Everyday experience creates preconceptions that fail drastically
when that experience is extended to the very massive and the very
fast, or when extended to the very small and the very cold. The
large scale requires relativity theory, and the small scale
requires
quantum mechanics. Quantum physics deals
with "Nature as She is—absurd."
Quantum physics deals with situations where the usual picture of
reality breaks down.
Photons (discrete units
of light) and other very small things have some behaviors that
resemble classical particles like billiard balls and other
behaviors that resemble classical waves like water waves. Radiators
of photons such as
neon lights have
spectra, but the spectra are chopped up
instead of being continuous. The
energies
carried by photons form a discontinuous and color coded series. The
energies, the colors, and the spectral
intensities of
electromagnetic radiation produced
are all interconnected by
law. But the
same laws ordain that the more closely one pins down one measure
(such as the position of a particle) the more wildly another
measure relating to the same thing must fluctuate (such as
momentum). Put another way, measuring position first and then
measuring momentum is
not the same as measuring momentum
first and then measuring position. Even more disconcerting,
particles can be created as twins and therefore as
entangled entities -- which means that
doing something that pins down one characteristic of one particle
will reveal something about its entangled twin even if it is
millions and millions of miles away.
An elegant example
The most elegant character on the quantum stage is the
double-slit experiment. It
demonstrates particle-wave duality, and it highlights several
features of quantum mechanics.
Photons
emitted from some source such as a laser will behave differently
depending on whether one or two slits lie in their path. When only
one slit is present, the light observed on the screen will appear
as a rather narrow
diffraction
pattern (see the photo below).
However, things get really strange when two slits
are introduced into the experiment. With two slits present, what
arrives at a remote detection screen will be a
superposition of two waves. As the
illustration shows, a wave from the top slit and another from the
bottom slit will fall on top of each other on the detection screen,
and so they are superimposed. The same basic experiment can be done
by shooting an electron at a double slit. The wave nature of light
causes the light waves passing through both slits to interfere with
themselves, creating an interference pattern of bright and dark
bands on the screen. However, at the screen, the light is always
found to be absorbed as discrete particles, called photons.
What is even stranger is what happens when the light source is
"turned down" to where only one photon will be emitted at a time.
Everyday experience leads one to expect that the photon will travel
through one or the other slit as a particle, and strike the screen
as a particle. However, any single photon travels through both
slits as a wave, and creates two wave patterns that interfere with
each other. Even more strangely, that photon is then detected as a
particle on the screen.
Where a photon or electron shows up on the detection screen will
depend on the probabilities as calculated by adding the amplitudes
of the two waves at each point, and squaring that sum. However,
where any individual photon or electron strikes the screen will be
the result of an entirely random process. The end result will be in
accord with the probabilities that can be calculated.
Photons function as though they are waves as they go through the
slits. When two slits are present, the "wave function" pertaining
to each photon goes through each slit. The wave functions are
superimposed all across the detection screen, yet at the detection
screen only one particle, a photon, shows up and its position is in
accord with strict probability laws. So what humans interpret as
the wave nature of photons and as the particle nature of photons
must both appear in the final results. All of these points will be
revisited below.
How the unexpected came to light
A black-body heated to 5000 kelvins
emits different wavelengths of radiation with different intensities
(blue).
Classical electromagnetism drastically overestimates these
intensities (black).
The 1827 experiments of
Thomas
Young and
Augustin Fresnel
demonstrated that
light is subject to
interference patterns. By the end of the
nineteenth century,
classical
electromagnetism treated light as a
wave
phenomenon: a combination of oscillating electric and magnetic
fields.
Around the turn of the twentieth century, it became clear that
classical electromagnetism fails to explain several physical
phenomena. The first is
black-body
radiation, for which classical electromagnetism predicts that a
heated object should produce enormous intensities of high-energy
ultraviolet light. The way out of this
"
ultraviolet catastrophe"
was prepared by
Max Planck, who
determined that the
energy content of
black-body radiation of frequency
f comes as multiples of
a quantum of energy
hf. He was not aware of the
ultraviolet catastrophe, however, and only was trying to fix the
problem he knew about—that the existing models did not yield the
right energy distribution for black-body radiation.
The second unexplainable phenomenon was the
photoelectric effect, wherein light
could sometimes eject electrons from a metal. Contrary to the
predictions of classical mechanics, however, it was not the
intensity but the
frequency of the light that determined if it could
eject electrons. In 1905,
Albert
Einstein successfully explained the effect by positing that
light is composed of particles called
photons.
Around the same time, scientists were trying to understand the
physics of
emission spectra. It
was known that a
gaseous atomic
hydrogen would glow when exposed to electric
discharge, and that this glow was composed of a small number of
wavelengths of light.
Johann Balmer
and
Johannes Rydberg had
empirically determined a formula to fit the observed wavelengths,
but in 1913
Niels Bohr developed a new
theory of the atom to explain the observed emissions. In the
Bohr model, the electron orbits the
nucleus only at certain allowed distances, and with certain allowed
energies. A photon is emitted when the electron jumps from a
higher-energy orbit to a lower-energy orbit.
Planck and the constant h
Max Planck's work established an
empirical equation to represent the observed energy distribution in
the frequencies present in
black-body radiation. According to
Planck's model, the radiating body is composed of an enormous
number of elementary oscillators, each having its own unique
frequency. According to this model, the energy
E emitted
by the black-body must be some integer multiple of
hf,
where
h is the
Planck
constant, a
proportionality
constant equal to , and
f is the frequency of the
light.
