Wave–particle duality is the concept in
quantum mechanics that every particle or quantum
entity may be partly described in terms not
only of particles, but also of waves. It expresses
the inability of the classical concepts "particle"
or "wave" to fully describe the behaviour
of quantum-scale objects. As Albert Einstein
wrote:
It seems as though we must use sometimes the
one theory and sometimes the other, while
at times we may use either. We are faced with
a new kind of difficulty. We have two contradictory
pictures of reality; separately neither of
them fully explains the phenomena of light,
but together they do.
Through the work of Max Planck, Albert Einstein,
Louis de Broglie, Arthur Compton, Niels Bohr,
and many others, current scientific theory
holds that all particles exhibit a wave nature
and vice versa. This phenomenon has been verified
not only for elementary particles, but also
for compound particles like atoms and even
molecules. For macroscopic particles, because
of their extremely short wavelengths, wave
properties usually cannot be detected.Although
the use of the wave-particle duality has worked
well in physics, the meaning or interpretation
has not been satisfactorily resolved; see
Interpretations of quantum mechanics.
Bohr regarded the "duality paradox" as a fundamental
or metaphysical fact of nature. A given kind
of quantum object will exhibit sometimes wave,
sometimes particle, character, in respectively
different physical settings. He saw such duality
as one aspect of the concept of complementarity.
Bohr regarded renunciation of the cause-effect
relation, or complementarity, of the space-time
picture, as essential to the quantum mechanical
account.Werner Heisenberg considered the question
further. He saw the duality as present for
all quantic entities, but not quite in the
usual quantum mechanical account considered
by Bohr. He saw it in what is called second
quantization, which generates an entirely
new concept of fields that exist in ordinary
space-time, causality still being visualizable.
Classical field values (e.g. the electric
and magnetic field strengths of Maxwell) are
replaced by an entirely new kind of field
value, as considered in quantum field theory.
Turning the reasoning around, ordinary quantum
mechanics can be deduced as a specialized
consequence of quantum field theory.
== History ==
=== 
Classical particle and wave theories of light
===
Democritus argued that all things in the universe,
including light, are composed of indivisible
sub-components. At the beginning of the 11th
Century, the Arabic scientist Ibn al-Haytham
wrote the first comprehensive Book of optics
describing reflection, refraction, and the
operation of a pinhole lens via rays of light
traveling from the point of emission to the
eye. He asserted that these rays were composed
of particles of light. In 1630, René Descartes
popularized and accredited the opposing wave
description in his treatise on light, The
World (Descartes), showing that the behavior
of light could be re-created by modeling wave-like
disturbances in a universal medium i.e. luminiferous
aether. Beginning in 1670 and progressing
over three decades, Isaac Newton developed
and championed his corpuscular theory, arguing
that the perfectly straight lines of reflection
demonstrated light's particle nature, only
particles could travel in such straight lines.
He explained refraction by positing that particles
of light accelerated laterally upon entering
a denser medium. Around the same time, Newton's
contemporaries Robert Hooke and Christiaan
Huygens, and later Augustin-Jean Fresnel,
mathematically refined the wave viewpoint,
showing that if light traveled at different
speeds in different media, refraction could
be easily explained as the medium-dependent
propagation of light waves. The resulting
Huygens–Fresnel principle was extremely
successful at reproducing light's behavior
and was subsequently supported by Thomas Young's
discovery of wave interference of light by
his double-slit experiment in 1801. The wave
view did not immediately displace the ray
and particle view, but began to dominate scientific
thinking about light in the mid 19th century,
since it could explain polarization phenomena
that the alternatives could not.
James Clerk Maxwell discovered that he could
apply his previously discovered Maxwell's
equations, along with a slight modification
to describe self-propagating waves of oscillating
electric and magnetic fields. It quickly became
apparent that visible light, ultraviolet light,
and infrared light were all electromagnetic
waves of differing frequency.
=== Black-body radiation and Planck's law
===
In 1901, Max Planck published an analysis
that succeeded in reproducing the observed
spectrum of light emitted by a glowing object.
To accomplish this, Planck had to make a mathematical
assumption of quantized energy of the oscillators
i.e. atoms of the black body that emit radiation.
Einstein later proposed that electromagnetic
radiation itself is quantized, not the energy
of radiating atoms.
Black-body radiation, the emission of electromagnetic
energy due to an object's heat, could not
be explained from classical arguments alone.
The equipartition theorem of classical mechanics,
the basis of all classical thermodynamic theories,
stated that an object's energy is partitioned
equally among the object's vibrational modes.
