Quantum
mechanics
From Wikipedia, the free encyclopedia
For a more accessible
and less technical introduction to this topic, see Introduction
to quantum mechanics.
Solution to Schrödinger's
equation for the hydrogen atom at different energy levels. The
brighter areas represent a higher probability of finding an electron
Quantum mechanics (QM; also known as quantum
physics, or quantum theory) is a fundamental branch of
physics which
deals with physical phenomena at nanoscopic
scales, where the action
is on the order of the Planck
constant. The name derives from the observation that some
physical quantities can change only in discrete amounts (Latin
quanta),
and not in a continuous (cf. analog)
way. It departs from classical
mechanics primarily at the quantum
realm of atomic
and subatomic
length scales. Quantum mechanics provides a mathematical description
of much of the dual particlelike and wavelike
behavior and interactions of energy
and matter. Quantum
mechanics provides a substantially useful framework for many features
of the modern periodic
table of elements, including the behavior of atoms
during chemical
bonding, and has played a significant role in the development of
many modern technologies.
In advanced topics of quantum mechanics, some of these behaviors
are macroscopic
(see macroscopic
quantum phenomena) and emerge at only extreme (i.e., very low or
very high) energies or temperatures
(such as in the use of superconducting
magnets). In the context of quantum mechanics, the wave–particle
duality of energy and matter and the uncertainty
principle provide a unified view of the behavior of photons,
electrons, and
other atomicscale objects.
The mathematical
formulations of quantum mechanics are abstract. A mathematical
function, the wavefunction,
provides information about the probability
amplitude of position, momentum, and other physical properties of
a particle. Mathematical manipulations of the wavefunction usually
involve bra–ket
notation, which requires an understanding of complex
numbers and linear
functionals. The wavefunction formulation treats the particle as
a quantum
harmonic oscillator, and the mathematics is akin to that
describing acoustic
resonance. Many of the results of quantum mechanics are not
easily visualized in terms of classical
mechanics. For instance, in a quantum mechanical model, the
lowest energy state of a system, the ground
state, is nonzero as opposed to a more "traditional"
ground state with zero kinetic
energy (all particles at rest). Instead of a traditional static,
unchanging zero energy state, quantum mechanics allows for far more
dynamic, chaotic possibilities, according to John
Wheeler.
The earliest
versions of quantum mechanics were formulated in the first decade of
the 20th century. About this time, the atomic
theory and the corpuscular
theory of light (as updated by Einstein)[1]
first came to be widely accepted as scientific fact; these latter
theories can be viewed as quantum theories of matter
and electromagnetic
radiation, respectively. Early
quantum theory was significantly reformulated in the mid1920s by
Werner
Heisenberg, Max
Born and Pascual
Jordan (matrix
mechanics); Louis
de Broglie and Erwin
Schrödinger (wave
mechanics); and Wolfgang
Pauli and Satyendra
Nath Bose (statistics of subatomic particles). Moreover, the
Copenhagen
interpretation of Niels
Bohr became widely accepted. By 1930, quantum mechanics had been
further unified and formalized by the work of David
Hilbert, Paul
Dirac and John
von Neumann[2]
with a greater emphasis placed on measurement
in quantum mechanics, the statistical nature of our knowledge of
reality, and philosophical
speculation about the role of the observer. Quantum mechanics has
since permeated throughout many aspects of 20thcentury physics and
other disciplines including quantum
chemistry, quantum
electronics, quantum
optics, and quantum
information science. Much 19thcentury physics has been
reevaluated as the "classical limit" of quantum mechanics
and its more advanced developments in terms of quantum
field theory, string
theory, and speculative quantum
gravity theories.
History
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
Huygens and Leonhard
Euler proposed a wave theory of light based on experimental
observations.[3]
In 1803, Thomas
Young, an English polymath,
performed the famous doubleslit
experiment that he later described in a paper entitled On the
nature of light and colours. This experiment played a major role
in the general acceptance of the wave
theory of light.
In 1838, Michael
Faraday discovered cathode
rays. These studies were followed by the 1859 statement of the
blackbody
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.[4]
Planck's hypothesis that energy is radiated and absorbed in discrete
"quanta" (or energy elements) precisely matched the
observed patterns of blackbody radiation.
In 1896, Wilhelm
Wien empirically determined a distribution law of blackbody
radiation,[5]
known as Wien's
law in his honor. Ludwig Boltzmann independently arrived at this
result by considerations of Maxwell's
equations. However, it was valid only at high frequencies and
underestimated the radiance at low frequencies. Later, Max
Planck corrected this model using Boltzmann's statistical
interpretation of thermodynamics and proposed what is now called
Planck's law,
which led to the development of quantum mechanics.
Among the first to study quantum
phenomena in nature were Arthur
Compton, C.V.
Raman, and Pieter
Zeeman, each of whom has a quantum effect named after him. Robert
A. Millikan studied the photoelectric
effect experimentally, and Albert
Einstein developed a theory for it. At the same time, Niels
Bohr developed his theory of the atomic structure, which was
later confirmed by the experiments of Henry
Moseley. In 1913, Peter
Debye extended Niels Bohr's theory of atomic structure,
introducing elliptical
orbits, a concept also introduced by Arnold
Sommerfeld.[6]
This phase is known as old
quantum theory.
According to Planck, each energy element (E) is
proportional to its frequency
(ν):