In classical physics, and in everyday experience of things like
guitar strings, a vibrating object has an amplitude that is
proportional to the force applied to it, e.g., to how hard the
guitar string is plucked. In his model, Planck's oscillators each
had its fundamental frequency, to that fundamental frequency was
associated a lowest energy given by
hf, and the oscillator
was then said to be able to radiate any integer multiple of that
fundamental energy, yielding the equation
- E = nhf \qquad \qquad n = 1,2,3,\ldots.
In terms of photons radiated by something like a neon light, this
law could be interpreted to explain that the change of energy state
by any electron yields a photon of a frequency and an energy
related to the change of energy of the electron, and that the
energy of the neon light at that frequency is the sum of the
n number of electrons making that change. In other words,
there is no single oscillator emitting energy at multiples of some
lowest energy, but multiples of single oscillators each having a
single energy and frequency. Rather than itself being some lowest
possible amount of energy,
h is a proportionality constant
that properly associates frequencies stated in some unit with
energies stated in some unit. In some unit systems such as
Planck units,
h is numerically equal
to one, and the number representing the frequency is the same as
the number representing the energy. The quantum of energy is then
written
f, and the energy of the radiation is .
The photoelectric effect
In 1887,
Heinrich Hertz observed that
light can eject electrons from metal. In 1902,
Philipp Lenard discovered that the maximum
possible energy of each ejected electron is related to the
frequency of the light, not its intensity. Moreover, if the
frequency of the light is too low, no electrons are ejected. The
lowest frequency which still ejects electrons from a metal is
called the
threshold frequency,
and it is unique to the metal.
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, each photon has an
energy equal to
hf. An electron is likely to be struck
only by a single photon; this photon imparts at most an energy
hf to the electron. Therefore, the intensity of the beam
has no effect Actually there can be intensity-dependent
effects
^{[187762]} , but at light intensities
achievable with non-laser sources these effects are unobservable. ;
only its frequency determines the maximum energy that can be
imparted to the electrons.
To explain the threshold frequency
f_{0}, Einstein
argued that it takes a certain amount of energy
φ ("phi")
simply to remove the electron from the metal. Called the
work function,
φ is
equal to
hf_{0} and is specific to the metal. If
the frequency
f of the photon is less than
f_{0} its energy
hf is less than the
energy
φ, and so the photon does not carry sufficient
energy to remove the electron from the metal. If
f is
greater than
f_{0}, the energy
hf is
enough to remove an electron; an ejected electron has a
kinetic energy K which is at most
equal to the difference of
hf and
φ:
- K = hf - \varphi = hf - hf_0.
For example, a photon of
ultraviolet
light, having a short wavelength, will deliver a high amount of
energy—enough to contribute to cellular
damage such as a sunburn. A photon of
infrared light, having a long wavelength, will
deliver a low amount of energy—only enough to warm one's skin. So a
very large infrared light can warm a large surface, perhaps large
enough to keep people comfortable in a cold room or even make
people too hot, but it cannot give anyone a sunburn. What is needed
for an electron to be freed from its original atom and jump a spark
gap or do something else requiring a certain voltage to be present
is for the frequency of the incident light to be high enough.
Einstein's description of light as being composed of photons
extended Planck's notion of quantized energy: 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. Though the
photon is a particle, it was still said to have the wave-like
property of frequency. Once again, the particle account of light
had been "compromised."
Despite this, though, Einstein's photoelectric effect equation
can be derived and explained
without resorting to
"photons". That is, the electromagnetic radiation can be treated as
a classical electromagnetic wave, as long as the electrons in the
material are treated by the laws of quantum mechanics, and the
results are quantitavely correct for thermal light sources (the
sun, incandescent lamps, etc.) both for the rate of electron
emission as well as their angular distribution. For more on this
point, see
^{[187763]}
Atomic emission spectra
When excited by an electric discharge,
atomic hydrogen produces several wavelengths of light.
By the end of the nineteenth century it was known that atomic
hydrogen would glow in an electric discharge. When light from this
glow was passed through a
diffraction grating, it was found to be
made up of only four wavelengths. These constitute the visible
portion of hydrogen's
emission
spectrum.
The four visible wavelengths of light
in the emission spectrum of atomic hydrogen.
In 1885 the Swedish mathematician
Johann
Balmer discovered that each wavelength
λ in the
visible spectrum of hydrogen is related to some integer
n
by the equation
- \lambda = B\left(\frac{n^2}{n^2-4}\right) \qquad\qquad n =
3,4,5,6
where
B is a constant which Balmer determined to be equal
to 364.56 nm.
In 1888,
Johannes Rydberg
generalized and greatly increased the explanatory utility of
Balmer's formula. He supposed that hydrogen will emit light of
wavelength
λ if
λ is related to two integers
n and
m according to what is now known as the
Rydberg formula:
- \frac{1}{\lambda} = R \left(\frac{1}{m^2} -
\frac{1}{n^2}\right),
where
R is the
Rydberg
constant, equal to 0.0110 nm
^{−1}, and
n must
be greater than
m.
Rydberg's formula accounts for the four visible wavelengths by
setting and . It also predicts additional wavelengths in the
emission spectrum: for and for , the emission spectrum should
contain certain ultraviolet wavelengths, and for and , it should
also contain certain infrared wavelengths. Experimental observation
of these wavelengths came several decades later: in 1908
Louis Paschen found some of the predicted
infrared wavelengths, and in 1914
Theodore Lyman found some of the predicted
ultraviolet wavelengths.
