Quantum mechanics is the science of the very
small. It explains the behavior of matter
and its interactions with energy on the scale
of atoms and subatomic particles. By contrast,
classical physics only explains matter and
energy on a scale familiar to human experience,
including the behavior of astronomical bodies
such as the Moon. Classical physics is still
used in much of modern science and technology.
However, towards the end of the 19th century,
scientists discovered phenomena in both the
large (macro) and the small (micro) worlds
that classical physics could not explain.
The desire to resolve inconsistencies between
observed phenomena and classical theory led
to two major revolutions in physics that created
a shift in the original scientific paradigm:
the theory of relativity and the development
of quantum mechanics. This article describes
how physicists discovered the limitations
of classical physics and developed the main
concepts of the quantum theory that replaced
it in the early decades of the 20th century.
It describes these concepts in roughly the
order in which they were first discovered.
For a more complete history of the subject,
see History of quantum mechanics.
Light behaves in some aspects like particles
and in other aspects like waves. Matter—the
"stuff" of the universe consisting of particles
such as electrons and atoms—exhibits wavelike
behavior too. Some light sources, such as
neon lights, give off only certain frequencies
of light. Quantum mechanics shows that light,
along with all other forms of electromagnetic
radiation, comes in discrete units, called
photons, and predicts its energies, colors,
and spectral intensities. A single photon
is a quantum, or smallest observable amount,
of the electromagnetic field because a partial
photon has never been observed. More broadly,
quantum mechanics shows that many quantities,
such as angular momentum, that appeared continuous
in the zoomed-out view of classical mechanics,
turn out to be (at the small, zoomed-in scale
of quantum mechanics) quantized. Angular momentum
is required to take on one of a set of discrete
allowable values, and since the gap between
these values is so minute, the discontinuity
is only apparent at the atomic level.
Many aspects of quantum mechanics are counterintuitive
and can seem paradoxical, because they describe
behavior quite different from that seen at
larger length scales. In the words of quantum
physicist Richard Feynman, quantum mechanics
deals with "nature as She is – absurd".
For example, the uncertainty principle of
quantum mechanics means that the more closely
one pins down one measurement (such as the
position of a particle), the less accurate
another measurement pertaining to the same
particle (such as its momentum) must become.
== The first quantum theory: Max Planck and
black-body radiation ==
Thermal radiation is electromagnetic radiation
emitted from the surface of an object due
to the object's internal energy. If an object
is heated sufficiently, it starts to emit
light at the red end of the spectrum, as it
becomes red hot.
Heating it further causes the colour to change
from red to yellow, white, and blue, as it
emits light at increasingly shorter wavelengths
(higher frequencies). A perfect emitter is
also a perfect absorber: when it is cold,
such an object looks perfectly black, because
it absorbs all the light that falls on it
and emits none. Consequently, an ideal thermal
emitter is known as a black body, and the
radiation it emits is called black-body radiation.
In the late 19th century, thermal radiation
had been fairly well characterized experimentally.
However, classical physics led to the Rayleigh-Jeans
law, which, as shown in the figure, agrees
with experimental results well at low frequencies,
but strongly disagrees at high frequencies.
Physicists searched for a single theory that
explained all the experimental results.
The first model that was able to explain the
full spectrum of thermal radiation was put
forward by Max Planck in 1900. He proposed
a mathematical model in which the thermal
radiation was in equilibrium with a set of
harmonic oscillators. To reproduce the experimental
results, he had to assume that each oscillator
emitted an integer number of units of energy
at its single characteristic frequency, rather
than being able to emit any arbitrary amount
of energy. In other words, the energy emitted
by an oscillator was quantized. The quantum
of energy for each oscillator, according to
Planck, was proportional to the frequency
of the oscillator; the constant of proportionality
is now known as the Planck constant. The Planck
constant, usually written as h, has the value
of 6.63×10−34 J s. So, the energy E of
an oscillator of frequency f is given by
E
=
n
h
f
,
where
n
=
1
,
2
,
3
,
…
{\displaystyle E=nhf,\quad {\text{where}}\quad
n=1,2,3,\ldots }
To change the color of such a radiating body,
it is necessary to change its temperature.
Planck's law explains why: increasing the
temperature of a body allows it to emit more
energy overall, and means that a larger proportion
of the energy is towards the violet end of
the spectrum.
Planck's law was the first quantum theory
in physics, and Planck won the Nobel Prize
in 1918 "in recognition of the services he
rendered to the advancement of Physics by
his discovery of energy quanta". At the time,
however, Planck's view was that quantization
was purely a heuristic mathematical construct,
rather than (as is now believed) a fundamental
change in our understanding of the world.
== Photons: the quantisation of light ==
In 1905, Albert Einstein took an extra step.
He suggested that quantisation was not just
a mathematical construct, but that the energy
in a beam of light actually occurs in individual
packets, which are now called photons. The
energy of a single photon is given by its
frequency multiplied by Planck's constant:
E
=
h
f
{\displaystyle E=hf}
For centuries, scientists had debated between
two possible theories of light: was it a wave
or did it instead comprise a stream of tiny
particles? By the 19th century, the debate
was generally considered to have been settled
in favor of the wave theory, as it was able
to explain observed effects such as refraction,
diffraction, interference and polarization.
