Elementary fermions:
-
Matter particles
-
Quarks:
-
up, down
-
charm, strange
-
top, bottom
-
Leptons:
-
electron, electron neutrino (a.k.a., "neutrino")
-
muon, muon neutrino
-
tau, tau neutrino
-
Antimatter particles
-
Antiquarks
-
Antileptons
Elementary bosons:
-
Force particles (gauge bosons):
-
photon
-
gluon (numbering eight)
-
W
+
, W
−
, and Z
0
bosons
-
graviton (hypothetical)
[1]
-
Scalar boson
-
Higgs boson
A particle's mass is quantified in units of energy versus the electron's
(electronvolts). Through conversion of energy into mass, any particle can be
produced through collision of other particles at high energy,
[1][11]
although the output
particle might not contain the input particles, for instance matter creation from
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colliding photons. Likewise, the composite fermions protons were collided at nearly
light speed to produce the relatively more massive Higgs boson.
[11]
The most massive
elementary particle, the top quark, rapidly decays, but apparently does not contain,
lighter particles.
When probed at energies available in experiments, particles exhibit spherical
sizes. In operating particle physics' Standard Model, elementary particles are usually
represented for predictive utility as point particles, which, as zero-dimensional, lack
spatial extension. Though extremely successful, the Standard Model is limited to the
microcosm by its omission of gravitation, and has some parameters arbitrarily added
but unexplained.
[12]
Seeking to resolve those shortcomings, string theory posits that
elementary particles are ultimately composed of one-dimensional energy strings
whose absolute minimum size is the Planck length.
Quantum physics
In the mathematically rigorous formulation of quantum mechanics developed
by Paul Dirac, David Hilbert, John von Neumann, and Hermann Weyl, the possible
states of a quantum mechanical system are symbolized as unit vectors (called state
vectors). Formally, these reside in a complex separable Hilbert space—variously
called the state space or the associated Hilbert space of the system—that is well
defined up to a complex number of norm 1 (the phase factor). In other words, the
possible states are points in the projective space of a Hilbert space, usually called the
complex projective space. The exact nature of this Hilbert space is dependent on the
system—for example, the state space for position and momentum states is the space
of square-integrable functions, while the state space for the spin of a single proton is
just the product of two complex planes. Each observable is represented by a
maximally Hermitian (precisely: by a self-adjoint) linear operator acting on the state
space. Each eigenstate of an observable corresponds to an eigenvector of the
operator, and the associated eigenvalue corresponds to the value of the observable in
that eigenstate. If the operator's spectrum is discrete, the observable can attain only
those discrete eigenvalues.
In the formalism of quantum mechanics, the state of a system at a given time is
described by a complex wave function, also referred to as state vector in a complex
vector space.
[22]
This abstract mathematical object allows for the calculation of
probabilities of outcomes of concrete experiments. For example, it allows one to
compute the probability of finding an electron in a particular region around the
nucleus at a particular time. Contrary to classical mechanics, one can never make
simultaneous predictions of conjugate variables, such as position and momentum,
with accuracy. For instance, electrons may be considered (to a certain probability) to
be located somewhere within a given region of space, but with their exact positions
unknown. Contours of constant probability, often referred to as "clouds", may be
drawn around the nucleus of an atom to conceptualize where the electron might be
located with the most probability. Heisenberg's uncertainty principle quantifies the
inability to precisely locate the particle given its conjugate momentum.
[23]
According to one interpretation, as the result of a measurement the wave
function containing the probability information for a system collapses from a given
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initial state to a particular eigenstate. The possible results of a measurement are the
eigenvalues of the operator representing the observable—which explains the choice
of Hermitian operators, for which all the eigenvalues are real. The probability
distribution of an observable in a given state can be found by computing the spectral
decomposition of the corresponding operator. Heisenberg's uncertainty principle is
represented by the statement that the operators corresponding to certain observables
do not commute.
The probabilistic nature of quantum mechanics thus stems from the act of
measurement. This is one of the most difficult aspects of quantum systems to
understand. It was the central topic in the famous Bohr–Einstein debates, in which the
two scientists attempted to clarify these fundamental principles by way of thought
experiments. In the decades after the formulation of quantum mechanics, the question
of what constitutes a "measurement" has been extensively studied. Newer
interpretations of quantum mechanics have been formulated that do away with the
concept of "wave function collapse" (see, for example, the relative state
interpretation). The basic idea is that when a quantum system interacts with a
measuring apparatus, their respective wave functions become entangled, so that the
original quantum system ceases to exist as an independent entity. For details, see the
article on measurement in quantum mechanics.
Generally, quantum mechanics does not assign definite values. Instead, it
makes a prediction using a probability distribution; that is, it describes the probability
of obtaining the possible outcomes from measuring an observable. Often these results
are skewed by many causes, such as dense probability clouds. Probability clouds are
approximate (but better than the Bohr model) whereby electron location is given by a
probability function, the wave function eigenvalue, such that the probability is the
squared modulus of thecomplex amplitude, or quantum state nuclear attraction.