But applying the same reasoning to the electromagnetic
emission of such a thermal object was not
so successful. That thermal objects emit light
had been long known. Since light was known
to be waves of electromagnetism, physicists
hoped to describe this emission via classical
laws. This became known as the black body
problem. Since the equipartition theorem worked
so well in describing the vibrational modes
of the thermal object itself, it was natural
to assume that it would perform equally well
in describing the radiative emission of such
objects. But a problem quickly arose if each
mode received an equal partition of energy,
the short wavelength modes would consume all
the energy. This became clear when plotting
the Rayleigh–Jeans law, which, while correctly
predicting the intensity of long wavelength
emissions, predicted infinite total energy
as the intensity diverges to infinity for
short wavelengths. This became known as the
ultraviolet catastrophe.
In 1900, Max Planck hypothesized that the
frequency of light emitted by the black body
depended on the frequency of the oscillator
that emitted it, and the energy of these oscillators
increased linearly with frequency (according
E = hf where h is Planck's constant and f
is the frequency). This was not an unsound
proposal considering that macroscopic oscillators
operate similarly when studying five simple
harmonic oscillators of equal amplitude but
different frequency, the oscillator with the
highest frequency possesses the highest energy
(though this relationship is not linear like
Planck's). By demanding that high-frequency
light must be emitted by an oscillator of
equal frequency, and further requiring that
this oscillator occupy higher energy than
one of a lesser frequency, Planck avoided
any catastrophe, giving an equal partition
to high-frequency oscillators produced successively
fewer oscillators and less emitted light.
And as in the Maxwell–Boltzmann distribution,
the low-frequency, low-energy oscillators
were suppressed by the onslaught of thermal
jiggling from higher energy oscillators, which
necessarily increased their energy and frequency.
The most revolutionary aspect of Planck's
treatment of the black body is that it inherently
relies on an integer number of oscillators
in thermal equilibrium with the electromagnetic
field. These oscillators give their entire
energy to the electromagnetic field, creating
a quantum of light, as often as they are excited
by the electromagnetic field, absorbing a
quantum of light and beginning to oscillate
at the corresponding frequency. Planck had
intentionally created an atomic theory of
the black body, but had unintentionally generated
an atomic theory of light, where the black
body never generates quanta of light at a
given frequency with an energy less than hν.
However, once realizing that he had quantized
the electromagnetic field, he denounced particles
of light as a limitation of his approximation,
not a property of reality.
=== Photoelectric effect ===
While Planck had solved the ultraviolet catastrophe
by using atoms and a quantized electromagnetic
field, most contemporary physicists agreed
that Planck's "light quanta" represented only
flaws in his model. A more-complete derivation
of black body radiation would yield a fully
continuous and "wave-like" electromagnetic
field with no quantization. However, in 1905
Albert Einstein took Planck's black body model
to produce his solution to another outstanding
problem of the day: the photoelectric effect,
wherein electrons are emitted from atoms when
they absorb energy from light. Since their
existence was theorized eight years previously,
phenomena had been studied with the electron
model in mind in physics laboratories worldwide.
In 1902 Philipp Lenard discovered that the
energy of these ejected electrons did not
depend on the intensity of the incoming light,
but instead on its frequency. So if one shines
a little low-frequency light upon a metal,
a few low energy electrons are ejected. If
one now shines a very intense beam of low-frequency
light upon the same metal, a whole slew of
electrons are ejected; however they possess
the same low energy, there are merely more
of them. The more light there is, the more
electrons are ejected. Whereas in order to
get high energy electrons, one must illuminate
the metal with high-frequency light. Like
blackbody radiation, this was at odds with
a theory invoking continuous transfer of energy
between radiation and matter. However, it
can still be explained using a fully classical
description of light, as long as matter is
quantum mechanical in nature.If one used Planck's
energy quanta, and demanded that electromagnetic
radiation at a given frequency could only
transfer energy to matter in integer multiples
of an energy quantum hν, then the photoelectric
effect could be explained very simply. Low-frequency
light only ejects low-energy electrons because
each electron is excited by the absorption
of a single photon. Increasing the intensity
of the low-frequency light (increasing the
number of photons) only increases the number
of excited electrons, not their energy, because
the energy of each photon remains low. Only
by increasing the frequency of the light,
and thus increasing the energy of the photons,
can one eject electrons with higher energy.
Thus, using Planck's constant h to determine
the energy of the photons based upon their
frequency, the energy of ejected electrons
should also increase linearly with frequency,
the gradient of the line being Planck's constant.
