the test of such
predictions, and it is first the combination of both kinds which
constitutes a well-defined phenomenon."
^ Bohr, N. (1948). On the notions of complementarity and causality,
Dialectica 2: 312–319. "As a more appropriate way of expression, one
may advocate limitation of the use of the word phenomenon to refer to
observations obtained under specified circumstances, including an
account of the whole experiment."
^ Ludwig, G. (1987). An Axiomatic Basis for
Quantum Mechanics, volume
Mechanics and Macrosystems, translated by K. Just,
Springer, Berlin, ISBN 978-3-642-71899-1, Chapter XIII, Special
Structures in Preparation and Registration Devices, §1, Measurement
chains, p. 132.
^ a b Heisenberg, W. (1927). Über den anschaulichen Inhalt der
quantentheoretischen Kinematik und Mechanik, Z. Phys. 43: 172–198.
Translation as 'The actual content of quantum theoretical kinematics
and mechanics' here , "But in the rigorous formulation of the law
of causality, — "If we know the present precisely, we can calculate
the future" — it is not the conclusion that is faulty, but the
^ Green, H.S. (1965). Matrix Mechanics, with a foreword by Max Born,
P. Noordhoff Ltd, Groningen. "It is not possible, therefore, to
provide 'initial conditions' for the prediction of the behaviour of
atomic systems, in the way contemplated by classical physics. This is
accepted by quantum theory, not merely as an experimental difficulty,
but as a fundamental law of nature", p. 32.
^ Rosenfeld, L. (1957). Misunderstandings about the foundations of
quantum theory, pp. 41–45 in Observation and Interpretation, edited
by S. Körner, Butterworths, London. "A phenomenon is therefore a
process (endowed with the characteristic quantal wholeness) involving
a definite type of interaction between the system and the apparatus."
^ Dirac, P.A.M. (1973). Development of the physicist's conception of
nature, pp. 1–55 in The Physicist's Conception of Nature, edited by
J. Mehra, D. Reidel, Dordrecht, ISBN 90-277-0345-0, p. 5: "That
led Heisenberg to his really masterful step forward, resulting in the
new quantum mechanics. His idea was to build up a theory entirely in
terms of quantities referring to two states."
^ Born, M. (1927). Physical aspects of quantum mechanics,
354–357, "These probabilities are thus dynamically determined. But
what the system actually does is not determined ..."
^ Messiah, A. (1961).
Quantum Mechanics, volume 1, translated by G.M.
Temmer from the French Mécanique Quantique, North-Holland, Amsterdam,
^ Bohr, N. (1928). "The
Quantum postulate and the recent development
of atomic theory". Nature. 121: 580–590.
^ Heisenberg, W. (1930). The Physical Principles of the Quantum
Theory, translated by C. Eckart and F.C. Hoyt, University of Chicago
^ Goldstein, H. (1950). Classical Mechanics, Addison-Wesley,
^ "There is as yet no logically consistent and complete relativistic
quantum field theory.", p. 4. — V. B. Berestetskii, E. M.
Lifshitz, L P Pitaevskii (1971). J. B. Sykes, J. S. Bell
Quantum Theory 4, part I. Course of
Physics (Landau and Lifshitz) ISBN 0-08-016025-5
^ "Stephen Hawking; Gödel and the end of physics". cam.ac.uk.
Retrieved 11 September 2015.
Space and Time". google.com. Retrieved 11 September
^ Tatsumi Aoyama; Masashi Hayakawa; Toichiro Kinoshita; Makiko Nio
QED Contribution to the
Electron g-2 and an
Improved Value of the Fine Structure Constant". Physical Review
Letters. 109 (11): 111807. arXiv:1205.5368v2 .
^ Parker, B. (1993). Overcoming some of the problems.
^ The Character of Physical Law (1965) Ch. 6; also quoted in The New
Quantum Universe (2003), by Tony Hey and Patrick Walters
^ Weinberg, S. "Collapse of the State Vector", Phys. Rev. A 85, 062116
^ Harrison, Edward (16 March 2000). Cosmology: The Science of the
Universe. Cambridge University Press. p. 239.
^ "Action at a Distance in
Mechanics (Stanford Encyclopedia of
Philosophy)". Plato.stanford.edu. 2007-01-26. Retrieved
^ "Everett's Relative-State Formulation of
Encyclopedia of Philosophy)". Plato.stanford.edu. Retrieved
^ The Transactional Interpretation of
Mechanics by John
Cramer. Reviews of Modern
Physics 58, 647-688, July (1986)
^ See, for example, the
Feynman Lectures on
Physics for some of the
technological applications which use quantum mechanics, e.g.,
transistors (vol III, pp. 14–11 ff), integrated circuits, which are
follow-on technology in solid-state physics (vol II, pp. 8–6), and
lasers (vol III, pp. 9–13).
^ Pauling, Linus; Wilson, Edgar Bright (1985-03-01). Introduction to
Mechanics with Applications to Chemistry.
ISBN 9780486648712. Retrieved 2012-08-18.
^ Schneier, Bruce (1993). Applied
Cryptography (2nd ed.). Wiley.
p. 554. ISBN 0471117099.