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.[7]
In fact, he considered his quantum
hypothesis a mathematical trick to get the right answer rather
than a sizable discovery.[8]
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.[9]
The 1927 Solvay
Conference in Brussels.
The foundations of quantum mechanics were established during the
first half of the 20th century by Max
Planck, Niels
Bohr, Werner
Heisenberg, Louis
de Broglie, Arthur
Compton, Albert
Einstein, Erwin
Schrödinger, Max
Born, John
von Neumann, Paul
Dirac, Enrico
Fermi, Wolfgang
Pauli, Max
von Laue, Freeman
Dyson, David
Hilbert, Wilhelm
Wien, Satyendra
Nath Bose, Arnold
Sommerfeld, and others.
In the mid1920s, 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 particlelike behavior in
certain processes and measurements, light quanta came to be called
photons (1926).
From Einstein's simple postulation was born a flurry of debating,
theorizing, and testing. Thus, the entire field of quantum
physics emerged, leading to its wider acceptance at the Fifth
Solvay
Conference in 1927.
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.
While quantum mechanics traditionally
described the world of the very small, it is also needed to explain
certain recently investigated macroscopic
systems such as superconductors,
superfluids,
and large organic molecules.[10]
The word quantum derives from the Latin,
meaning "how great" or "how much".[11]
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 wavelike 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, solidstate
physics, atomic
physics, molecular
physics, computational
physics, computational
chemistry, quantum
chemistry, particle
physics, nuclear
chemistry, and nuclear
physics.[12]
Some fundamental aspects of the theory are still actively
studied.[13]
Quantum mechanics is essential to
understanding the behavior of systems at atomic
length scales and smaller. If the physical nature of an atom was
solely described by classical
mechanics, electrons would not orbit the nucleus, since
orbiting electrons emit radiation (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, nondeterministic,
smeared, probabilistic
wave–particle orbital
about the nucleus, defying the traditional assumptions of classical
mechanics and electromagnetism.[14]
Quantum mechanics was initially developed to provide a better
explanation and description of the atom, especially the differences
in the spectra of
light emitted by different isotopes
of the same element,
as well as subatomic particles. In short, the quantummechanical
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:
Mathematical
formulations
Main article: Mathematical
formulations of quantum mechanics
See also: Quantum
logic
In the mathematically rigorous formulation of quantum mechanics
developed by Paul
Dirac,[15]
David
Hilbert,[16]
John von
Neumann,[17]
and Hermann
Weyl,[18]
the possible states of a quantum mechanical system are represented by
unit vectors
(called state vectors). Formally, these reside in a complex
separable
Hilbert
space—variously called the state
space or the associated Hilbert space of the
system—that is well defined up to a complex number of norm 1
(the phase factor). In other words, the possible states are points in
the projective
space of a Hilbert space, usually called the complex
projective space. The exact nature of this Hilbert space is
dependent on the system—for example, the state space for
position and momentum states is the space of squareintegrable
functions, while the state space for the spin of a single proton is
just the product of two complex planes. Each observable is
represented by a maximally Hermitian
(precisely: by a selfadjoint)
linear operator
acting on the state space. Each eigenstate
of an observable corresponds to an eigenvector
of the operator, and the associated eigenvalue
corresponds to the value of the observable in that eigenstate. If the
operator's spectrum is discrete, the observable can 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.[19]
This abstract mathematical object allows for the calculation of
probabilities
of outcomes of concrete experiments. For example, it allows one to
compute the probability of finding an electron in a particular region
around the nucleus at a particular time. Contrary to classical
mechanics, one can never make simultaneous predictions of conjugate
variables, such as position and momentum, with accuracy. For
instance, electrons may be considered (to a certain probability) to
be located somewhere within a given region of space, but with their
exact positions unknown. Contours of constant probability, often
referred to as "clouds", may be drawn around the nucleus of
an atom to conceptualize where the electron might be located with the
most probability. Heisenberg's uncertainty
principle quantifies the inability to precisely locate the
particle given its conjugate momentum.[20]
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 BohrEinstein
debates, in which the two scientists attempted to clarify these
fundamental principles by way of thought
experiments. In the decades after the formulation of quantum
mechanics, the question of what constitutes a "measurement"
has been extensively studied. Newer interpretations
of quantum mechanics have been formulated that do away with the
concept of "wavefunction collapse" (see, for example, the
relative
state interpretation). The basic idea is that when a quantum
system interacts with a measuring apparatus, their respective
wavefunctions become entangled,
so that the original quantum system ceases to exist as an independent
entity. For details, see the article on measurement
in quantum mechanics.[21]
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.[22][23]
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").[24]
In the everyday world, it is natural and intuitive to think of
everything (every observable) as being in an eigenstate. Everything
appears to have a definite position, a definite momentum, a definite
energy, and a definite time of occurrence. However, quantum mechanics
does not pinpoint the exact values of a particle's position and
momentum (since they are conjugate
pairs) or its energy and time (since they too are conjugate
pairs); rather, it provides only a range of probabilities in which
that particle might be given its momentum and momentum probability.
Therefore, it is helpful to use different words to describe states
having uncertain
values and states having definite values (eigenstates).
Usually, a system will not be in an eigenstate
of the observable (particle) we are interested in. However, if one
measures the observable, the wavefunction will instantaneously be an
eigenstate (or "generalized" eigenstate) of that
observable. This process is known as wavefunction
collapse, a controversial and muchdebated process[25]
that involves expanding the system under study to include the
measurement device. If one knows the corresponding wave function at
the instant before the measurement, one will be able to compute the
probability of the wavefunction collapsing into each of the possible
eigenstates. For example, the free particle in the previous example
will usually have a wavefunction that is a wave
packet centered around some mean position x_{0}
(neither an eigenstate of position nor of momentum). When one
measures the position of the particle, it is impossible to predict