These results were not confirmed until 1915,
when Robert Andrews Millikan produced experimental
results in perfect accord with Einstein's
predictions.
While energy of ejected electrons reflected
Planck's constant, the existence of photons
was not explicitly proven until the discovery
of the photon antibunching effect, of which
a modern experiment can be performed in undergraduate-level
labs. This phenomenon could only be explained
via photons. Einstein's "light quanta" would
not be called photons until 1925, but even
in 1905 they represented the quintessential
example of wave-particle duality. Electromagnetic
radiation propagates following linear wave
equations, but can only be emitted or absorbed
as discrete elements, thus acting as a wave
and a particle simultaneously.
==== Einstein's explanation of photoelectric
effect ====
In 1905, Albert Einstein provided an explanation
of the photoelectric effect, an experiment
that the wave theory of light failed to explain.
He did so by postulating the existence of
photons, quanta of light energy with particulate
qualities.
In the photoelectric effect, it was observed
that shining a light on certain metals would
lead to an electric current in a circuit.
Presumably, the light was knocking electrons
out of the metal, causing current to flow.
However, using the case of potassium as an
example, it was also observed that while a
dim blue light was enough to cause a current,
even the strongest, brightest red light available
with the technology of the time caused no
current at all. According to the classical
theory of light and matter, the strength or
amplitude of a light wave was in proportion
to its brightness: a bright light should have
been easily strong enough to create a large
current. Yet, oddly, this was not so.
Einstein explained this enigma by postulating
that the electrons can receive energy from
electromagnetic field only in discrete units
(quanta or photons): an amount of energy E
that was related to the frequency f of the
light by
E
=
h
f
{\displaystyle E=hf\,}
where h is Planck's constant (6.626 × 10−34
Js). Only photons of a high enough frequency
(above a certain threshold value) could knock
an electron free. For example, photons of
blue light had sufficient energy to free an
electron from the metal, but photons of red
light did not. One photon of light above the
threshold frequency could release only one
electron; the higher the frequency of a photon,
the higher the kinetic energy of the emitted
electron, but no amount of light below the
threshold frequency could release an electron.
To violate this law would require extremely
high-intensity lasers that had not yet been
invented. Intensity-dependent phenomena have
now been studied in detail with such lasers.Einstein
was awarded the Nobel Prize in Physics in
1921 for his discovery of the law of the photoelectric
effect.
=== de Broglie's hypothesis ===
In 1924, Louis-Victor de Broglie formulated
the de Broglie hypothesis, claiming that all
matter has a wave-like nature, he related
wavelength and momentum:
λ
=
h
p
{\displaystyle \lambda ={\frac {h}{p}}}
This is a generalization of Einstein's equation
above, since the momentum of a photon is given
by p =
E
c
{\displaystyle {\tfrac {E}{c}}}
and the wavelength (in a vacuum) by λ =
c
f
{\displaystyle {\tfrac {c}{f}}}
, where c is the speed of light in vacuum.
De Broglie's formula was confirmed three years
later for electrons with the observation of
electron diffraction in two independent experiments.
At the University of Aberdeen, George Paget
Thomson passed a beam of electrons through
a thin metal film and observed the predicted
interference patterns. At Bell Labs, Clinton
Joseph Davisson and Lester Halbert Germer
guided the electron beam through a crystalline
grid in their experiment popularly known as
Davisson–Germer experiment.
De Broglie was awarded the Nobel Prize for
Physics in 1929 for his hypothesis. Thomson
and Davisson shared the Nobel Prize for Physics
in 1937 for their experimental work.
=== Heisenberg's Uncertainty principle ===
In his work on formulating quantum mechanics,
Werner Heisenberg postulated his uncertainty
principle, which states:
Δ
x
Δ
p
≥
1
2
ℏ
{\displaystyle \Delta x\,\Delta p\geq {\tfrac
{1}{2}}\hbar }
where
Δ
{\displaystyle \Delta }
here indicates standard deviation, a measure
of spread or uncertainty;
x and p are a particle's position and linear
momentum respectively.
ℏ
{\displaystyle \hbar }
is the reduced Planck's constant (Planck's
constant divided by 2
π
{\displaystyle \pi }
).Heisenberg originally explained this as
a consequence of the process of measuring:
Measuring position accurately would disturb
momentum and vice versa, offering an example
(the "gamma-ray microscope") that depended
crucially on the de Broglie hypothesis. The
thought is now, however, that this only partly
explains the phenomenon, but that the uncertainty
also exists in the particle itself, even before
the measurement is made.