Naturally, these probabilities will depend on the quantum state at the "instant" of the
measurement. Hence, uncertainty is involved in the value. There are, however,
certain states that are associated with a definite value of a particular observable.
These are known as eigenstates of the observable ("eigen" can be translated from
German as meaning "inherent" or "characteristic").
In the everyday world, it is natural and intuitive to think of everything (every
observable) as being in an eigenstate. Everything appears to have a definite position,
a definite momentum, a definite energy, and a definite time of occurrence. However,
quantum mechanics does not pinpoint the exact values of a particle's position and
momentum (since they are conjugate pairs) or its energy and time (since they too are
conjugate pairs); rather, it provides only a range of probabilities in which that particle
might be given its momentum and momentum probability. Therefore, it is helpful to
use different words to describe states having uncertain values and states having
definite values (eigenstates). Usually, a system will not be in an eigenstate of the
observable (particle) we are interested in. However, if one measures the observable,
the wave function will instantaneously be an eigenstate (or "generalized" eigenstate)
of that observable. This process is known as wave function collapse, a controversial
and much-debated process
[28]
that involves expanding the system under study to
include the measurement device. If one knows the corresponding wave function at the
369
instant before the measurement, one will be able to compute the probability of the
wave function collapsing into each of the possible eigenstates. For example, the free
particle in the previous example will usually have a wave function that is a wave
packet centered around some mean position x
0
(neither an eigenstate of position nor
of momentum). When one measures the position of the particle, it is impossible to
predict with certainty the result. It is probable, but not certain, that it will be near x
0
,
where the amplitude of the wave function is large. After the measurement is
performed, having obtained some result x, the wave function collapses into a position
eigenstate centered at x.
The time evolution of a quantum state is described by the Schrödinger
equation, in which the Hamiltonian (the operator corresponding to the total energy of
the system) generates the time evolution. The time evolution of wave functions is
deterministic in the sense that - given a wave function at an initial time - it makes a
definite prediction of what the wave function will be at any later time.
During a measurement, on the other hand, the change of the initial wave
function into another, later wave function is not deterministic, it is unpredictable (i.e.,
random). A time-evolution simulation can be seen here.
Wave functions change as time progresses. The Schrödinger equation describes
how wave functions change in time, playing a role similar to Newton's second law in
classical mechanics. The Schrödinger equation, applied to the aforementioned
example of the free particle, predicts that the center of a wave packet will move
through space at a constant velocity (like a classical particle with no forces acting on
it). However, the wave packet will also spread out as time progresses, which means
that the position becomes more uncertain with time. This also has the effect of
turning a position eigenstate (which can be thought of as an infinitely sharp wave
packet) into a broadened wave packet that no longer represents a (definite, certain)
position eigenstate.
Fig. 1: Probability densities corresponding to the wave functions of an electron
in a hydrogen atom possessing definite energy levels (increasing from the top of the
image to the bottom: n = 1, 2, 3, ...) and angular momenta (increasing across from left
to right: s, p, d, ...). Brighter areas correspond to higher probability density in a
position measurement. Such wave functions are directly comparable to Chladni's
figures of acoustic modes of vibration in classical physics, and are modes of
oscillation as well, possessing a sharp energy and, thus, a definitefrequency. The
angular momentum and energy are quantized, and take only discrete values like those
shown (as is the case for resonant frequencies in acoustics)
370
Some wave functions produce probability distributions that are constant, or
independent of time—such as when in a stationary state of constant energy, time
vanishes in the absolute square of the wave function. Many systems that are treated
dynamically in classical mechanics are described by such "static" wave functions. For
example, a single electron in an unexcited atom is pictured classically as a particle
moving in a circular trajectory around the atomic nucleus, whereas in quantum
mechanics it is described by a static, spherically symmetric wave function
surrounding the nucleus (Fig. 1) (note, however, that only the lowest angular
momentum states, labeled s, are spherically symmetric).
[34]
The Schrödinger equation acts on the entire probability amplitude, not merely
its absolute value. Whereas the absolute value of the probability amplitude encodes
information about probabilities, its phase encodes information about the interference
between quantum states. This gives rise to the "wave-like" behavior of quantum
states. As it turns out, analytic solutions of the Schrödinger equation are available for
only a very small number of relatively simple model Hamiltonians, of which the
quantum harmonic oscillator, the particle in a box, the dihydrogen cation, and the
hydrogen atom are the most important representatives. Even thehelium atom—which
contains just one more electron than does the hydrogen atom—has defied all attempts
at a fully analytic treatment.
There exist several techniques for generating approximate solutions, however.
In the important method known as perturbation theory, one uses the analytic result for
a simple quantum mechanical model to generate a result for a more complicated
model that is related to the simpler model by (for one example) the addition of a
weak potential energy. Another method is the "semi-classical equation of motion"
approach, which applies to systems for which quantum mechanics produces only
weak (small) deviations from classical behavior. These deviations can then be
computed based on the classical motion. This approach is particularly important in
the field of quantum chaos.