In 1908,
Walter Ritz discovered what has
come to be known as the
Ritz combination
principle that demonstrates how new intervals among frequencies
in a bright line spectrum can be discovered because there are
several differences of frequencies between the energy states (or
orbits) of electrons that keep repeating themselves. This principle
is implicit in Heisenberg's breakthrough formulation of the new
quantum mechanics in 1925.
The Bohr model of the atom
The Bohr model of the atom, showing
electron quantum jumping to ground state .
In 1897, a research team headed by
J J
Thompson discovered and named the
electron, the carrier of negative charge. By means
of the
gold foil
experiment, physicists discovered that matter is mostly empty
space. Once that was clear, it was hypothesized that negatively
charged electrons orbit a positively charged
nucleus, so that all atoms resemble a
miniature solar system. But that simple analogy predicted that
electrons would take only about one hundredth of a microsecond to
crash into the nucleus. Hence the great question of early 20th
century physics was: "How do electrons normally remain in stable
orbits around the nucleus?"
In 1913,
Niels Bohr solved this
substantial problem by applying the notion of discrete
(non-continuous) quanta to electron orbits.Bohr theorized that an
electron can only inhabit certain orbits around the nucleus, and
that they may have only certain energies. He arrived at this
conclusion by assuming that the
angular
momentum L of an electron must be quantized:
- L = n\frac{h}{2\pi},
where
n is a positive integer and
h is the Planck
constant. Together with
Coulomb's law
and the equations of
circular
motion, this assumption implies that an electron orbits a
proton at a distance
r given by
- r = n^2 a_{\mathrm{B}},\!
where
n is a positive integer and
a_{B}
is the
Bohr radius, equal to
0.0529 nm. Additionally, the energy of the electron is given
by
- E = -\frac{k_{\mathrm{e}}e^2}{2a_{\mathrm{B}}}
\frac{1}{n^2},
where
k_{e} is the
Coulomb constant and
e is the
elementary charge.
A consequence of these constraints is that the electron will not
crash into the nucleus: it cannot continuously emit energy, and it
cannot come closer to the nucleus than
a_{B}. An
electron changes its energy by suddenly disappearing from its
original orbit and reappearing in another orbit. An electron that
absorbs a photon gains a quantum of energy, so it jumps to an orbit
that is farther from the nucleus, while an electron that emits a
photon loses a quantum of energy and so jumps to an orbital closer
to the nucleus. An electron cannot gain or lose a fractional
quantum of energy, and hence it cannot be found at some fraction of
the distance between allowed orbits.
With these assumptions, the Bohr model explains why the Rydberg
formula is true: each photon from glowing atomic hydrogen is due to
an electron moving from a higher orbit
r_{n} to a
lower orbit
r_{m}. The energy
E_{γ} of this photon is the difference in the
energies
E_{n} and
E_{m} of the
electron:
- E_{\gamma} = E_n - E_m =
\frac{k_{\mathrm{e}}e^2}{2a_{\mathrm{B}}}\left(\frac{1}{m^2}-\frac{1}{n^2}\right)
Then, since the photon's energy is related to its wavelength by ,
- \frac{1}{\lambda} =
\frac{k_{\mathrm{e}}e^2}{2a_{\mathrm{B}}hc}\left(\frac{1}{m^2}-\frac{1}{n^2}\right).
The quantity
k_{e}e^{2}/2
a_{B}hc
is equal to the Rydberg constant. Therefore, the Bohr model of the
atom explains the emission spectrum of hydrogen in terms of
fundamental constants and the behavior of the electron. The model
can be easily modified to account of the emission spectrum of any
hydrogen-like atom (that is,
ions such as
He
^{+} or O
^{7+} which contain only one
electron).
Refinements were added by other researchers shortly after Bohr's
work appeared.
Arnold Sommerfeld
showed that not all orbits could be perfectly circular, so a new
"atomic number" was added for the shape of the orbit,
k.Sommerfeld also showed that the orientation of orbit
could be influenced by magnetic fields imposed on the radiating
gas, which added a third quantum number,
m.
Bohr's theory represented electrons as orbiting the nucleus of an
atom, much as planets orbit around the sun. However, we now
envision electrons circulating around the nuclei of atoms in a way
that is strikingly different from Bohr's atom, and what we see in
the world of our everyday experience. Instead of orbits, electrons
are said to inhabit "
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.
Bohr's model of the atom was essentially two-dimensional: an
electron orbiting in a plane around its nuclear "sun." Modern
theory describes a three-dimensional arrangement of electronic
shells and orbitals around atomic nuclei. The orbitals are
spherical (s-type) or lobular (p, d and f-types) in shape. It is
the underlying structure and symmetry of atomic orbitals, and the
way that electrons fill them, that determines the structure and
strength of chemical bonds between atoms. Thus, the bizarre quantum
nature of the atomic and sub-atomic world finds natural expression
in the macroscopic world with which we are more familiar.
Consequences of contemporary research
Red: (high amperage) low
voltage. Violet: high voltage.
Other experiments showed that when light shone upon a metal surface
it would drive electrons away. The result can be seen in the light
meters designed for photographic use. A beam of light creates an
electrical potential (a
voltage), and that voltage will cause a
current with a certain
amperage to flow through an external part of the
circuit. In a light meter, the voltage induced by a beam of light
induces an electric current of a certain voltage and amperage that
powers a little
electromagnet that
moves the needle in the light meter. Contrary to classical
electromagnetism, the voltage produced by a beam of light did not
change when the intensity of the beam of light was changed. Only
the current in the circuit changed. But if one substituted beams of
a single frequency, red beams and violet beams of equal intensity
would produce different voltages. These experiments showed that
longer wavelength light produces lower voltage, i.e., it puts less
of a "kick" on individual electrons, and shorter wavelength light
produces a greater force on individual electrons.