James Clerk Maxwell had shown that electricity,
magnetism and light are all manifestations
of the same phenomenon: the electromagnetic
field. Maxwell's equations, which are the
complete set of laws of classical electromagnetism,
describe light as waves: a combination of
oscillating electric and magnetic fields.
Because of the preponderance of evidence in
favor of the wave theory, Einstein's ideas
were met initially with great skepticism.
Eventually, however, the photon model became
favored. One of the most significant pieces
of evidence in its favor was its ability to
explain several puzzling properties of the
photoelectric effect, described in the following
section. Nonetheless, the wave analogy remained
indispensable for helping to understand other
characteristics of light: diffraction, refraction
and interference.
=== The photoelectric effect ===
In 1887, Heinrich Hertz observed that when
light with sufficient frequency hits a metallic
surface, it emits electrons. In 1902, Philipp
Lenard discovered that the maximum possible
energy of an ejected electron is related to
the frequency of the light, not to its intensity:
if the frequency is too low, no electrons
are ejected regardless of the intensity. Strong
beams of light toward the red end of the spectrum
might produce no electrical potential at all,
while weak beams of light toward the violet
end of the spectrum would produce higher and
higher voltages. The lowest frequency of light
that can cause electrons to be emitted, called
the threshold frequency, is different for
different metals. This observation is at odds
with classical electromagnetism, which predicts
that the electron's energy should be proportional
to the intensity of the radiation. So when
physicists first discovered devices exhibiting
the photoelectric effect, they initially expected
that a higher intensity of light would produce
a higher voltage from the photoelectric device.
Einstein explained the effect by postulating
that a beam of light is a stream of particles
("photons") and that, if the beam is of frequency
f, then each photon has an energy equal to
hf. An electron is likely to be struck only
by a single photon, which imparts at most
an energy hf to the electron. Therefore, the
intensity of the beam has no effect and only
its frequency determines the maximum energy
that can be imparted to the electron.To explain
the threshold effect, Einstein argued that
it takes a certain amount of energy, called
the work function and denoted by φ, to remove
an electron from the metal. This amount of
energy is different for each metal. If the
energy of the photon is less than the work
function, then it does not carry sufficient
energy to remove the electron from the metal.
The threshold frequency, f0, is the frequency
of a photon whose energy is equal to the work
function:
φ
=
h
f
0
.
{\displaystyle \varphi =hf_{0}.}
If f is greater than f0, the energy hf is
enough to remove an electron. The ejected
electron has a kinetic energy, EK, which is,
at most, equal to the photon's energy minus
the energy needed to dislodge the electron
from 
the metal:
E
K
=
h
f
−
φ
=
h
(
f
−
f
0
)
.
{\displaystyle E_{K}=hf-\varphi =h(f-f_{0}).}
Einstein's description of light as being composed
of particles extended Planck's notion of quantised
energy, which is that a single photon of a
given frequency, f, delivers an invariant
amount of energy, hf. In other words, individual
photons can deliver more or less energy, but
only depending on their frequencies. In nature,
single photons are rarely encountered. The
Sun and emission sources available in the
19th century emit vast numbers of photons
every second, and so the importance of the
energy carried by each individual photon was
not obvious. Einstein's idea that the energy
contained in individual units of light depends
on their frequency made it possible to explain
experimental results that had hitherto seemed
quite counterintuitive. However, although
the photon is a particle, it was still being
described as having the wave-like property
of frequency. Effectively, the account of
light as a particle is insufficient, and its
wave-like nature is still required.
=== Consequences of light being quantised
===
The relationship between the frequency of
electromagnetic radiation and the energy of
each individual photon is why ultraviolet
light can cause sunburn, but visible or infrared
light cannot. A photon of ultraviolet light
delivers a high amount of energy—enough
to contribute to cellular damage such as occurs
in a sunburn. A photon of infrared light delivers
less energy—only enough to warm one's skin.
So, an infrared lamp can warm a large surface,
perhaps large enough to keep people comfortable
in a cold room, but it cannot give anyone
a sunburn.All photons of the same frequency
have identical energy, and all photons of
different frequencies have proportionally
(order 1, Ephoton = hf ) different energies.
However, although the energy imparted by photons
is invariant at any given frequency, the initial
energy state of the electrons in a photoelectric
device prior to absorption of light is not
necessarily uniform. Anomalous results may
occur in the case of individual electrons.
For instance, an electron that was already
excited above the equilibrium level of the
photoelectric device might be ejected when
it absorbed uncharacteristically low frequency
illumination. Statistically, however, the
characteristic behavior of a photoelectric
device reflects the behavior of the vast majority
of its electrons, which are at their equilibrium
level. This point is helpful in comprehending
the distinction between the study of individual
particles in quantum dynamics and the study
of massed particles in classical physics.