with certainty the result.[21]
It is probable, but not certain, that it will be near x_{0},
where the amplitude of the wave function is large. After the
measurement is performed, having obtained some result x, the
wave function collapses into a position eigenstate centered at x.[26]
The time evolution of a quantum state is
described by the Schrödinger
equation, in which the Hamiltonian
(the operator
corresponding to the total
energy of the system) generates the time evolution. The time
evolution of wave functions is deterministic
in the sense that  given a wavefunction at an initial time 
it makes a definite prediction of what the wavefunction will be at
any later time.[27]
During a
measurement,
on the other hand, the change of the initial wavefunction into
another, later wavefunction is not deterministic, it is unpredictable
(i.e., random). A
timeevolution simulation can be seen here.[28][29]
Wave functions change as time
progresses. The Schrödinger
equation describes how wavefunctions change in time, playing a
role similar to Newton's
second law in classical
mechanics. The Schrödinger equation, applied to the
aforementioned example of the free particle, predicts that the center
of a wave packet will move through space at a constant velocity (like
a classical particle with no forces acting on it). However, the wave
packet will also spread out as time progresses, which means that the
position becomes more uncertain with time. This also has the effect
of turning a position eigenstate (which can be thought of as an
infinitely sharp wave packet) into a broadened wave packet that no
longer represents a (definite, certain) position eigenstate.[30]
Fig. 1: Probability
densities corresponding to the wavefunctions of an electron in a
hydrogen atom possessing definite energy levels (increasing from the
top of the image to the bottom: n = 1, 2, 3, ...) and angular
momenta (increasing across from left to right: s, p, d,
...). Brighter areas correspond to higher probability density in a
position measurement. Such wavefunctions are directly comparable to
Chladni's
figures of acoustic
modes of vibration in classical
physics, and are modes of oscillation as well, possessing a sharp
energy and, thus, a
definite frequency.
The angular
momentum and energy are quantized,
and take only discrete values like those shown (as is the case
for resonant
frequencies in acoustics)
Some wave functions produce probability
distributions that are constant, or independent of time—such as
when in a stationary
state of constant energy, time vanishes in the absolute square of
the wave function. Many systems that are treated dynamically in
classical mechanics are described by such "static" wave
functions. For example, a single electron
in an unexcited atom
is pictured classically as a particle moving in a circular trajectory
around the atomic
nucleus, whereas in quantum mechanics it is described by a
static, spherically
symmetric wavefunction surrounding the nucleus (Fig.
1) (note, however, that only the lowest angular momentum states,
labeled s, are spherically symmetric).[31]
The Schrödinger equation acts on the entire
probability amplitude, not merely its absolute value. Whereas the
absolute value of the probability amplitude encodes information about
probabilities, its phase
encodes information about the interference
between quantum states. This gives rise to the "wavelike"
behavior of quantum states. As it turns out, analytic solutions of
the Schrödinger equation are available for only a
very small number of relatively simple model Hamiltonians, of
which the quantum
harmonic oscillator, the particle
in a box, the hydrogen
molecular ion, and the hydrogen
atom are the most important representatives. Even the helium
atom—which contains just one more electron than does the
hydrogen atom—has defied all attempts at a fully analytic
treatment.
There exist several techniques for generating approximate
solutions, however. In the important method known as perturbation
theory, one uses the analytic result for a simple quantum
mechanical model to generate a result for a more complicated model
that is related to the simpler model by (for one example) the
addition of a weak potential
energy. Another method is the "semiclassical 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).[32]
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.[33]
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).[34]
An alternative formulation of quantum mechanics is Feynman's
path
integral formulation, in which a quantummechanical amplitude is
considered as a sum over all possible classical and nonclassical
paths between the initial and final states. This is the
quantummechanical counterpart of the action
principle in classical mechanics.
Copenhagen interpretation of quantum versus classical kinematics
The phenomena of the quantum and
classical mechanics are conceived with different kinematic
descriptions.[35]
In the Copenhagen view of quantum mechanics, phenomena are required
to be experiments, with complete descriptions of the initial
preparatory device for the system, of the potential or possible
intermediate processes, and of the final registering device or
detector.[36][37][38][citation
needed] 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, a quantum phenomenon is a process,
a passage from initial to final condition, not an instantaneous
"state" in the classical sense of that word.[39]
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.[40]
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.[41][citation
needed]
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 adventure its way
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.[42][43]
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.[44]
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.
Interactions with other scientific theories
The rules of quantum mechanics are fundamental. They assert that
the state space of a system is a Hilbert
space and that observables of that system are Hermitian
operators acting on that space—although they do not tell us
which Hilbert space or which operators. These can be chosen
appropriately in order to obtain a quantitative description of a
quantum system. An important guide for making these choices is the
correspondence
principle, which states that the predictions of quantum mechanics
reduce to those of classical mechanics when a system moves to higher
energies or, equivalently, larger quantum numbers, i.e. whereas a
single particle exhibits a degree of randomness, in systems
incorporating millions of particles averaging takes over and, at the
high energy limit, the statistical probability of random behaviour
approaches zero. In other words, classical mechanics is simply a
quantum mechanics of large systems. This "high energy"
limit is known as the classical or correspondence limit.
One can even start from an established classical model of a
particular system, then attempt to guess the underlying quantum model
that would give rise to the classical model in the correspondence
limit.
List
of unsolved problems in physics
In the correspondence
limit of quantum mechanics: Is there a
preferred interpretation of quantum mechanics? How does the
quantum description of reality, which includes elements such as
the "superposition
of states" and "wavefunction
collapse", give rise to the reality we perceive?