Infrared light warms; ultraviolet
penetrates.
The double-slit experiment
Scientists were forced to draw a seemingly very self-contradictory
conclusion. Light behaves like a wave in some situations, and yet
it performs like particles in other situations. The quantum
physicists enunciated the principle of
complementarity, i.e., the idea that light
cannot be adequately characterized by the wave interpretation, but
it also cannot be adequately characterized by the particle
interpretation. One cannot stand without the other, at least not
when talking about things on an atomic scale. In the quantum world,
a photon may be emitted as the result of an interaction within a
single atom, and end up by being absorbed by a single atom in the
detection screen. But where on that detection screen it appears
depends very strongly on whether there is a single path between the
point of origin and the screen, or there are two or more paths. If
there are two or more paths then
something
wavelike passes through both slits and then interferes with itself.
The fact of self interference determines the probabilities of the
photon's potential points of appearance.
The double-slit experiment is a very compelling example of quantum
effects and complementarity. Originally performed by Young and
Fresnel in 1827, a beam of light is directed through two narrow,
closely spaced slits. The two resulting beams
interfere with each other and produce light and
dark bands on a detector—a consequence of the wave nature of light.
Later repetitions of the experiment involved an apparatus which
could shoot only one photon at a time at the slits. One would
naively expect that each photon must pass through either one slit
or the other; thus, one should observe only two bands on the
detector. In reality, the same interference pattern of light and
dark bands emerges. Therefore, the photon cannot adequately be
explained as being simply a particle: it is also some kind of wave
phenomenon which interferes with itself as it passes through the
slits.
This same behavior is exhibited by not just photons, but also
electrons, atoms, and even some molecules. Thus all matter
possesses both particle and wave characteristics.
Wave-particle duality
Both the idea of a wave and the idea of a particle are
models derived from our everyday experience. We
cannot see individual photons, and can only investigate their
properties indirectly. Take, for example, the rainbow of colours we
see reflected from a puddle of water when a thin film of oil rests
on its surface. We can explain that phenomenon by modelling light
as waves. Other phenomena, such as the working of the
photoelectric meters in our cameras, may be
explained by thinking in terms of particles of light colliding with
the detection screen inside the meter. In both cases, we take
concepts from our everyday experience and apply them to a world we
will never see or otherwise experience directly.
Neither wave nor particle is an entirely satisfactory explanation.
In general, any
model can only
approximate that which it models. A model is useful only within the
range of conditions where it makes accurate predictions.
Newtonian physics remains a good predictor
of most everyday (macroscopic) phenomena. To remind us that both
"wave" and "particle" are concepts imported from our macro world to
explain atomic-scale phenomena, some physicists have used the term
"
wavicle" to refer to whatever it is that is
"really there." Astrophysicist
A.S. Eddington proposed in 1927 that "We
can scarcely describe such an entity as a wave or as a particle;
perhaps as a compromise we had better call it a 'wavicle' ". In the
following discussion, "wave" and "particle" may both be used
depending on which aspect of quantum mechanical phenomena is under
discussion.
Niels Bohr showed that neither the wave
analogy nor the particle analogy, taken individually, fully
describe the empirical properties of light. All forms of
electromagnetic radiation were
found to behave in certain experiments as though they were
particles, and in other experiments as though they were waves. With
these facts in mind, Bohr enunciated the
principle of complementarity,
which pairs concepts such as wave and particle, or position and
momentum.
In 1923,
Louis de Broglie explored
the mathematical consequences of Bohr's findings and discovered the
theory of
wave-particle
duality, which states that subatomic particles also have
simultaneous wave and particle properties. 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 will be observed only in situations that
permit a
standing wave around a
nucleus. An example of a standing
wave is a string fixed at both ends and made to vibrate (as in a
string instrument). Hence a
standing wave must have zero amplitude at each fixed end. The waves
created by a stringed instrument also appear to oscillate in place,
moving from crest to trough in an up-and-down motion. A standing
wave requires that the wavelength be an integer fraction of the
length of the vibrating object. (In other words, a harmonic
frequency must be an integer multiple of the fundamental frequency
of the vibrating object.) In a vibrating medium that traces out a
simple closed curve, the wave
must be a continuous formation of crests and troughs all around the
curve. Since electron
orbitals are simple
closed curves, each electron must be its own standing wave,
occupying a unique orbital.
In 1924, de Broglie extended his treatment of the wave-particle
duality and derived a relationship between the momentum
p
and wavelength
λ of a particle:
- p = \frac{h}{\lambda}.
The relationship, called the de Broglie hypothesis, holds for all
types of matter. Thus all matter exhibits properties of both
particles and waves.
De Broglie's treatment of quantum events served as a jumping off
point for Schrödinger when he set about to construct a wave
equation to describe quantum theoretical events.
Development of modern quantum mechanics
Full quantum mechanical theory
An electron falling from energy state
3 to energy state 2 emits one red photon.
To make a long and rather complicated story short,
Werner Heisenberg used the idea that since
classical physics is correct when
it applies to phenomena in the world of things larger than atoms
and molecules, it must stand as a special case of a more inclusive
quantum theoretical model. So he hoped that he could modify quantum
physics in such a way that when the parameters were on the scale of
everyday objects it would look just like classical physics, but
when the parameters were pulled down to the atomic scale the
discontinuities seen in things like the widely spaced frequencies
of the visible hydrogen bright line spectrum would come back into
sight.