== The quantisation of matter: the Bohr model
of the atom ==
By the dawn of the 20th century, evidence
required a model of the atom with a diffuse
cloud of negatively charged electrons surrounding
a small, dense, positively charged nucleus.
These properties suggested a model in which
electrons circle around the nucleus like planets
orbiting a sun. However, it was also known
that the atom in this model would be unstable:
according to classical theory, orbiting electrons
are undergoing centripetal acceleration, and
should therefore give off electromagnetic
radiation, the loss of energy also causing
them to spiral toward the nucleus, colliding
with it in a fraction of a second.
A second, related, puzzle was the emission
spectrum of atoms. When a gas is heated, it
gives off light only at discrete frequencies.
For example, the visible light given off by
hydrogen consists of four different colors,
as shown in the picture below. The intensity
of the light at different frequencies is also
different. By contrast, white light consists
of a continuous emission across the whole
range of visible frequencies. By the end of
the nineteenth century, a simple rule known
as Balmer's formula showed how the frequencies
of the different lines related to each other,
though without explaining why this was, or
making any prediction about the intensities.
The formula also predicted some additional
spectral lines in ultraviolet and infrared
light that had not been observed at the time.
These lines were later observed experimentally,
raising confidence in the value of the formula.
In 1913 Niels Bohr proposed a new model of
the atom that included quantized electron
orbits: electrons still orbit the nucleus
much as planets orbit around the sun, but
they are only permitted to inhabit certain
orbits, not to orbit at any distance. When
an atom emitted (or absorbed) energy, the
electron did not move in a continuous trajectory
from one orbit around the nucleus to another,
as might be expected classically. Instead,
the electron would jump instantaneously from
one orbit to another, giving off the emitted
light in the form of a photon. The possible
energies of photons given off by each element
were determined by the differences in energy
between the orbits, and so the emission spectrum
for each element would contain a number of
lines.
Starting from only one simple assumption about
the rule that the orbits must obey, the Bohr
model was able to relate the observed spectral
lines in the emission spectrum of hydrogen
to previously known constants. In Bohr's model
the electron simply wasn't allowed to emit
energy continuously and crash into the nucleus:
once it was in the closest permitted orbit,
it was stable forever. Bohr's model didn't
explain why the orbits should be quantised
in that way, nor was it able to make accurate
predictions for atoms with more than one electron,
or to explain why some spectral lines are
brighter than others.
Some fundamental assumptions of the Bohr model
were soon proven wrong—but the key result
that the discrete lines in emission spectra
are due to some property of the electrons
in atoms being quantised is correct. The way
that the electrons actually behave is strikingly
different from Bohr's atom, and from what
we see in the world of our everyday experience;
this modern quantum mechanical model of the
atom is discussed below.
== Wave-particle duality ==
Just as light has both wave-like and particle-like
properties, matter also has wave-like properties.Matter
behaving as a wave was first demonstrated
experimentally for electrons: a beam of electrons
can exhibit diffraction, just like a beam
of light or a water wave. Similar wave-like
phenomena were later shown for atoms and even
molecules.
The wavelength, λ, associated with any object
is related to its momentum, p, through the
Planck constant, h:
p
=
h
λ
.
{\displaystyle p={\frac {h}{\lambda }}.}
The relationship, called the de Broglie hypothesis,
holds for all types of matter: all matter
exhibits properties of both particles and
waves.
The concept of wave–particle duality says
that neither the classical concept of "particle"
nor of "wave" can fully describe the behavior
of quantum-scale objects, either photons or
matter. Wave–particle duality is an example
of the principle of complementarity in quantum
physics. An elegant example of wave–particle
duality, the double slit experiment, is discussed
in the section below.
=== The double-slit experiment ===
In the double-slit experiment, as originally
performed by Thomas Young and Augustin Fresnel
in 1827, a beam of light is directed through
two narrow, closely spaced slits, producing
an interference pattern of light and dark
bands on a screen. If one of the slits is
covered up, one might naively expect that
the intensity of the fringes due to interference
would be halved everywhere. In fact, a much
simpler pattern is seen, a simple diffraction
pattern. Closing one slit results in a much
simpler pattern diametrically opposite the
open slit. Exactly the same behavior can be
demonstrated in water waves, and so the double-slit
experiment was seen as a demonstration of
the wave nature of light.
Variations of the double-slit experiment have
been performed using electrons, atoms, and
even large molecules, and the same type of
interference pattern is seen. Thus it has
been demonstrated that all matter possesses
both particle and wave characteristics.
Even if the source intensity is turned down,
so that only one particle (e.g. photon or
electron) is passing through the apparatus
at a time, the same interference pattern develops
over time. The quantum particle acts as a
wave when passing through the double slits,
but as a particle when it is detected. This
is a typical feature of quantum complementarity:
a quantum particle acts as a wave in an experiment
to measure its wave-like properties, and like
a particle in an experiment to measure its
particle-like properties. The point on the
detector screen where any individual particle
shows up is the result of a random process.