^ "Applications of
Quantum Computing". research.ibm.com. Retrieved 28
^ Chen, Xie; Gu, Zheng-Cheng; Wen, Xiao-Gang (2010). "Local unitary
transformation, long-range quantum entanglement, wave function
renormalization, and topological order". Phys. Rev. B. 82: 155138.
arXiv:1004.3835 . Bibcode:2010PhRvB..82o5138C.
^ Anderson, Mark (2009-01-13). "Is
Mechanics Controlling Your
Thoughts? Subatomic Particles". DISCOVER Magazine. Retrieved
Quantum mechanics boosts photosynthesis". physicsworld.com.
^ Davies, P. C. W.; Betts, David S. (1984).
Quantum Mechanics, Second
edition. Chapman and Hall. p. 79. ISBN 0-7487-4446-0. ,
Chapter 6, p. 79
^ Baofu, Peter (2007-12-31). The Future of Complexity: Conceiving a
Better Way to Understand Order and Chaos. ISBN 9789812708991.
^ Derivation of particle in a box, chemistry.tidalswan.com
^ N.B. on precision: If
displaystyle delta x
displaystyle delta p
are the precisions of position and momentum obtained in an individual
displaystyle sigma _ x
displaystyle sigma _ p
their standard deviations in an ensemble of individual measurements
on similarly prepared systems, then "There are, in principle, no
restrictions on the precisions of individual measurements
displaystyle delta x
displaystyle delta p
, but the standard deviations will always satisfy
displaystyle sigma _ x sigma _ p geq hbar /2
The following titles, all by working physicists, attempt to
communicate quantum theory to lay people, using a minimum of technical
Chester, Marvin (1987) Primer of
Quantum Mechanics. John Wiley.
Cox, Brian; Forshaw, Jeff (2011). The
Quantum Universe: Everything
That Can Happen Does Happen:. Allen Lane.
Richard Feynman, 1985. QED: The Strange Theory of
Light and Matter,
Princeton University Press. ISBN 0-691-08388-6. Four elementary
lectures on quantum electrodynamics and quantum field theory, yet
containing many insights for the expert.
Ghirardi, GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald
Malsbary, trans. Princeton Univ. Press. The most technical of the
works cited here. Passages using algebra, trigonometry, and bra–ket
notation can be passed over on a first reading.
N. David Mermin, 1990, "Spooky actions at a distance: mysteries of the
QT" in his Boojums all the way through. Cambridge University Press:
Victor Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and
Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 5-8. Includes
cosmological and philosophical considerations.
Bryce DeWitt, R. Neill Graham, eds., 1973. The Many-Worlds
Quantum Mechanics, Princeton Series in Physics,
Princeton University Press. ISBN 0-691-08131-X
Dirac, P. A. M. (1930). The Principles of
ISBN 0-19-852011-5. The beginning chapters make up a very
clear and comprehensible introduction.
Everett, Hugh (1957). "Relative State Formulation of Quantum
Mechanics". Reviews of Modern Physics. 29: 454–62.
Feynman, Richard P.; Leighton, Robert B.; Sands, Matthew (1965). The
Feynman Lectures on Physics. 1–3. Addison-Wesley.
Griffiths, David J. (2004). Introduction to
ed.). Prentice Hall. ISBN 0-13-111892-7.
OCLC 40251748. A standard undergraduate text.
Max Jammer, 1966. The Conceptual Development of
Hagen Kleinert, 2004. Path Integrals in
Quantum Mechanics, Statistics,
Polymer Physics, and Financial Markets, 3rd ed. Singapore: World
Scientific. Draft of 4th edition.
L.D. Landau, E.M. Lifshitz (1977).
Quantum Mechanics: Non-Relativistic
Theory. Vol. 3 (3rd ed.). Pergamon Press.
ISBN 978-0-08-020940-1. Online copy
Gunther Ludwig, 1968.
Wave Mechanics. London: Pergamon Press.
George Mackey (2004). The mathematical foundations of quantum
mechanics. Dover Publications. ISBN 0-486-43517-2.
Albert Messiah, 1966.
Mechanics (Vol. I), English translation
from French by G. M. Temmer. North Holland, John Wiley & Sons. Cf.
chpt. IV, section III. online
Omnès, Roland (1999). Understanding
Quantum Mechanics. Princeton
University Press. ISBN 0-691-00435-8. OCLC 39849482.
Scerri, Eric R., 2006. The Periodic Table: Its Story and Its
Significance. Oxford University Press. Considers the extent to which
chemistry and the periodic system have been reduced to quantum
mechanics. ISBN 0-19-530573-6
Transnational College of Lex (1996). What is
Quantum Mechanics? A
Physics Adventure. Language Research Foundation, Boston.
ISBN 0-9643504-1-6. OCLC 34661512.
von Neumann, John (1955).
Mathematical Foundations of Quantum
Mechanics. Princeton University Press. ISBN 0-691-02893-1.