By means of an intense series of mathematical analogies that some
physicists have termed "magical," Heisenberg wrote out an equation
that is the quantum mechanical analog for the classical computation
of intensities. Remember that the one thing that people at that
time most wanted to understand about hydrogen radiation was how to
predict or account for the intensities of the lines in its
spectrum. Although Heisenberg did not know it at the time, the
general format he worked out to express his new way of working with
quantum theoretical calculations can serve as a recipe for two
matrices and how to multiply them.
Heisenberg's groundbreaking paper of 1925 neither uses nor even
mentions matrices. Heisenberg's great advance was the "scheme which
was capable in principle of determining uniquely the relevant
physical qualities (transition frequencies and amplitudes)" of
hydrogen radiation.
After Heisenberg wrote his groundbreaking paper, he turned it over
to one of his senior colleagues for any needed corrections and went
on a well-deserved vacation.
Max Born
puzzled over the equations and the non-commuting equations that
Heisenberg had found troublesome and disturbing. After several days
he realized that these equations amounted to directions for writing
out matrices. Matrices were a bit off the beaten track, even for
mathematicians of that time, but how to do math with them was
already clearly established. He and a few others had the job of
working everything out in matrix form before Heisenberg returned
from his time off, and within a few months the new quantum
mechanics in matrix form formed the basis for another paper.
When quantities such as position and momentum are mentioned in the
context of Heisenberg's matrix mechanics, it is essential to keep
in mind that a statement such as
pq ≠
qp does not refer to a single value of
p and a single value
q but to a
matrix (grid of values arranged in a defined way) of values of
position and a matrix of values of momentum. So multiplying
p times
q or
q
times
p is really talking about the
matrix multiplication of the two
matrices. When two matrices are multiplied, the answer is a third
matrix.
Max Born saw that when the matrices that represent
pq and
qp were calculated they
would not be equal. Heisenberg had already seen the same thing in
terms of his original way of formulating things, and Heisenberg may
have guessed what was almost immediately obvious to Born — that the
difference between the answer matrices for
pq and
for
qp would always involve two factors that came
out of Heisenberg's original math: Plank's constant
h and
i, which is the square root
of negative one. So the very idea of what Heisenberg preferred to
call the "indeterminacy principle" (usually known as the
uncertainty principle) was lurking in Heisenberg's original
equations.
Paul Dirac decided that the essence of
Heisenberg's work lay in the very feature that Heisenberg had
originally found problematical — the fact of non-commutativity such
as that between multiplication of a momentum matrix by a
displacement matrix and multiplication of a displacement matrix by
a momentum matrix. That insight led Dirac in new and productive
directions.
Schrödinger wave equation
In 1925, building on
De Broglie's
theoretical model of particles as waves,
Erwin Schrödinger analyzed how an
electron would behave if it were assumed to
be a wave surrounding a nucleus. Rather than explaining the atom by
an analogy to satellites orbiting a planet, he treated electrons as
waves with each electron having a unique wavefunction. The
mathematical wavefunction is called the "Schrödinger equation"
after its creator. Schrödinger's equation describes a wavefunction
by three properties (
Wolfgang Pauli
later added a fourth:
spin):
- 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 collective name for these three properties is the "wavefunction
of the electron," describing the
quantum
state of the electron. The
quantum
state of an electron refers to its collective properties, which
describe what can be said about the electron at a point in time.
The quantum state of the electron is described by its wavefunction,
denoted by the Greek letter \psi ("psi," pronounced "sigh").
The three properties of Schrödinger's equation describing the
wavefunction of the electron (and thus its quantum state) are each
called
quantum numbers. The first
property describing the orbital is the
principal quantum number, numbered
according to Bohr's model, in which
n denotes the
energy of each orbital.
The next quantum number, the
azimuthal quantum number, denoted
l (lower case 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
l represents the
orbital angular momentum of an electron around its nucleus.
However, the shape of each orbital has its own letter as well. The
first shape is denoted by the letter
s (for
"spherical"). 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 Orbitals),
and are denoted by the letters
d,
f, and
g.
The third quantum number in Schrödinger's equation describes the
magnetic moment of the electron.
This number is denoted by either
m or
m with a subscript
l, because the
magnetic moment depends on the second quantum number
l.
In May 1926, Schrödinger proved that Heisenberg's
matrix mechanics and his own
wave mechanics made the same predictions
about the properties and behaviour of the electron; mathematically,
the two theories were identical. Yet both men disagreed on the
physical interpretations of their respective theories. Heisenberg
saw no problem in the existence of discontinuous quantum jumps,
while Schrödinger hoped that a theory based on continuous wave-like
properties could avoid what he called (in the words of
Wilhelm Wien), "this nonsense about quantum
jumps."
Uncertainty principle
One of Heisenberg's seniors,
Max Born
explained how he took his strange "recipe" given above and
discovered something ground breaking:
By consideration of ...examples...[Heisenberg] found
this rule....
This was in the summer of 1925.
Heisenberg...took leave of absence...and handed over
his paper to me for publication....
Heisenberg's rule of multiplication left me no peace,
and after a week of intensive thought and trial, I suddenly
remembered an algebraic theory....Such quadratic arrays are quite
familiar to mathematicians and are called matrices, in association
with a definite rule of multiplication.
I applied this rule to Heisenberg's quantum condition
and found that it agreed for the diagonal elements.