However, the distribution pattern of many
individual particles mimics the diffraction
pattern produced by waves.
=== Application to the Bohr model ===
De Broglie expanded the Bohr model of the
atom by showing that an electron in orbit
around a nucleus could be thought of as having
wave-like properties. In particular, an electron
is observed only in situations that permit
a standing wave around a nucleus. An example
of a standing wave is a violin string, which
is fixed at both ends and can be made to vibrate.
The waves created by a stringed instrument
appear to oscillate in place, moving from
crest to trough in an up-and-down motion.
The wavelength of a standing wave is related
to the length of the vibrating object and
the boundary conditions. For example, because
the violin string is fixed at both ends, it
can carry standing waves of wavelengths
2
l
n
{\displaystyle {\frac {2l}{n}}}
, where l is the length and n is a positive
integer. De Broglie suggested that the allowed
electron orbits were those for which the circumference
of the orbit would be an integer number of
wavelengths. The electron's wavelength therefore
determines that only Bohr orbits of certain
distances from the nucleus are possible. In
turn, at any distance from the nucleus smaller
than a certain value it would be impossible
to establish an orbit. The minimum possible
distance from the nucleus is called the Bohr
radius.De Broglie's treatment of quantum events
served as a starting point for Schrödinger
when he set out to construct a wave equation
to describe quantum theoretical events.
== Spin ==
In 1922, Otto Stern and Walther Gerlach shot
silver atoms through an (inhomogeneous) magnetic
field. In classical mechanics, a magnet thrown
through a magnetic field may be, depending
on its orientation (if it is pointing with
its northern pole upwards or down, or somewhere
in between), deflected a small or large distance
upwards or downwards. The atoms that Stern
and Gerlach shot through the magnetic field
acted in a similar way. However, while the
magnets could be deflected variable distances,
the atoms would always be deflected a constant
distance either up or down. This implied that
the property of the atom that corresponds
to the magnet's orientation must be quantised,
taking one of two values (either up or down),
as opposed to being chosen freely from any
angle.
Ralph Kronig originated the theory that particles
such as atoms or electrons behave as if they
rotate, or "spin", about an axis. Spin would
account for the missing magnetic moment, and
allow two electrons in the same orbital to
occupy distinct quantum states if they "spun"
in opposite directions, thus satisfying the
exclusion principle. The quantum number represented
the sense (positive or negative) of spin.
The choice of orientation of the magnetic
field used in the Stern-Gerlach experiment
is arbitrary. In the animation shown here,
the field is vertical and so the atoms are
deflected either up or down. If the magnet
is rotated a quarter turn, the atoms are deflected
either left or right. Using a vertical field
shows that the spin along the vertical axis
is quantised, and using a horizontal field
shows that the spin along the horizontal axis
is quantised.
If, instead of hitting a detector screen,
one of the beams of atoms coming out of the
Stern-Gerlach apparatus is passed into another
(inhomogeneous) magnetic field oriented in
the same direction, all of the atoms are deflected
the same way in this second field. However,
if the second field is oriented at 90° to
the first, then half of the atoms are deflected
one way and half the other, so that the atom's
spin about the horizontal and vertical axes
are independent of each other. However, if
one of these beams (e.g. the atoms that were
deflected up then left) is passed into a third
magnetic field, oriented the same way as the
first, half of the atoms go one way and half
the other, even though they all went in the
same direction originally. The action of measuring
the atoms' spin with respect to a horizontal
field has changed their spin with respect
to a vertical field.
The Stern-Gerlach experiment demonstrates
a number of important features of quantum
mechanics:
a feature of the natural world has been demonstrated
to be quantised, and only able to take certain
discrete values
particles possess an intrinsic angular momentum
that is closely analogous to the angular momentum
of a classically spinning object
measurement changes the system being measured
in quantum mechanics. Only the spin of an
object in one direction can be known, and
observing the spin in another direction destroys
the original information about the spin.
quantum mechanics is probabilistic: whether
the spin of any individual atom sent into
the apparatus is positive or negative is random.
== Development of modern quantum mechanics
==
In 1925, Werner Heisenberg attempted to solve
one of the problems that the Bohr model left
unanswered, explaining the intensities of
the different lines in the hydrogen emission
spectrum. Through a series of mathematical
analogies, he wrote out the quantum mechanical
analogue for the classical computation of
intensities. Shortly afterwards, Heisenberg's
colleague Max Born realised that Heisenberg's
method of calculating the probabilities for
transitions between the different energy levels
could best be expressed by using the mathematical
concept of matrices.In the same year, building
on de Broglie's hypothesis, Erwin Schrödinger
developed the equation that describes the
behavior of a quantum mechanical wave. The
mathematical model, called the Schrödinger
equation after its creator, is central to
quantum mechanics, defines the permitted stationary
states of a quantum system, and describes
how the quantum state of a physical system
changes in time. The wave itself is described
by a mathematical function known as a "wave
function". Schrödinger said that the wave
function provides the "means for predicting
probability of measurement results".Schrödinger
was able to calculate the energy levels of
hydrogen by treating a hydrogen atom's electron
as a classical wave, moving in a well of electrical
potential created by the proton. This calculation
accurately reproduced the energy levels of
the Bohr model.