Hermann Weyl, 1950. The Theory of Groups and
Quantum Mechanics, Dover
D. Greenberger, K. Hentschel, F. Weinert, eds., 2009. Compendium of
quantum physics, Concepts, experiments, history and philosophy,
Springer-Verlag, Berlin, Heidelberg.
Bernstein, Jeremy (2009).
Quantum Leaps. Cambridge, Massachusetts:
Belknap Press of Harvard University Press.
Bohm, David (1989).
Quantum Theory. Dover Publications.
Eisberg, Robert; Resnick, Robert (1985).
Physics of Atoms,
Molecules, Solids, Nuclei, and Particles (2nd ed.). Wiley.
ISBN 0-471-87373-X. CS1 maint: Multiple names: authors list
Liboff, Richard L. (2002). Introductory
Addison-Wesley. ISBN 0-8053-8714-5.
Merzbacher, Eugen (1998).
Quantum Mechanics. Wiley, John & Sons,
Inc. ISBN 0-471-88702-1.
Sakurai, J. J. (1994). Modern
Quantum Mechanics. Addison Wesley.
Shankar, R. (1994). Principles of
Quantum Mechanics. Springer.
Stone, A. Douglas (2013). Einstein and the Quantum. Princeton
University Press. ISBN 978-0-691-13968-5.
Martinus J. G. Veltman
Martinus J. G. Veltman (2003), Facts and Mysteries in Elementary
Zucav, Gary (1979, 2001). The Dancing Wu Li Masters: An overview of
the new physics (Perennial Classics Edition) HarperCollins.
Find more about
Quantum mechanicsat's sister projects
Definitions from Wiktionary
Media from Wikimedia Commons
News from Wikinews
Quotations from Wikiquote
Texts from Wikisource
Textbooks from Wikibooks
Learning resources from Wikiversity
3D animations, applications and research for basic quantum effects
(animations also available in commons.wikimedia.org (Université paris
Quantum Cook Book by R. Shankar, Open Yale PHYS 201 material (4pp)
The Modern Revolution in
Physics - an online textbook.
J. O'Connor and E. F. Robertson: A history of quantum mechanics.
Quantum Theory at Quantiki.
Physics Made Relatively Simple: three video lectures by Hans
H is for h-bar.
Mechanics Books Collection: Collection of free books
A collection of lectures on
Physics Database - Fundamentals and Historical Background of
Doron Cohen: Lecture notes in
Mechanics (comprehensive, with
MIT OpenCourseWare: Chemistry.
MIT OpenCourseWare: Physics. See 8.04
Stanford Continuing Education PHY 25:
Mechanics by Leonard
Susskind, see course description[permanent dead link] Fall 2007
5½ Examples in
Spark Notes -
Physics Online : interactive introduction to quantum
mechanics (RS applets).
Experiments to the foundations of quantum physics with single photons.
AQME : Advancing
Mechanics for Engineers — by
T.Barzso, D.Vasileska and G.Klimeck online learning resource with
simulation tools on nanohub
Mechanics by Martin Plenio
Mechanics by Richard Fitzpatrick
Online course on
Many-worlds or relative-state interpretation.
PHYS 201: Fundamentals of
Physics II by Ramamurti Shankar, Open Yale
Mechanics by Leonard Susskind
Everything you wanted to know about the quantum world — archive
of articles from New Scientist.
Physics Research from Science Daily
Overbye, Dennis (December 27, 2005). "
Quantum Trickery: Testing
Einstein's Strangest Theory". The New York Times. Retrieved April 12,
Audio: Astronomy Cast
Quantum Mechanics — June 2009. Fraser
Cain interviews Pamela L. Gay.
Physics of Reality", BBC Radio 4 discussion with Roger Penrose,
Fay Dowker & Tony Sudbery (In Our Time, May 2, 2002).
Ismael, Jenann. "
Quantum Mechanics". In Zalta, Edward N. Stanford
Encyclopedia of Philosophy.
Krips, Henry. "Measurement in
Quantum Theory". In Zalta, Edward N.
Stanford Encyclopedia of Philosophy.
Old quantum theory
Heisenberg uncertainty principle
Spontaneous parametric down-conversion
Von Neumann entropy
Symmetry in quantum mechanics
Spontaneous symmetry breaking
Wave function collapse
Path integral formulation
Delayed choice quantum eraser
Quantum suicide and immortality
Wheeler's delayed choice
Quantum differential calculus
Quantum measurement problem
Quantum stochastic calculus
Quantum cellular automata
Quantum finite automata
Quantum logic gates
Quantum dot display
Quantum dot solar cell
Quantum dot cellular automaton
Quantum dot single-photon source
Quantum dot laser
Quantum complexity theory
Quantum error correction
Quantum image processing
Quantum key distribution
Quantum machine learning
Quantum neural network
Quantum statistical mechanics
Relativistic quantum mechanics
Fractional quantum mechanics
Quantum field theory
Axiomatic quantum field theory
Quantum field theory in curved spacetime
Thermal quantum field theory
Topological quantum field theory
Local quantum field theory
Conformal field theory
Two-dimensional conformal field theory
Liouville field theory
Branches of physics
Quantum field theory
Physics in life science
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