In fact, the modern explanation of the uncertainty
principle, extending the Copenhagen interpretation
first put forward by Bohr and Heisenberg,
depends even more centrally on the wave nature
of a particle. Just as it is nonsensical to
discuss the precise location of a wave on
a string, particles do not have perfectly
precise positions; likewise, just as it is
nonsensical to discuss the wavelength of a
"pulse" wave traveling down a string, particles
do not have perfectly precise momenta that
corresponds to the inverse of wavelength.
Moreover, when position is relatively well
defined, the wave is pulse-like and has a
very ill-defined wavelength, and thus momentum.
And conversely, when momentum, and thus wavelength,
is relatively well defined, the wave looks
long and sinusoidal, and therefore it has
a very ill-defined position.
=== de Broglie–Bohm theory ===
De Broglie himself had proposed a pilot wave
construct to explain the observed wave-particle
duality. In this view, each particle has a
well-defined position and momentum, but is
guided by a wave function derived from Schrödinger's
equation. The pilot wave theory was initially
rejected because it generated non-local effects
when applied to systems involving more than
one particle. Non-locality, however, soon
became established as an integral feature
of quantum theory and David Bohm extended
de Broglie's model to explicitly include it.
In the resulting representation, also called
the de Broglie–Bohm theory or Bohmian mechanics,
the wave-particle duality vanishes, and explains
the wave behaviour as a scattering with wave
appearance, because the particle's motion
is subject to a guiding equation or quantum
potential.This idea seems to me so natural
and simple, to resolve the wave–particle
dilemma in such a clear and ordinary way,
that it is a great mystery to me that it was
so generally ignored. – J.S.Bell
The best illustration of the pilot-wave model
was given by Couder's 2010 "walking droplets"
experiments, demonstrating the pilot-wave
behaviour in a macroscopic mechanical analog.
== Wave nature of large objects ==
Since the demonstrations of wave-like properties
in photons and electrons, similar experiments
have been conducted with neutrons and protons.
Among the most famous experiments are those
of Estermann and Otto Stern in 1929.
Authors of similar recent experiments with
atoms and molecules, described below, claim
that these larger particles also act like
waves.
A dramatic series of experiments emphasizing
the action of gravity in relation to wave–particle
duality was conducted in the 1970s using the
neutron interferometer. Neutrons, one of the
components of the atomic nucleus, provide
much of the mass of a nucleus and thus of
ordinary matter. In the neutron interferometer,
they act as quantum-mechanical waves directly
subject to the force of gravity. While the
results were not surprising since gravity
was known to act on everything, including
light (see tests of general relativity and
the Pound–Rebka falling photon experiment),
the self-interference of the quantum mechanical
wave of a massive fermion in a gravitational
field had never been experimentally confirmed
before.
In 1999, the diffraction of C60 fullerenes
by researchers from the University of Vienna
was reported. Fullerenes are comparatively
large and massive objects, having an atomic
mass of about 720 u. The de Broglie wavelength
of the incident beam was about 2.5 pm, whereas
the diameter of the molecule is about 1 nm,
about 400 times larger. In 2012, these far-field
diffraction experiments could be extended
to phthalocyanine molecules and their heavier
derivatives, which are composed of 58 and
114 atoms respectively. In these experiments
the build-up of such interference patterns
could be recorded in real time and with single
molecule sensitivity.In 2003, the Vienna group
also demonstrated the wave nature of tetraphenylporphyrin—a
flat biodye with an extension of about 2 nm
and a mass of 614 u. For this demonstration
they employed a near-field Talbot Lau interferometer.
In the same interferometer they also found
interference fringes for C60F48., a fluorinated
buckyball with a mass of about 1600 u, composed
of 108 atoms. Large molecules are already
so complex that they give experimental access
to some aspects of the quantum-classical interface,
i.e., to certain decoherence mechanisms. In
2011, the interference of molecules as heavy
as 6910 u could be demonstrated in a Kapitza–Dirac–Talbot–Lau
interferometer. In 2013, the interference
of molecules beyond 10,000 u has been demonstrated.Whether
objects heavier than the Planck mass (about
the weight of a large bacterium) have a de
Broglie wavelength is theoretically unclear
and experimentally unreachable; above the
Planck mass a particle's Compton wavelength
would be smaller than the Planck length and
its own Schwarzschild radius, a scale at which
current theories of physics may break down
or need to be replaced by more general ones.Recently
Couder, Fort, et al. showed that we can use
macroscopic oil droplets on a vibrating surface
as a model of wave–particle duality—localized
droplet creates periodical waves around and
interaction with them leads to quantum-like
phenomena: interference in double-slit experiment,
unpredictable tunneling (depending in complicated
way on practically hidden state of field),
orbit quantization (that particle has to 'find
a resonance' with field perturbations it creates—after
one orbit, its internal phase has to return
to the initial state) and Zeeman effect.