It was easy to guess what the remaining elements must
be, namely, null; and immediately there stood before me the strange
formula
- :{QP - PQ = \frac{ih}{2\pi}}
[The symbol Q is the matrix for displacement,
P is the matrix for momentum, i
stands for the square root of negative one, and h
is Planck's constant.]
That is the Heisenberg uncertainty principle, and it came out of
the math! Quantum mechanics strongly limits the precision with
which the properties of moving subatomic particles can be measured.
An observer can precisely measure either position or momentum, but
not both. In the limit, measuring either variable with complete
precision would entail a complete absence of precision in the
measurement of the other.
Wavefunction collapse
Wavefunction collapse is a forced term for whatever 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 an electron "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.
Eigenstates and eigenvalues
For a more detailed introduction to this subject, see:
Introduction to
eigenstates
Because of the
uncertainty
principle, statements about both the position and momentum of
particles can only assign a
probability
that the position or momentum will have 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.
The Pauli exclusion principle
Wolfgang Pauli proposed the following
concise statement of
his
principle: "There cannot exist an atom in such a quantum state
that two electrons within have the same set of quantum
numbers."
He developed the exclusion principle from what he called a
"two-valued quantum degree of freedom" to account for the
observation of a doublet, meaning a pair of lines differing by a
small amount (e.g., on the order of 0.15Å), in the
spectrum of atomic hydrogen. The existence
of these closely spaced lines in the bright-line spectrum meant
that there was more energy in the electron orbital from
magnetic moments than had previously been
described.
In early 1925,
Uhlenbeck and
Goudsmit proposed that electrons
rotate about an axis in the same way that the earth rotates on its
axis. They proposed to call this property
spin. 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. A new
quantum number was
then needed, one to represent the momentum embodied in the rotation
of each electron.
By this time an electron was recognized to have four kinds of
fundamental characteristics that came to be identified by the four
quantum numbers:
The chemist
Linus Pauling wrote, by
way of example:
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 many-particle
quantum field theory.
Quantum entanglement
Superposition of two quantum
characteristics, and two resolution possibilities.
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 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 probabilities that are
the squares of the 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 that can be mentally
labeled as blue and another state that can be
mentally labeled as red will then appear (in imagination, of
course) 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 will determine
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, it would necessarily
show up in the opposite state to whatever its twin had
revealed.
Suppose that some species of animal life carries both male and
female characteristics in its genetic potential. It will become
either male or female depending on some environmental change.
Perhaps it will remain indeterminate until the weather either turns
very hot or very cold. Then it will show one set of sexual
characteristics and will be locked into that sexual status by
epigenetic changes, the presence in its system of high levels of
androgen or estrogen, etc. There are actually situations in nature
that are similar to this scenario, but now imagine that if twins
are born, then they are forbidden by nature to both manifest the
same sex. So if one twin goes to Antarctica and changes to become a
female, then the other twin will turn into a male despite the fact
that local weather has done nothing special to it. Such a world
would be very hard to explain. How can something that happens to
one animal in Antarctica affect its twin in Redwood, California? Is
it mental telepathy? What? How can the change be instantaneous?
Even a radio message from Antarctica would take a certain amount of
time.
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
which 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 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 which 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 magnetic field, in turn causes
electric current (moving electrons). The
interacting electric and magnetic field is called an
electromagnetic field.
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 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 showed all possible
interactions of a given event. The diagrams showed that the
electromagnetic force is the interactions of photons between
interacting particles.
An example of a prediction of quantum electrodynamics which has
been verified experimentally is the
Lamb
shift. This refers to an effect whereby the quantum nature of
the electromagnetic field causes the energy levels in an atom or
ion to deviate slightly from what they would otherwise be. As a
result, spectral lines may shift or split.
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. The
Standard Model unifies
the electromagnetic and
weak
interactions into one theory. This is called the
electroweak theory.
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.
Summary
In classical mechanics, the energy of an oscillation or vibration
can take on any value. However, in quantum mechanics, energy of an
oscillator is gained and lost in "chunks" whose size is given by
Planck's constant
h times the frequency at which
the oscillator would oscillate according to classical mechanics.
Atomic electrons can exist in states with discrete "energy levels"
or in a superposition of such states. Just as the velocity with
which an object approaches a sun will determine the distance from
the sun at which it can establish a stable orbit, so too the energy
carried by an electron will automatically assign it to a given
orbital around the nucleus of an atom. Moving from one energy state
to another either requires that more energy be supplied to the
electron (moving it to a higher energy state) or the electron must
lose a certain amount of energy as a photon (moving it to a lower
energy state). Four such transitions from a higher to a lower
energy state give visible lines in the bright line spectrum of
hydrogen. Other transitions give lines outside the visible
spectrum.
The four visible lines in the spectrum of hydrogen were originally
observed, but for a time scientists did not know anything more than
their wavelengths. Then, Balmer figured out a mathematical rule by
which he could make quantum theoretical predictions of the observed
wavelengths. The same basic rule was improved in two stages, first
by writing it in terms of the inverse values of all of the numbers
involved, and second by generalizing the rule and replacing
{\left( \frac{1}{4} - \frac{1}{n^2} \right)} with {\left(
\frac{1}{m^2} - \frac{1}{n^2} \right)}
This additional level of generality permitted the entire hydrogen
bright-line spectrum, from infrared through the visible colors to
ultraviolet and higher frequencies, to be predicted because
m could be a whole range of integers as long as
any
m was always larger than the corresponding
n. Using Planck's constant, one could assign
energies to individual frequencies (or wavelengths) of
electromagnetic radiation. To predict the intensities of these
bright lines, physicists needed to use matrix mathematics,
Schrödinger's equation, or some other computational scheme
involving higher mathematics. There were not only the basic six
energy levels of hydrogen, but also other factors that created
additional energy levels. The very first calculation that
Heisenberg made in his new theory involved an infinite series, and
the more factors involved (the more "quantum numbers" were
involved) the more complex the mathematics. But the basic insight
into the structure of the hydrogen atom was encoded in the simple
formula that Balmer guessed from a list of wavelengths.