When quantum mechanics was originally formulated, it was applied
to models whose correspondence limit was nonrelativistic
classical
mechanics. For instance, the wellknown model of the quantum
harmonic oscillator uses an explicitly nonrelativistic
expression for the kinetic
energy of the oscillator, and is thus a quantum version of the
classical
harmonic oscillator.
Early attempts to merge quantum mechanics with special
relativity involved the replacement of the Schrödinger
equation with a covariant equation such as the Klein–Gordon
equation or the Dirac
equation. While these theories were successful in explaining many
experimental results, they had certain unsatisfactory qualities
stemming from their neglect of the relativistic creation and
annihilation of particles. A fully relativistic quantum theory
required the development of quantum
field theory, which applies quantization to a field (rather than
a fixed set of particles). The first complete quantum field theory,
quantum
electrodynamics, provides a fully quantum description of the
electromagnetic
interaction. The full apparatus of quantum field theory is often
unnecessary for describing electrodynamic systems. A simpler
approach, one that has been employed since the inception of quantum
mechanics, is to treat charged
particles as quantum mechanical objects being acted on by a classical
electromagnetic
field. For example, the elementary quantum model of the hydrogen
atom describes the electric
field of the hydrogen atom using a classical
Coulomb
potential. This "semiclassical" approach fails if
quantum fluctuations in the electromagnetic field play an important
role, such as in the emission of photons
by charged
particles.
Quantum
field theories for the strong
nuclear force and the weak
nuclear force have 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.[45]
It has proven difficult to construct quantum models of gravity,
the remaining fundamental
force. Semiclassical approximations are workable, and have led
to predictions such as Hawking
radiation. However, the formulation of a complete theory of
quantum
gravity is hindered by apparent incompatibilities between general
relativity (the most accurate theory of gravity currently known)
and some of the fundamental assumptions of quantum theory. The
resolution of these incompatibilities is an area of active research,
and theories such as string
theory are among the possible candidates for a future theory of
quantum gravity.
Classical mechanics has also been
extended into the complex
domain, with complex classical mechanics exhibiting behaviors
similar to quantum mechanics.[46]
Quantum mechanics and classical physics
Predictions of quantum mechanics have been verified experimentally to
an extremely high degree of accuracy.[47]
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).[48]
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.[49]
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.[50]
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 longrange entanglement and other
nonlocal phenomena characteristic of quantum systems.[51]
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.[52]
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.[53]
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.[54]
Relativity
and quantum mechanics
Main article: Relativistic
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.[55]
Einstein himself is well known for rejecting some of the claims of
quantum mechanics. While clearly contributing to the field, he did
not accept many of the more "philosophical consequences and
interpretations" of quantum mechanics, such as the lack of
deterministic causality.
He is famously quoted as saying, in response to this aspect, "My
God does not play with dice". He also had difficulty with the
assertion that a single subatomic
particle can occupy numerous areas of space at one time. However,
he was also the first to notice some of the apparently exotic
consequences of entanglement,
and used them to formulate the Einstein–Podolsky–Rosen
paradox in the hope of showing that quantum mechanics had
unacceptable implications if taken as a complete description of
physical reality. This was 1935, but in 1964 it was shown by John
Bell (see Bell
inequality) that  although Einstein was correct in identifying
seemingly paradoxical implications of quantum
mechanical nonlocality  these implications could be
experimentally tested. Alain Aspect's initial experiments in 1982,
and many subsequent experiments since, have definitively verified
quantum entanglement.
According to the paper of J. Bell and the Copenhagen
interpretation—the common interpretation of quantum
mechanics by physicists since 1927  and contrary to Einstein's
ideas, quantum mechanics was not, at the same time a
"realistic" theory and a "local"
theory.
The Einstein–Podolsky–Rosen
paradox shows in any case that there exist experiments by which
one can measure the state of one particle and instantaneously change
the state of its entangled partner  although the two particles can
be an arbitrary distance apart. However, this effect does not violate
causality, since
no transfer of information happens. Quantum entanglement forms the
basis of quantum
cryptography, which is used in highsecurity commercial
applications in banking and government.
Gravity is negligible in many areas of
particle physics, so that unification between general relativity and
quantum mechanics is not an urgent issue in those particular
applications. However, the lack of a correct theory of quantum
gravity is an important issue in cosmology
and the search by physicists for an elegant "Theory
of Everything" (TOE). Consequently, resolving the
inconsistencies between both theories has been a major goal of 20th
and 21st century physics. Many prominent physicists, including
Stephen
Hawking, have labored for many years in the attempt to discover a
theory underlying everything. This TOE would combine not only
the different models of subatomic physics, but also derive the four
fundamental
forces of nature  the strong
force, electromagnetism,
the weak
force, and gravity
 from a single force or phenomenon. While Stephen Hawking was
initially a believer in the Theory of Everything, after considering
Gödel's
Incompleteness Theorem, he has concluded that one is not
obtainable, and has stated so publicly in his lecture "Gödel
and the End of Physics" (2002).[56]
Attempts
at a unified field theory
Main article: Grand
unified 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,[57][58][unreliable
source?]^{(blog)} has been successfully merged with
the weak nuclear force into the electroweak
force and work is currently being done to merge the electroweak
and strong force into the electrostrong
force. Current predictions state that at around 10^{14}
GeV the three aforementioned forces are fused into a single unified
field.[59]
Beyond this "grand unification", it is speculated that it
may be possible to merge gravity with the other three gauge
symmetries, expected to occur at roughly 10^{19} GeV.
However — and while special relativity is parsimoniously
incorporated into quantum electrodynamics — the expanded
general
relativity, currently the best theory describing the gravitation
force, has not been fully incorporated into quantum theory. One of
those searching for a coherent TOE is Edward
Witten, a theoretical physicist who formulated the Mtheory,
which is an attempt at describing the supersymmetrical based string
theory. Mtheory posits that our apparent 4dimensional spacetime
is, in reality, actually an 11dimensional spacetime containing 10
spatial dimensions and 1 time dimension, although 7 of the spatial
dimensions are  at lower energies  completely "compactified"
(or infinitely curved) and not readily amenable to measurement or
probing.
Another popular theory is Loop
quantum gravity (LQG), a theory that describes the quantum
properties of gravity. It is also a theory of quantum
space and quantum
time, because in general relativity the geometry of spacetime is
a manifestation of gravity.