== Importance ==
Wave–particle duality is deeply embedded
into the foundations of quantum mechanics.
In the formalism of the theory, all the information
about a particle is encoded in its wave function,
a complex-valued function roughly analogous
to the amplitude of a wave at each point in
space. This function evolves according to
Schrödinger equation. For particles with
mass this equation has solutions that follow
the form of the wave equation. Propagation
of such waves leads to wave-like phenomena
such as interference and diffraction. Particles
without mass, like photons, have no solutions
of the Schrödinger equation so have another
wave.
The particle-like behaviour is most evident
due to phenomena associated with measurement
in quantum mechanics. Upon measuring the location
of the particle, the particle will be forced
into a more localized state as given by the
uncertainty principle. When viewed through
this formalism, the measurement of the wave
function will randomly lead to wave function
collapse, or rather quantum decoherence, to
a sharply peaked function at some location.
For particles with mass, the likelihood of
detecting the particle at any particular location
is equal to the squared amplitude of the wave
function there. The measurement will return
a well-defined position, and is subject to
Heisenberg's uncertainty principle. A measurement
is only a particular type of interaction where
some data is recorded and the measured quantity
is forced into a particular quantum state.
The act of measurement is therefore not fundamentally
different from any other interaction.
Following the development of quantum field
theory the ambiguity disappeared. The field
permits solutions that follow the wave equation,
which are referred to as the wave functions.
The term particle is used to label the irreducible
representations of the Lorentz group that
are permitted by the field. An interaction
as in a Feynman diagram is accepted as a calculationally
convenient approximation where the outgoing
legs are known to be simplifications of the
propagation and the internal lines are for
some order in an expansion of the field interaction.
Since the field is non-local and quantized,
the phenomena that previously were thought
of as paradoxes are explained. Within the
limits of the wave-particle duality the quantum
field theory gives the same results.
== Visualization ==
There are two ways to visualize the wave-particle
behaviour by the standard model and by the
de Broglie–Bohr theory.
Below is an illustration of wave–particle
duality as it relates to de Broglie's hypothesis
and Heisenberg's Uncertainty principle, in
terms of the position and momentum space wavefunctions
for one spinless particle with mass in one
dimension. These wavefunctions are Fourier
transforms of each other.
The more localized the position-space wavefunction,
the more likely the particle is to be found
with the position coordinates in that region,
and correspondingly the momentum-space wavefunction
is less localized so the possible momentum
components the particle could have are more
widespread.
Conversely the more localized the momentum-space
wavefunction, the more likely the particle
is to be found with those values of momentum
components in that region, and correspondingly
the less localized the position-space wavefunction,
so the position coordinates the particle could
occupy are more widespread.
== Alternative views ==
Wave–particle duality is an ongoing conundrum
in modern physics. Most physicists accept
wave-particle duality as the best explanation
for a broad range of observed phenomena; however,
it is not without controversy. Alternative
views are also presented here. These views
are not generally accepted by mainstream physics,
but serve as a basis for valuable discussion
within the community.
=== 1. Both-particle-and-wave view ===
The pilot wave model, originally developed
by Louis de Broglie and further developed
by David Bohm into the hidden variable theory
proposes that there is no duality, but rather
a system exhibits both particle properties
and wave properties simultaneously, and particles
are guided, in a deterministic fashion, by
the pilot wave (or its "quantum potential"),
which will direct them to areas of constructive
interference in preference to areas of destructive
interference. This idea is held by a significant
minority within the physics community.At least
one physicist considers the "wave-duality"
as not being an incomprehensible mystery.
L.E. Ballentine, Quantum Mechanics, A Modern
Development, p. 4, explains:
When first discovered, particle diffraction
was a source of great puzzlement. Are "particles"
really "waves?" In the early experiments,
the diffraction patterns were detected holistically
by means of a photographic plate, which could
not detect individual particles. As a result,
the notion grew that particle and wave properties
were mutually incompatible, or complementary,
in the sense that different measurement apparatuses
would be required to observe them. That idea,
however, was only an unfortunate generalization
from a technological limitation. Today it
is possible to detect the arrival of individual
electrons, and to see the diffraction pattern
emerge as a statistical pattern made up of
many small spots (Tonomura et al., 1989).