In May 1926, Schrödinger proved that Heisenberg's
matrix mechanics and his own wave mechanics
made the same predictions about the properties
and behavior of the electron; mathematically,
the two theories had an underlying common
form. Yet the two men disagreed on the interpretation
of their mutual theory. For instance, Heisenberg
accepted the theoretical prediction of jumps
of electrons between orbitals in an atom,
but Schrödinger hoped that a theory based
on continuous wave-like properties could avoid
what he called (as paraphrased by Wilhelm
Wien) "this nonsense about quantum jumps."
== 
Copenhagen interpretation ==
Bohr, Heisenberg and others tried to explain
what these experimental results and mathematical
models really mean. Their description, known
as the Copenhagen interpretation of quantum
mechanics, aimed to describe the nature of
reality that was being probed by the measurements
and described by the mathematical formulations
of quantum mechanics.
The main principles of the Copenhagen interpretation
are:
A system is completely described by a wave
function, usually represented by the Greek
letter
ψ
{\displaystyle \psi }
("psi"). (Heisenberg)
How
ψ
{\displaystyle \psi }
changes over time is given by the Schrödinger
equation.
The description of nature is essentially probabilistic.
The probability of an event – for example,
where on the screen a particle shows up in
the double-slit experiment – is related
to the square of the absolute value of the
amplitude of its wave function. (Born rule,
due to Max Born, which gives a physical meaning
to the wave function in the Copenhagen interpretation:
the probability amplitude)
It is not possible to know the values of all
of the properties of the system at the same
time; those properties that are not known
with precision must be described by probabilities.
(Heisenberg's uncertainty principle)
Matter, like energy, exhibits a wave–particle
duality. An experiment can demonstrate the
particle-like properties of matter, or its
wave-like properties; but not both at the
same time. (Complementarity principle due
to Bohr)
Measuring devices are essentially classical
devices, and measure classical properties
such as position and momentum.
The quantum mechanical description of large
systems should closely approximate the classical
description. (Correspondence principle of
Bohr and Heisenberg)Various consequences of
these principles are discussed in more detail
in the following subsections.
=== Uncertainty principle ===
Suppose it is desired to measure the position
and speed of an object – for example a car
going through a radar speed trap. It can be
assumed that the car has a definite position
and speed at a particular moment in time.
How accurately these values can be measured
depends on the quality of the measuring equipment.
If the precision of the measuring equipment
is improved, it provides a result closer to
the true value. It might be assumed that the
speed of the car and its position could be
operationally defined and measured simultaneously,
as precisely as might be desired.
In 1927, Heisenberg proved that this last
assumption is not correct. Quantum mechanics
shows that certain pairs of physical properties,
such as for example position and speed, cannot
be simultaneously measured, nor defined in
operational terms, to arbitrary precision:
the more precisely one property is measured,
or defined in operational terms, the less
precisely can the other. This statement is
known as the uncertainty principle. The uncertainty
principle isn't only a statement about the
accuracy of our measuring equipment, but,
more deeply, is about the conceptual nature
of the measured quantities – the assumption
that the car had simultaneously defined position
and speed does not work in quantum mechanics.
On a scale of cars and people, these uncertainties
are negligible, but when dealing with atoms
and electrons they become critical.Heisenberg
gave, as an illustration, the measurement
of the position and momentum of an electron
using a photon of light. In measuring the
electron's position, the higher the frequency
of the photon, the more accurate is the measurement
of the position of the impact of the photon
with the electron, but the greater is the
disturbance of the electron. This is because
from the impact with the photon, the electron
absorbs a random amount of energy, rendering
the measurement obtained of its momentum increasingly
uncertain (momentum is velocity multiplied
by mass), for one is necessarily measuring
its post-impact disturbed momentum from the
collision products and not its original momentum.
With a photon of lower frequency, the disturbance
(and hence uncertainty) in the momentum is
less, but so is the accuracy of the measurement
of the position of the impact.The uncertainty
principle shows mathematically that the product
of the uncertainty in the position and momentum
of a particle (momentum is velocity multiplied
by mass) could never be less than a certain
value, and that this value is related to Planck's
constant.
=== Wave function collapse ===
Wave function collapse is a forced expression
for whatever just happened when it becomes
appropriate to replace the description of
an uncertain state of a system by a description
of the system in a definite state. Explanations
for the nature of the process of becoming
certain are controversial. At any time before
a photon "shows up" on a detection screen
it can only be described by a set of probabilities
for where it might show up. When it does show
up, for instance in the CCD of an electronic
camera, the time and the space where it interacted
with the device are known within very tight
limits. However, the photon has disappeared,
and the wave function has disappeared with
it. In its place some physical change in the
detection screen has appeared, e.g., an exposed
spot in a sheet of photographic film, or a
change in electric potential in some cell
of a CCD.