LQG is an attempt to merge and adapt standard quantum mechanics and
standard general
relativity. The main output of the theory is a physical picture
of space where space is granular. The granularity is a direct
consequence of the quantization. It has the same nature of the
granularity of the photons in the quantum theory of electromagnetism
or the discrete levels of the energy of the atoms. But here it is
space itself which is discrete. More precisely, space can be viewed
as an extremely fine fabric or network "woven" of finite
loops. These networks of loops are called spin
networks. The evolution of a spin network over time, is called a
spin foam. The predicted size of this structure is the Planck
length, which is approximately 1.616×10^{−35}
m. According to theory, there is no meaning to length shorter than
this (cf. Planck
scale energy). Therefore LQG predicts that not just matter, but
also space itself, has an atomic structure. Loop quantum Gravity was
first proposed by Carlo
Rovelli.
Philosophical
implications
Main article: Interpretations
of quantum mechanics
Since its
inception, the many counterintuitive
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."[60]
According to Steven
Weinberg, "There is now in my opinion no entirely
satisfactory interpretation of quantum mechanics."[61]
The Copenhagen
interpretation  due largely to the Danish theoretical physicist
Niels Bohr 
remains the quantum mechanical formalism that is currently most
widely accepted amongst physicists, some 75 years after its
enunciation. According to this interpretation, the probabilistic
nature of quantum mechanics is not a temporary feature which
will eventually be replaced by a deterministic theory, but instead
must be considered a final renunciation of the classical idea
of "causality." It is also believed therein that any
welldefined 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, disliked
this loss of determinism in measurement. Einstein held that there
should be a local
hidden variable theory underlying quantum mechanics and,
consequently, that the present theory was incomplete. He produced a
series of objections to quantum theory, the most famous of which has
become known as the Einstein–Podolsky–Rosen
paradox. John
Bell showed that this "EPR" paradox led to
experimentally
testable differences between quantum mechanics and local
realistic theories. Experiments
have been performed confirming the accuracy of quantum mechanics,
thereby demonstrating that the physical world cannot be described by
any local realistic theory.[62]
The BohrEinstein
debates provide a vibrant critique of the Copenhagen
Interpretation from an epistemological
point of view.
The
Everett
manyworlds interpretation, formulated in 1956, holds that all
the possibilities described by quantum theory simultaneously
occur in a multiverse
composed of mostly independent parallel universes.[63]
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
nondeterministic 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.[64]
Relational
quantum mechanics appeared in the late 1990s as the modern
derivative of the Copenhagen
Interpretation.
Applications
Quantum mechanics has had enormous[65]
success in explaining many of the features of our universe. Quantum
mechanics is often the only tool available that can reveal the
individual behaviors of the subatomic
particles that make up all forms of matter (electrons,
protons, neutrons,
photons, and
others). Quantum mechanics has strongly influenced string
theories, candidates for a Theory
of Everything (see reductionism).
Quantum mechanics is also critically
important for understanding how individual atoms combine covalently
to form molecules.
The application of quantum mechanics to chemistry
is known as quantum
chemistry. Relativistic quantum mechanics can, in principle,
mathematically describe most of chemistry. Quantum mechanics can also
provide quantitative insight into ionic
and covalent
bonding processes by explicitly showing which molecules are
energetically favorable to which others and the magnitudes of the
energies involved.[66]
Furthermore, most of the calculations performed in modern
computational
chemistry rely on quantum mechanics.
A working mechanism of a resonant tunneling diode device, based on
the phenomenon of quantum tunneling through potential barriers
A great deal of modern technological inventions operate at a scale
where quantum effects are significant. Examples include the laser,
the transistor
(and thus the microchip),
the electron
microscope, and magnetic
resonance imaging (MRI). The study of semiconductors
led to the invention of the diode
and the transistor,
which are indispensable parts of modern electronics
systems and devices.
Researchers are currently seeking robust methods of directly
manipulating quantum states. Efforts are being made to more fully
develop quantum
cryptography, which will theoretically allow guaranteed secure
transmission of information.
A more distant goal is the development of quantum
computers, which are expected to perform certain computational
tasks exponentially faster than classical computers.
Instead of using classical bits, quantum computers use qubits,
which can be in superpositions
of states. Another active research topic is quantum
teleportation, which deals with techniques to transmit quantum
information over arbitrary distances.
Quantum
tunneling is vital to the operation of many devices. Even in the
simple light
switch, the electrons in the electric
current could not penetrate the potential barrier made up of a
layer of oxide without quantum tunneling. Flash
memory chips found in USB
drives use quantum tunneling to erase their memory cells.
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 wellknown 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.
Quantum theory
also provides accurate descriptions for many previously unexplained
phenomena, such as blackbody
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.[67]
Recent work on photosynthesis
has provided evidence that quantum correlations play an essential
role in this fundamental process of plants and many other
organisms.[68]
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, there is wave–particle
duality, so the properties of the particle can be described as
the properties of a wave. Therefore, its quantum
state can be represented as a wave
of arbitrary shape and extending over space as a wave
function. The position and momentum of the particle are
observables.
The Uncertainty
Principle states that both the position and the momentum cannot
simultaneously be measured with complete precision. However, one can
measure the position (alone) of a moving free particle, creating an
eigenstate of position with a wavefunction that is very large (a
Dirac delta)
at a particular position x, and zero everywhere else. If one
performs a position measurement on such a wavefunction, the resultant
x will be obtained with 100% probability (i.e., with full
certainty, or complete precision). This is called an eigenstate of
position—or, stated in mathematical terms, a generalized
position eigenstate (eigendistribution).
If the particle is in an eigenstate of position, then its momentum is
completely unknown. On the other hand, if the particle is in an
eigenstate of momentum, then its position is completely unknown.[69]
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.[70]
3D confined electron wave functions for each eigenstate in a
Quantum Dot. Here, rectangular and triangularshaped quantum dots are
shown. Energy states in rectangular dots are more ‘stype’
and ‘ptype’. However, in a triangular dot, the wave
functions are mixed due to confinement symmetry.
Step potential
Main article: Solution
of Schrödinger equation for a step potential
Scattering at a finite potential step of height V_{0},
shown in green. The amplitudes and direction of left and
rightmoving waves are indicated. Yellow is the incident wave, blue
are reflected and transmitted waves, red does not occur. E >
V_{0} for this figure.
The potential in this case is given by:

The solutions are superpositions of left and rightmoving waves:


where the wave
vectors are related to the energy via
 ,
and

with coefficients A and B determined from the boundary
conditions and by imposing a continuous derivative
on the solution.
Each term of the solution can be interpreted as an incident,
reflected, or transmitted component of the wave, allowing the
calculation of transmission and reflection coefficients. Notably, in
contrast to classical mechanics, incident particles with energies
greater than the potential step are partially reflected.
Rectangular
potential barrier
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.
Particle in a box
1dimensional potential energy box (or
infinite potential well)
Main article: Particle
in a box
The particle in a onedimensional
potential energy box is the most mathematically simple example where
restraints lead to the quantization of energy levels. The box is
defined as having zero potential energy everywhere inside a
certain region, and infinite potential energy everywhere outside
that region. For the onedimensional case in the
direction, the timeindependent Schrödinger equation may be
written[71]

With the differential operator defined by

the previous equation is evocative of the classic
kinetic energy analogue,

with state
in this case having energy
coincident with the kinetic energy of the particle.
The general solutions of the Schrödinger equation for the
particle in a box are

or, from Euler's
formula,

The infinite potential walls of the box determine the values of C,
D, and k at x = 0 and x = L where
ψ must be zero. Thus, at x = 0,

and D = 0. At x = L,

in which C cannot be zero as this would conflict with the Born
interpretation. Therefore, since sin(kL) = 0, kL must
be an integer multiple of π,

The quantization of energy levels follows from this constraint on k,
since

Finite potential well
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 particleinabox problem as the
wavefunction is not pinned to zero at the walls of the well. Instead,
the wavefunction must satisfy more complicated mathematical boundary
conditions as it is nonzero in regions outside the well.
Harmonic
oscillator
Main article: Quantum
harmonic oscillator
Some trajectories of a harmonic
oscillator (i.e. a ball attached to a spring)
in classical
mechanics (AB) and quantum mechanics (CH). In quantum
mechanics, the position of the ball is represented by a wave
(called the wavefunction),
with the real part
shown in blue and the imaginary
part shown in red. Some of the trajectories (such as C,D,E,and F)
are standing
waves (or "stationary
states"). Each standingwave frequency is proportional to a
possible energy
level of the oscillator. This "energy quantization"
does not occur in classical physics, where the oscillator can have
any energy.
As in the classical case, the potential for the quantum harmonic
oscillator is given by

This problem can either be treated by directly solving the
Schrödinger, which is not trivial, or by using the more elegant
"ladder method" first proposed by Paul Dirac. The
eigenstates are
given by