The photoelectric effect was discovered soon after Balmer made his
rule, and in 1905 Einstein first depicted light as being made of
photons to account for that effect.
Bohr explained the Rydberg formula in terms of atomic structure.
Two years later, in 1925, Heisenberg removed the last traces of
classical physics from the new quantum theory by making the
breakthrough that led to the matrix formulation of quantum
mechanics, and Pauli enunciated his exclusion principle. Further
advances came by closing in on some of the elusive details: (1)
adding the
m_{l} (spin) quantum numbers
(discovered by Pauli), (2) adding the
M_{s} quantum numbers (discovered by
Goudsmit and Uhlenbeck), (3) broadening the quantum picture to
account for relativistic effects (Dirac's work), (4) showing that
particles such as electrons, and even larger entities, have a wave
nature (the matter waves of de Broglie). Dirac introduced a new
theoretical formulation "which if interrogated in a particle-like
way gave particle behavior and if interrogated in a wave-like way
gave wave behavior."
Several improvements in mathematical formulation have also
furthered quantum mechanics:
De Broglie's quantum theoretical description based on waves was
followed upon by Schrödinger. Schrödinger's method of representing
the state of each atomic entity is a generally more practical
scheme to use than Heisenberg's. It makes it possible to
conceptualize a "wave function" that passes through both sides of a
double-slit experiment and then arrives at the detection screen as
two parts of itself that are superimposed but a little shifted (a
little out of phase). It also makes it possible to understand how
two photons or other things of that order of magnitude might be
created in the same event or otherwise closely linked in history
and so carry identical copies of superimposed wave functions. That
mental picture can then be used to explain how when one of them is
coerced into revealing itself, it must manifest one or the other
superimposed wave nature, and its twin (regardless of its distance
away in space or time) must manifest the complementary wave
nature.
Prominent among later scientists who increased the elegance and
accuracy of quantum-theoretical formulations was
Richard Feynman who followed up on Dirac's
work. The basic picture given in the original Balmer formula
remained true, but it has been qualified by revelation of many
details, such as angular momentum and spin, and extended to
descriptions that go beyond a mere explanation of the electron and
its behavior while bound to an atomic nucleus. Active research
still continues to resolve some remaining issues.
See also
Persons important for discovering and elaborating quantum
theory:
Further reading
The following titles, all by working physicists, attempt to
communicate quantum theory to lay people, using a minimum of
technical apparatus.
- Richard Feynman, 1985.
QED: The
Strange Theory of Light and Matter, Princeton University
Press. ISBN 0-691-08388-6
- 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 Univ. Press: 110-176. The author is a rare
physicist who tries to communicate to philosophers and
humanists.
- Roland Omnes (1999)
Understanding Quantum Mechanics. Princeton Univ.
Press.
- Victor Stenger, 2000.
Timeless Reality: Symmetry, Simplicity, and Multiple
Universes. Buffalo NY: Prometheus Books. Chpts. 5-8.
- Veltman, M. J. G., 2003. Facts and Mysteries in Elementary
Particle Physics. World Scientific Publishing Company.
Notes
- Richard P. Feynman, QED, p. 10
- Aezel, Amir D., Entanglement, p. 35. (Penguin, 2003)
ISBN 0-452-28457
- Dicke and Wittke, Introduction to Quantum Mechanics,
p. 12
- Robert
H. Dicke and James P. Wittke, 1960. Introduction to Quantum
Mechanics. Addison-Wesley: 9f.
- For the length of time involved, see George Gamow's One,
Two, Three...Infinity, p. 140.
- In this case, the energy of the electron is the sum of its
kinetic
and potential energies. The electron has
kinetic energy by virtue of its motion around the nucleus, and
potential energy because of its electromagnetic interaction with
the nucleus.
- This follows from Planck's equation , since the wavelength of
light is related to its frequency by .
- See Linus
Pauling, The Nature of the Chemical Bond,
- A very clear explanation of interference in thin films may be
found in Sears, op. cit., p. 203ff.
- A.S. Eddington, The Nature of the Physical
World, the course of Gifford Lectures that Eddington delivered
in the University of Edinburgh in January to March 1927, Kessinger
Publishing, 2005, p. 201.
- Aezel, Amir D., Entanglrment, p. 51f. (Penguin, 2003)
ISBN 0-452-28457
- Heisenberg's paper of 1925 is translated in B. L. Van der
Waerden's Sources of Quantum Mechanics, where it appears
as chapter 12.
- Aitchison, et al., "Understanding Heisenberg's 'magical' paper
of July 1925: a new look at the calculational details," p. 2
- Thomas F. Jordan, Quantum Mechanics in Simmple Matrix
Form, p. 149
- W. Moore, Schrödinger: Life and Thought, Cambridge
University Press (1989), p. 222.
- Born's Nobel lecture quoted in Thomas F. Jordan's Quantum
Mechanics in Simple Matrix Form, p. 6
- See Introduction to quantum mechanics. by Henrik
Smith, p. 58 for a readable introduction. See Ian J. R. Aitchison,
et al., "Understanding Heisenberg's 'magical' paper of July 1925,"
Appendix A, for a mathematical derivation of this
relationship.