=== Eigenstates and eigenvalues ===
For a more detailed introduction to this subject,
see: Introduction to eigenstatesBecause of
the uncertainty principle, statements about
both the position and momentum of particles
can only assign a probability that the position
or momentum has some numerical value. Therefore,
it is necessary to formulate clearly the difference
between the state of something that is indeterminate,
such as an electron in a probability cloud,
and the state of something having a definite
value. When an object can definitely be "pinned-down"
in some respect, it is said to possess an
eigenstate.
In the Stern-Gerlach experiment discussed
above, the spin of the atom about the vertical
axis has two eigenstates: up and down. Before
measuring it, we can only say that any individual
atom has equal probability of being found
to have spin up or spin down. The measurement
process causes the wavefunction to collapse
into one of the two states.
The eigenstates of spin about the vertical
axis are not simultaneously eigenstates of
spin about the horizontal axis, so this atom
has equal probability of being found to have
either value of spin about the horizontal
axis. As described in the section above, measuring
the spin about the horizontal axis can allow
an atom that was spun up to spin down: measuring
its spin about the horizontal axis collapses
its wave function into one of the eigenstates
of this measurement, which means it is no
longer in an eigenstate of spin about the
vertical axis, so can take either value.
=== The Pauli exclusion principle ===
In 1924, Wolfgang Pauli proposed a new quantum
degree of freedom (or quantum number), with
two possible values, to resolve inconsistencies
between observed molecular spectra and the
predictions of quantum mechanics. In particular,
the spectrum of atomic hydrogen had a doublet,
or pair of lines differing by a small amount,
where only one line was expected. Pauli formulated
his exclusion principle, stating that "There
cannot exist an atom in such a quantum state
that two electrons within [it] have the same
set of quantum numbers."A year later, Uhlenbeck
and Goudsmit identified Pauli's new degree
of freedom with the property called spin whose
effects were observed in the Stern–Gerlach
experiment.
=== Application to the hydrogen atom ===
Bohr's model of the atom was essentially a
planetary one, with the electrons orbiting
around the nuclear "sun." However, the uncertainty
principle states that an electron cannot simultaneously
have an exact location and velocity in the
way that a planet does. Instead of classical
orbits, electrons are said to inhabit atomic
orbitals. An orbital is the "cloud" of possible
locations in which an electron might be found,
a distribution of probabilities rather than
a precise location. Each orbital is three
dimensional, rather than the two dimensional
orbit, and is often depicted as a three-dimensional
region within which there is a 95 percent
probability of finding the electron.Schrödinger
was able to calculate the energy levels of
hydrogen by treating a hydrogen atom's electron
as a wave, represented by the "wave function"
Ψ, in an electric potential well, V, created
by the proton. The solutions to Schrödinger's
equation are distributions of probabilities
for electron positions and locations. Orbitals
have a range of different shapes in three
dimensions. The energies of the different
orbitals can be calculated, and they accurately
match the energy levels of the Bohr model.
Within Schrödinger's picture, each electron
has four properties:
An "orbital" designation, indicating whether
the particle wave is one that is closer to
the nucleus with less energy or one that is
farther from the nucleus with more energy;
The "shape" of the orbital, spherical or otherwise;
The "inclination" of the orbital, determining
the magnetic moment of the orbital around
the z-axis.
The "spin" of the electron.
The collective name for these properties is
the quantum state of the electron. The quantum
state can be described by giving a number
to each of these properties; these are known
as the electron's quantum numbers. The quantum
state of the electron is described by its
wave function. The Pauli exclusion principle
demands that no two electrons within an atom
may have the same values of all four numbers.
The first property describing the orbital
is the principal quantum number, n, which
is the same as in Bohr's model. n denotes
the energy level of each orbital. The possible
values for n are integers:
n
=
1
,
2
,
3
…
{\displaystyle n=1,2,3\ldots }
The next quantum number, the azimuthal quantum
number, denoted l, describes the shape of
the orbital. The shape is a consequence of
the angular momentum of the orbital. The angular
momentum represents the resistance of a spinning
object to speeding up or slowing down under
the influence of external force. The azimuthal
quantum number represents the orbital angular
momentum of an electron around its nucleus.
The possible values for l are integers from
0 to n − 1 (where n is the principal quantum
number of the electron):
l
=
0
,
1
,
…
,
n
−
1.
{\displaystyle l=0,1,\ldots ,n-1.}
The shape of each orbital is usually referred
to by a letter, rather than by its azimuthal
quantum number. The first shape (l=0) is denoted
by the letter s (a mnemonic being "sphere").
The next shape is denoted by the letter p
and has the form of a dumbbell. The other
orbitals have more complicated shapes (see
atomic orbital), and are denoted by the letters
d, f, g, etc.
The third quantum number, the magnetic quantum
number, describes the magnetic moment of the
electron, and is denoted by ml (or simply
m). The possible values for ml are integers
from −l to l (where l is the azimuthal quantum
number of the electron):
m
l
=
−
l
,
−
(
l
−
1
)
,
…
,
0
,
1
,
…
,
l
.