Evidently, quantum particles are indeed particles,
but whose behaviour is very different from
classical physics would have us to expect.
The Afshar experiment (2007) may suggest that
it is possible to simultaneously observe both
wave and particle properties of photons. This
claim is, however, disputed by other scientists.
=== 2. Wave-only view ===
Carver Mead, an American scientist and professor
at Caltech, proposes that the duality can
be replaced by a "wave-only" view. In his
book Collective Electrodynamics: Quantum Foundations
of Electromagnetism (2000), Mead purports
to analyze the behavior of electrons and photons
purely in terms of electron wave functions,
and attributes the apparent particle-like
behavior to quantization effects and eigenstates.
According to reviewer David Haddon:
Mead has cut the Gordian knot of quantum complementarity.
He claims that atoms, with their neutrons,
protons, and electrons, are not particles
at all but pure waves of matter. Mead cites
as the gross evidence of the exclusively wave
nature of both light and matter the discovery
between 1933 and 1996 of ten examples of pure
wave phenomena, including the ubiquitous laser
of CD players, the self-propagating electrical
currents of superconductors, and the Bose–Einstein
condensate of atoms.
Albert Einstein, who, in his search for a
Unified Field Theory, did not accept wave-particle
duality, wrote:
This double nature of radiation (and of material
corpuscles) ... has been interpreted by quantum-mechanics
in an ingenious and amazingly successful fashion.
This interpretation ... appears to me as only
a temporary way out...
The many-worlds interpretation (MWI) is sometimes
presented as a waves-only theory, including
by its originator, Hugh Everett who referred
to MWI as "the wave interpretation".The three
wave hypothesis of R. Horodecki relates the
particle to wave. The hypothesis implies that
a massive particle is an intrinsically spatially,
as well as temporally extended, wave phenomenon
by a nonlinear law.
The deterministic collapse theory considers
collapse and measurement as two independent
physical processes. Collapse occurs when two
wavepackets spatially overlap and satisfy
a mathemetical criterion, which depends on
the parameters of both wavepackets. It is
a contraction to the overlap volume. In a
measurement apparatus one of the two wavepackets
is one of the atomic clusters, which constitute
the apparatus, and the wavepackets collapse
to at most the volume of such a cluster. This
mimics the action of a point particle.
=== 3. Particle-only view ===
Still in the days of the old quantum theory,
a pre-quantum-mechanical version of wave–particle
duality was pioneered by William Duane, and
developed by others including Alfred Landé.
Duane explained diffraction of x-rays by a
crystal in terms solely of their particle
aspect. The deflection of the trajectory of
each diffracted photon was explained as due
to quantized momentum transfer from the spatially
regular structure of the diffracting crystal.
=== 4. Neither-wave-nor-particle view ===
It has been argued that there are never exact
particles or waves, but only some compromise
or intermediate between them. For this reason,
in 1928 Arthur Eddington coined the name "wavicle"
to describe the objects although it is not
regularly used today. One consideration
is that zero-dimensional mathematical points
cannot be observed. Another is that the formal
representation of such points, the Dirac delta
function is unphysical, because it cannot
be normalized. Parallel arguments apply to
pure wave states. Roger Penrose states:
"Such 'position states' are idealized wavefunctions
in the opposite sense from the momentum states.
Whereas the momentum states are infinitely
spread out, the position states are infinitely
concentrated. Neither is normalizable [...]."
=== 5. Relational approach to wave–particle
duality ===
Relational quantum mechanics has been developed
as a point of view that regards the event
of particle detection as having established
a relationship between the quantized field
and the detector. The inherent ambiguity associated
with applying Heisenberg’s uncertainty principle
is consequently avoided; hence there is no
wave-particle duality.
== Uses ==
Although it is difficult to draw a line separating
wave–particle duality from the rest of quantum
mechanics, it is nevertheless possible to
list some applications of this basic idea.
Wave–particle duality is exploited in electron
microscopy, where the small wavelengths associated
with the electron can be used to view objects
much smaller than what is visible using visible
light.
Similarly, neutron diffraction uses neutrons
with a wavelength of about 0.1 nm, the typical
spacing of atoms in a solid, to determine
the structure of solids.
Photos are now able to show this dual nature,
which may lead to new ways of examining and
recording this behaviour.
== See also