Mathematically equivalent formulations of quantum mechanics[edit]
There are numerous mathematically equivalent formulations of quantum
mechanics. One of the oldest and most commonly used formulations is the
"transformation theory" proposed by Paul Dirac, which unifies and generalizes the
two earliest formulations of quantum mechanics - matrix mechanics (invented by
Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).
[35]
Especially since Werner Heisenberg was awarded the Nobel Prize in Physics in
1932 for the creation of quantum mechanics, the role of Max Born in the
development of QM was overlooked until the 1954 Nobel award. The role is noted in
a 2005 biography of Born, which recounts his role in the matrix formulation of
quantum mechanics, and the use of probability amplitudes. Heisenberg himself
acknowledges having learned matrices from Born, as published in a 1940
festschrifthonoring Max Planck.
[36]
In the matrix formulation, the instantaneous state
of a quantum system encodes the probabilities of its measurable properties, or
"observables". Examples of observables include energy, position, momentum, and
angular momentum. Observables can be either continuous (e.g., the position of a
particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).
[37]
An
371
alternative formulation of quantum mechanics is Feynman's path integral
formulation, in which a quantum-mechanical amplitude is considered as a sum over
all possible classical and non-classical paths between the initial and final states. This
is the quantum-mechanical counterpart of the action principle in classical mechanics.
Interactions with other scientific theories[edit]
The rules of quantum mechanics are fundamental. They assert that the state
space of a system is a Hilbert space and that observables of that system are Hermitian
operatorsacting on that space—although they do not tell us which Hilbert space or
which operators. These can be chosen appropriately in order to obtain a quantitative
description of a quantum system. An important guide for making these choices is the
correspondence principle, which states that the predictions of quantum mechanics
reduce to those of classical mechanics when a system moves to higher energies or,
equivalently, larger quantum numbers, i.e. whereas a single particle exhibits a degree
of randomness, in systems incorporating millions of particles averaging takes over
and, at the high energy limit, the statistical probability of random behaviour
approaches zero. In other words, classical mechanics is simply a quantum mechanics
of large systems. This "high energy" limit is known as the classical or
correspondence limit. One can even start from an established classical model of a
particular system, then attempt to guess the underlying quantum model that would
give rise to the classical model in the correspondence limit.
Unsolved problem in physics:
In the correspondence limit of quantum
mechanics: Is there a preferred interpretation
of quantum mechanics? How does the quantum
description of reality, which includes elements
such as the "superpositionof states" and "wave
function collapse", give rise to the reality we
perceive?
(more unsolved problems in physics)
When quantum mechanics was originally formulated, it was applied to models
whose correspondence limit was non-relativistic classical mechanics. For instance,
the well-known model of the quantum harmonic oscillator uses an explicitly non-
relativistic expression for the kinetic energy of the oscillator, and is thus a quantum
version of the classical harmonic oscillator.
Early attempts to merge quantum mechanics with special relativity involved
the replacement of the Schrödinger equation with a covariant equation such as the
Klein–Gordon equation or the Dirac equation. While these theories were successful
in explaining many experimental results, they had certain unsatisfactory qualities
stemming from their neglect of the relativistic creation and annihilation of particles.
A fully relativistic quantum theory required the development ofquantum field theory,
which applies quantization to a field (rather than a fixed set of particles). The first
complete quantum field theory, quantum electrodynamics, provides a fully quantum
description of the electromagnetic interaction. The full apparatus of quantum field
theory is often unnecessary for describing electrodynamic systems. A simpler
372
approach, one that has been employed since the inception of quantum mechanics, is
to treat charged particles as quantum mechanical objects being acted on by a
classicalelectromagnetic field. For example, the elementary quantum model of the
hydrogen atom describes the electric field of the hydrogen atom using a classical
Coulomb potential. This "semi-classical" approach fails if quantum fluctuations in the
electromagnetic field play an important role, such as in the emission of photons by
charged particles.
Quantum field theories for the strong nuclear force and the weak nuclear force
have also been developed. The quantum field theory of the strong nuclear force is
called quantum chromodynamics, and describes the interactions of subnuclear
particles such as quarks and gluons. The weak nuclear force and the electromagnetic
force were unified, in their quantized forms, into a single quantum field theory
(known as electroweak theory), by the physicists Abdus Salam, Sheldon Glashow and
Steven Weinberg. These three men shared the Nobel Prize in Physics in 1979 for this
work.
[38]
It has proven difficult to construct quantum models of gravity, the remaining
fundamental force. Semi-classical approximations are workable, and have led to
predictions such asHawking radiation. However, the formulation of a complete
theory of quantum gravity is hindered by apparent incompatibilities between general
relativity (the most accurate theory of gravity currently known) and some of the
fundamental assumptions of quantum theory. The resolution of these
incompatibilities is an area of active research, and theories such as string theory are
among the possible candidates for a future theory of quantum gravity.
Classical mechanics has also been extended into the complex domain, with
complex classical mechanics exhibiting behaviors similar to quantum mechanics.
[39]
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