{\displaystyle m_{l}=-l,-(l-1),\ldots ,0,1,\ldots
,l.}
The magnetic quantum number measures the component
of the angular momentum in a particular direction.
The choice of direction is arbitrary, conventionally
the z-direction is chosen.
The fourth quantum number, the spin quantum
number (pertaining to the "orientation" of
the electron's spin) is denoted ms, with values
+​1⁄2 or −​1⁄2.
The chemist Linus Pauling wrote, by way of
example:
In the case of a helium atom with two electrons
in the 1s orbital, the Pauli Exclusion Principle
requires that the two electrons differ in
the value of one quantum number. Their values
of n, l, and ml are the same. Accordingly
they must differ in the value of ms, which
can have the value of +​1⁄2 for one electron
and −​1⁄2 for the other."
It is the underlying structure and symmetry
of atomic orbitals, and the way that electrons
fill them, that leads to the organisation
of the periodic table. The way the atomic
orbitals on different atoms combine to form
molecular orbitals determines the structure
and strength of chemical bonds between atoms.
== Dirac wave equation ==
In 1928, Paul Dirac extended the Pauli equation,
which described spinning electrons, to account
for special relativity. The result was a theory
that dealt properly with events, such as the
speed at which an electron orbits the nucleus,
occurring at a substantial fraction of the
speed of light. By using the simplest electromagnetic
interaction, Dirac was able to predict the
value of the magnetic moment associated with
the electron's spin, and found the experimentally
observed value, which was too large to be
that of a spinning charged sphere governed
by classical physics. He was able to solve
for the spectral lines of the hydrogen atom,
and to reproduce from physical first principles
Sommerfeld's successful formula for the fine
structure of the hydrogen spectrum.
Dirac's equations sometimes yielded a negative
value for energy, for which he proposed a
novel solution: he posited the existence of
an antielectron and of a dynamical vacuum.
This led to the many-particle quantum field
theory.
== Quantum entanglement ==
The Pauli exclusion principle says that two
electrons in one system cannot be in the same
state. Nature leaves open the possibility,
however, that two electrons can have both
states "superimposed" over each of them. Recall
that the wave functions that emerge simultaneously
from the double slits arrive at the detection
screen in a state of superposition. Nothing
is certain until the superimposed waveforms
"collapse". At that instant an electron shows
up somewhere in accordance with the probability
that is the square of the absolute value of
the sum of the complex-valued amplitudes of
the two superimposed waveforms. The situation
there is already very abstract. A concrete
way of thinking about entangled photons, photons
in which two contrary states are superimposed
on each of them in the same event, is as follows:
Imagine that the superposition of a state
labeled blue, and another state labeled red
then appear (in imagination) as a purple state.
Two photons are produced as the result of
the same atomic event. Perhaps they are produced
by the excitation of a crystal that characteristically
absorbs a photon of a certain frequency and
emits two photons of half the original frequency.
So the two photons come out purple. If the
experimenter now performs some experiment
that determines whether one of the photons
is either blue or red, then that experiment
changes the photon involved from one having
a superposition of blue and red characteristics
to a photon that has only one of those characteristics.
The problem that Einstein had with such an
imagined situation was that if one of these
photons had been kept bouncing between mirrors
in a laboratory on earth, and the other one
had traveled halfway to the nearest star,
when its twin was made to reveal itself as
either blue or red, that meant that the distant
photon now had to lose its purple status too.
So whenever it might be investigated after
its twin had been measured, it would necessarily
show up in the opposite state to whatever
its twin had revealed.
In trying to show that quantum mechanics was
not a complete theory, Einstein started with
the theory's prediction that two or more particles
that have interacted in the past can appear
strongly correlated when their various properties
are later measured. He sought to explain this
seeming interaction in a classical way, through
their common past, and preferably not by some
"spooky action at a distance." The argument
is worked out in a famous paper, Einstein,
Podolsky, and Rosen (1935; abbreviated EPR),
setting out what is now called the EPR paradox.
Assuming what is now usually called local
realism, EPR attempted to show from quantum
theory that a particle has both position and
momentum simultaneously, while according to
the Copenhagen interpretation, only one of
those two properties actually exists and only
at the moment that it is being measured. EPR
concluded that quantum theory is incomplete
in that it refuses to consider physical properties
that objectively exist in nature. (Einstein,
Podolsky, & Rosen 1935 is currently Einstein's
most cited publication in physics journals.)
In the same year, Erwin Schrödinger used
the word "entanglement" and declared: "I would
not call that one but rather the characteristic
trait of quantum mechanics." The question
of whether entanglement is a real condition
is still in dispute. The Bell inequalities
are the most powerful challenge to Einstein's
claims.
== Quantum field theory ==
The idea of quantum field theory began in
the late 1920s with British physicist Paul
Dirac, when he attempted to quantise the electromagnetic
field – a procedure for constructing a quantum
theory starting from a classical theory.
A field in physics is "a region or space in
which a given effect (such as magnetism) exists."
Other effects that manifest themselves as
fields are gravitation and static electricity.