where H_{n} are the Hermite
polynomials,

and the corresponding energy levels are
 .
This is another example illustrating the quantization of energy for
bound states.
See also
Notes
Einstein,
A. (1905). "Über einen die Erzeugung
und Verwandlung des Lichtes betreffenden heuristischen
Gesichtspunkt" [On a heuristic point of view concerning
the production and transformation of light]. Annalen
der Physik 17 (6): 132–148.
Bibcode:1905AnP...322..132E.
doi:10.1002/andp.19053220607.
Reprinted in The collected papers of Albert Einstein, John
Stachel, editor, Princeton University Press, 1989, Vol. 2, pp.
149166, in German; see also Einstein's early work on the quantum
hypothesis, ibid. pp. 134148.
P.A.M.
Dirac, The Principles of Quantum Mechanics, Clarendon Press,
Oxford, 1930.
J. von
Neumann, Mathematische Grundlagen der Quantenmechanik,
Springer, Berlin, 1932 (English translation: Mathematical
Foundations of Quantum Mechanics, Princeton University Press,
1955).
Nancy
Thorndike Greenspan, "The End of the Certain World: The Life
and Science of Max Born" (Basic Books, 2005), pp. 1248 and
2856.
Born,
M., Heisenberg,
W., Jordan,
Pascual (1926). Z. Phys. 35: 557–615.
Translated as 'On quantum mechanics II', pp. 321–385 in Van
der Waerden, B.L. (1967), Sources of Quantum Mechanics,
NorthHolland, Amsterdam, "The basic difference between the
theory proposed here and that used hitherto ... lies in the
characteristic kinematics ...", p. 385.
Bohr,
N. (1939). The Causality Problem in Atomic Physics, in New
Theories in Physics, Conference organized in collaboration with the
International Union of Physics and the Polish Intellectual
Cooperation Committee, Warsaw, May 30th – June 3rd 1938,
International Institute of Intellectual Cooperation, Paris, 1939,
pp. 11–30, reprinted in Neils Bohr, Collected Works,
volume 7 (1933 – 1958) edited by J. Kalckar, Elsevier,
Amsterdam, ISBN
0444898921, 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 welldefined
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
2, Quantum Mechanics and Macrosystems, translated by K. Just,
Springer, Berlin, ISBN
9783642718991, Chapter XIII, Special Structures in
Preparation and Registration Devices, §1, Measurement chains,
p. 132.
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."
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 premise."
Born,
M. (1927). Physical aspects of quantum mechanics, Nature
119: 354–357, "These probabilities are thus
dynamically determined. But what the system actually does is not
determined ..."
Heisenberg,
W. (1930). The Physical Principles of the Quantum Theory,
translated by C. Eckart and F.C. Hoyt, University of Chicago Press.
"There
is as yet no logically consistent and complete relativistic quantum
field theory.", p. 4. — V. B. Berestetskii, E.
M. Lifshitz, L P Pitaevskii (1971). J. B. Sykes, J. S. Bell
(translators). Relativistic Quantum Theory 4, part I.
Course of Theoretical Physics (Landau and Lifshitz) ISBN
0080160255
See,
for example, the
Feynman Lectures on Physics for some of the technological
applications which use quantum mechanics, e.g., transistors
(vol III, pp. 1411 ff), integrated
circuits, which are followon technology in solidstate physics
(vol II, pp. 86), and lasers
(vol III, pp. 913).
Derivation of particle in
a box, chemistry.tidalswan.com
References
The following titles, all by working physicists, attempt to
communicate quantum theory to lay people, using a minimum of
technical apparatus.
Chester, Marvin (1987) Primer
of Quantum Mechanics. John Wiley. ISBN
0486428788
Cox, Brian; Forshaw, Jeff (2011).
The