- Linus Pauling, The Nature of the Chemical
Bond, p. 47
- E. Schrödinger, Proceedings of the Cambridge Philosophical
Society, 31 (1935), p. 555says: "When two systems, of which we
know the states by their respective representation, enter into a
temporary physical interaction due to known forces between them and
when after a time of mutual influence the systems separate again,
then they can no longer be described as before, viz., by endowing
each of them with a representative of its own. I would not call
that one but rather the characteristic trait of
quantum mechanics."
References
- Bernstein, Jeremy, 2005, "Max Born and
the quantum theory," Am. J. Phys.
73(11).
- Beller, Mara, 2001. Quantum Dialogue: The Making of a
Revolution. University of Chicago Press.
- Louis de Broglie, 1953. The
Revolution in Physics. Noonday Press.
- Albert Einstein, 1934.
Essays in Science. Philosophical Library.
- Herbert Feigl and May Brodbeck,
1953. Readings in the Philosophy of Science,
Appleton-Century-Crofts.
- Fowler, Michael, 1999. The Bohr Atom. Lecture series,
University of Virginia.
- Werner Heisenberg, 1958.
Physics and Philosophy. Harper and Brothers.
- Lakshmibala, S., 2004, "Heisenberg, Matrix Mechanics and the
Uncertainty Principle," Resonance, Journal of Science
Education 9(8).
- Richard L. Liboff, 1992. Introductory Quantum
Mechanics, 2nd ed.
- Lindsay, Robert Bruce and Henry Margenau, 1936. Foundations
of Physics. Dover.
- McEvoy, J.P., and Zarate, Oscar. Introducing Quantum
Theory, ISBN 1874166374
- Nave, Carl Rod, 2005. Hyperphysics-Quantum Physics,
Department of Physics and Astronomy, Georgia State University,
CD.
- Peat, F. David, 2002. From Certainty to Uncertainty: The
Story of Science and Ideas in the Twenty-First Century. Joseph
Henry Press.
- Hans Reichenbach, 1944.
Philosophic Foundations of Quantum Mechanics. University
of California Press.
- Paul Arthur Schilpp, 1949.
Albert Einstein: Philosopher-Scientist. Tudor Publishing
Company.
- Scientific American Reader, 1953.
- Sears, Francis Weston, 1949. Optics.
Addison-Wesley.
- ; cited in:
- Van Vleck, J. H.,1928, "The Correspondence Principle in the
Statistical Interpretation of Quantum Mechanics," Proc.
Nat. Acad. Sci. 14: 179.
- Wieman, Carl, and Perkins, Katherine, 2005, "Transforming
Physics Education," Physics Today.
- Westmoreland, M. D., and Schumacher, B., 1998, " Quantum
Entanglement and the Nonexistence of Superluminal
Signals."
- Bronner, P., Strunz, A. et al. (2009). " Demonstrating quantum random with single photons."
European Journal of Physics 30(5): 1189-1200
- Richard P. Feynman, QED, p. 10
- Aezel, Amir D., Entanglement, p. 35. (Penguin, 2003)
ISBN 0-452-28457
- Dicke and Wittke, Introduction to Quantum Mechanics,
p. 12
- Robert
H. Dicke and James P. Wittke, 1960. Introduction to Quantum
Mechanics. Addison-Wesley: 9f.
- For the length of time involved, see George Gamow's One,
Two, Three...Infinity, p. 140.
- In this case, the energy of the electron is the sum of its
kinetic
and potential energies. The electron has
kinetic energy by virtue of its motion around the nucleus, and
potential energy because of its electromagnetic interaction with
the nucleus.
- This follows from Planck's equation , since the wavelength of
light is related to its frequency by .
- See Linus
Pauling, The Nature of the Chemical Bond,
- A very clear explanation of interference in thin films may be
found in Sears, op. cit., p. 203ff.
- A.S. Eddington, The Nature of the Physical
World, the course of Gifford Lectures that Eddington delivered
in the University of Edinburgh in January to March 1927, Kessinger
Publishing, 2005, p. 201.
- Aezel, Amir D., Entanglrment, p. 51f. (Penguin, 2003)
ISBN 0-452-28457
- Heisenberg's paper of 1925 is translated in B. L. Van der
Waerden's Sources of Quantum Mechanics, where it appears
as chapter 12.
- Aitchison, et al., "Understanding Heisenberg's 'magical' paper
of July 1925: a new look at the calculational details," p. 2
- Thomas F. Jordan, Quantum Mechanics in Simmple Matrix
Form, p. 149
- W. Moore, Schrödinger: Life and Thought, Cambridge
University Press (1989), p. 222.
- Born's Nobel lecture quoted in Thomas F. Jordan's Quantum
Mechanics in Simple Matrix Form, p. 6
- See Introduction to quantum mechanics. by Henrik
Smith, p. 58 for a readable introduction. See Ian J. R. Aitchison,
et al., "Understanding Heisenberg's 'magical' paper of July 1925,"
Appendix A, for a mathematical derivation of this
relationship.
- Linus Pauling, The Nature of the Chemical
Bond, p. 47
- E. Schrödinger, Proceedings of the Cambridge Philosophical
Society, 31 (1935), p. 555says: "When two systems, of which we
know the states by their respective representation, enter into a
temporary physical interaction due to known forces between them and
when after a time of mutual influence the systems separate again,
then they can no longer be described as before, viz., by endowing
each of them with a representative of its own. I would not call
that one but rather the characteristic trait of
quantum mechanics."
External links