In 2008, physicist Richard Hammond wrote that
Sometimes we distinguish between quantum mechanics
(QM) and quantum field theory (QFT). QM refers
to a system in which the number of particles
is fixed, and the fields (such as the electromechanical
field) are continuous classical entities.
QFT ... goes a step further and allows for
the creation and annihilation of particles
. . . .
He added, however, that quantum mechanics
is often used to refer to "the entire notion
of quantum view".In 1931, Dirac proposed the
existence of particles that later became known
as antimatter. Dirac shared the Nobel Prize
in Physics for 1933 with Schrödinger "for
the discovery of new productive forms of atomic
theory".On its face, quantum field theory
allows infinite numbers of particles, and
leaves it up to the theory itself to predict
how many and with which probabilities or numbers
they should exist. When developed further,
the theory often contradicts observation,
so that its creation and annihilation operators
can be empirically tied down. Furthermore,
empirical conservation laws like that of mass-energy
suggest certain constraints on the mathematical
form of the theory, which are mathematically
speaking finicky. The latter fact both serves
to make quantum field theories difficult to
handle, but has also lead to further restrictions
on admissible forms of the theory; the complications
are mentioned below under the rubrik of renormalization.
== Quantum electrodynamics ==
Quantum electrodynamics (QED) is the name
of the quantum theory of the electromagnetic
force. Understanding QED begins with understanding
electromagnetism. Electromagnetism can be
called "electrodynamics" because it is a dynamic
interaction between electrical and magnetic
forces. Electromagnetism begins with the electric
charge.
Electric charges are the sources of, and create,
electric fields. An electric field is a field
that exerts a force on any particles that
carry electric charges, at any point in space.
This includes the electron, proton, and even
quarks, among others. As a force is exerted,
electric charges move, a current flows and
a magnetic field is produced. The changing
magnetic field, in turn causes electric current
(often moving electrons). The physical description
of interacting charged particles, electrical
currents, electrical fields, and magnetic
fields is called electromagnetism.
In 1928 Paul Dirac produced a relativistic
quantum theory of electromagnetism. This was
the progenitor to modern quantum electrodynamics,
in that it had essential ingredients of the
modern theory. However, the problem of unsolvable
infinities developed in this relativistic
quantum theory. Years later, renormalization
largely solved this problem. Initially viewed
as a suspect, provisional procedure by some
of its originators, renormalization eventually
was embraced as an important and self-consistent
tool in QED and other fields of physics. Also,
in the late 1940s Feynman's diagrams depicted
all possible interactions pertaining to a
given event. The diagrams showed in particular
that the electromagnetic force is the exchange
of photons between interacting particles.The
Lamb shift is an example of a quantum electrodynamics
prediction that has been experimentally verified.
It is an effect whereby the quantum nature
of the electromagnetic field makes the energy
levels in an atom or ion deviate slightly
from what they would otherwise be. As a result,
spectral lines may shift or split.
Similarly, within a freely propagating electromagnetic
wave, the current can also be just an abstract
displacement current, instead of involving
charge carriers. In QED, its full description
makes essential use of short lived virtual
particles. There, QED again validates an earlier,
rather mysterious concept.
== Standard Model ==
In the 1960s physicists realized that QED
broke down at extremely high energies. From
this inconsistency the Standard Model of particle
physics was discovered, which remedied the
higher energy breakdown in theory. It is another
extended quantum field theory that unifies
the electromagnetic and weak interactions
into one theory. This is called the electroweak
theory.
Additionally the Standard Model contains a
high energy unification of the electroweak
theory with the strong force, described by
quantum chromodynamics. It also postulates
a connection with gravity as yet another gauge
theory, but the connection is as of 2015 still
poorly understood. The theory's successful
prediction of the Higgs particle to explain
inertial mass was confirmed by the Large Hadron
Collider, and thus the Standard model is now
considered the basic and more or less complete
description of particle physics as we know
it.
== Interpretations ==
The physical measurements, equations, and
predictions pertinent to quantum mechanics
are all consistent and hold a very high level
of confirmation. However, the question of
what these abstract models say about the underlying
nature of the real world has received competing
answers. These interpretations are widely
varying and sometimes somewhat abstract. For
instance, the Copenhagen interpretation states
that before a measurement, statements about
a particles' properties are completely meaningless,
while in the Many-worlds interpretation describes
the existence of a multiverse made up of every
possible universe.
== Applications ==
Applications of quantum mechanics include
the laser, the transistor, the electron microscope,
and magnetic resonance imaging. A special
class of quantum mechanical applications is
related to macroscopic quantum phenomena such
as superfluid helium and superconductors.
The study of semiconductors led to the invention
of the diode and the transistor, which are
indispensable for modern electronics.
In even the simple light switch, quantum tunnelling
is absolutely vital, as otherwise the electrons
in the electric current could not penetrate
the potential barrier made up of a layer of
oxide. Flash memory chips found in USB drives
also use quantum tunnelling, to erase their
memory cells.
== See also ==
== Notes