Quantum Universe: Everything That Can Happen Does Happen:.
Allen Lane. ISBN 1846144329.
Richard
Feynman, 1985. QED:
The Strange Theory of Light and Matter, Princeton
University Press. ISBN
0691083886. Four elementary lectures on quantum
electrodynamics and quantum
field theory, yet containing many insights for the expert.
Ghirardi,
GianCarlo, 2004. Sneaking a Look at God's Cards, Gerald
Malsbary, trans. Princeton Univ. Press. The most technical of the
works cited here. Passages using algebra,
trigonometry,
and 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: 11076.
Victor
Stenger, 2000. Timeless Reality: Symmetry, Simplicity, and
Multiple Universes. Buffalo NY: Prometheus Books. Chpts. 58.
Includes cosmological
and philosophical
considerations.
More technical:
Bryce
DeWitt, R. Neill Graham, eds., 1973. The ManyWorlds
Interpretation of Quantum Mechanics, Princeton Series in
Physics, Princeton
University Press. ISBN
069108131X
Dirac,
P. A. M. (1930). The
Principles of Quantum Mechanics. ISBN 0198520115.
The beginning chapters make up a very clear and comprehensible
introduction.
Hugh
Everett, 1957, "Relative State Formulation of Quantum
Mechanics", Reviews of Modern Physics 29: 45462.
Feynman,
Richard P.; Leighton,
Robert B.; Sands, Matthew (1965). The
Feynman Lectures on Physics 1–3.
AddisonWesley.
ISBN 0738200085.
Griffiths, David J. (2004).
Introduction to Quantum Mechanics (2nd ed.). Prentice Hall.
ISBN 0131118927.
OCLC 40251748.
A standard undergraduate text.
Max
Jammer, 1966. The Conceptual Development of Quantum
Mechanics. McGraw Hill.
Hagen
Kleinert, 2004. Path Integrals in Quantum Mechanics,
Statistics, Polymer Physics, and Financial Markets, 3rd ed.
Singapore: World Scientific. Draft
of 4th edition.
Gunther Ludwig, 1968. Wave
Mechanics. London: Pergamon Press. ISBN
0082032041
George
Mackey (2004). The mathematical foundations of quantum
mechanics. Dover Publications. ISBN
0486435172.
Albert
Messiah, 1966. Quantum Mechanics (Vol. I), English
translation from French by G. M. Temmer. North Holland, John Wiley &
Sons. Cf. chpt. IV, section III.
Omnès,
Roland (1999). Understanding Quantum Mechanics. Princeton
University Press. ISBN 0691004358.
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
0195305736
Transnational
College of Lex (1996). What is Quantum Mechanics? A Physics
Adventure. Language Research Foundation, Boston.
ISBN 0964350416.
OCLC 34661512.
von
Neumann, John (1955). Mathematical Foundations of Quantum
Mechanics. Princeton University Press. ISBN 0691028931.
Hermann
Weyl, 1950. The Theory of Groups and Quantum Mechanics,
Dover Publications.
D. Greenberger, K. Hentschel, F. Weinert, eds., 2009.
Compendium of quantum physics, Concepts, experiments, history and
philosophy, SpringerVerlag, Berlin, Heidelberg.
Further reading
Bernstein, Jeremy (2009). Quantum
Leaps. Cambridge, Massachusetts: Belknap Press of Harvard
University Press. ISBN 9780674035416.
Bohm,
David (1989). Quantum Theory. Dover
Publications. ISBN 0486659690.
Eisberg, Robert; Resnick,
Robert (1985). Quantum Physics of Atoms, Molecules, Solids,
Nuclei, and Particles (2nd ed.). Wiley. ISBN 047187373X.
Liboff,
Richard L. (2002). Introductory Quantum Mechanics.
AddisonWesley. ISBN 0805387145.
Merzbacher, Eugen (1998). Quantum
Mechanics. Wiley, John & Sons, Inc. ISBN 0471887021.
Sakurai,
J. J. (1994). Modern Quantum Mechanics. Addison Wesley.
ISBN 0201539292.
Shankar, R. (1994). Principles
of Quantum Mechanics. Springer. ISBN 0306447908.
Stone, A. Douglas (2013). Einstein and the Quantum.
Princeton University Press. ISBN 9780691139685.
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