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Spin (physics)
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In physics, spin refers to the angular momentum intrinsic to a body, as
opposed to orbital angular momentum, which is generated by the motion of
its center of mass about an external point.
In classical mechanics, the spin angular momentum of a body is
associated with the rotation of the body around its own center of mass.
For instance, the spin angular momentum of the Earth is associated with
its 24-hourly rotation about the polar axis, which gives rise to the
day-night cycle. On the other hand, the orbital angular momentum of the
Earth is associated with its motion around the Sun. The orbital period
of this motion defines the year.
Spin angular momentum is particularly important for systems at atomic
length scales or smaller, such as individual atoms, protons, or
electrons. The effects of quantum mechanics are important when
describing such particles. Quantum mechanical spin possesses several
unusual features, which will be described in the remainder of this
article.
(We will use the term "particle" to refer to such quantum mechanical
systems, with the understanding that they actually exhibit wave-particle
duality, and thus display both particle-like and wave-like behaviors.)
Contents
1 Spin of elementary and composite particles
2 Spin direction
3 Spin and magnetic moment
4 The spin-statistics connection
5 Applications
6 History
7 See also
8 References
9 External links
[edit]
Spin of elementary and composite particles
One of the most remarkable discoveries associated with quantum physics
is the fact that elementary particles can possess non-zero spin.
Elementary particles are particles that cannot be divided into any
smaller units, such as the photon, the electron, and the various quarks.
Theoretical and experimental studies have shown that the spin possessed
by these particles cannot be explained by postulating that they are made
up of even smaller particles rotating about a common center of mass (see
classical electron radius); as far as we can tell, these elementary
particles are true point particles. The spin that they carry is a truly
intrinsic physical property.
The concept of elementary particle spin was first proposed in 1925 by
Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit. A later section
covers the history of this hypothesis and its subsequent developments.
According to quantum mechanics, the angular momentum of any system is
quantized. The magnitude of angular momentum can only take on the values
where is Planck's constant divided by 2Ï? (sometimes called Dirac's
constant), and s is a non-negative integer or half-integer (0, 1/2, 1,
3/2, 2, etc.). For instance, electrons (which are elementary particles)
are called "spin-1/2" particles because their intrinsic spin angular
momentum has s = 1/2.
The spin carried by each elementary particle has a fixed s value that
depends only by the type of particle, and cannot be altered in any known
way (although, as we will see, it is possible to change the direction in
which the spin "points".) Every electron in existence possesses s = 1/2.
Other elementary spin-1/2 particles include neutrinos and quarks. On the
other hand, photons are spin-1 particles, whereas the hypothetical
graviton is a spin-2 particle.
The spin of composite particles, such as protons, neutrons, atomic
nuclei, and atoms, is made up of the spins of the constituent particles,
plus the orbital angular momentum of their motions around one another.
The angular momentum quantization condition applies to both elementary
and composite particles. Composite particles are often referred to as
having a definite spin, just like elementary particles; for example, the
proton is a spin-1/2 particle. This is understood to refer to the spin
of the lowest-energy internal state of the composite particle (i.e., a
given spin and orbital configuration of the constituents). It is not
always easy to deduce the spin of a composite particle from first
principles; for example, even though we know that the proton is a
spin-1/2 particle, the question of how this spin is distributed among
the three internal quarks and the surrounding gluons is an active area
of research. [1]
[edit]
Spin direction
In classical mechanics, the angular momentum of a particle possesses not
only a magnitude (how fast the body is rotating), but also a direction
(the axis of rotation of the particle). Quantum mechanical spin also
contains information about direction, albeit in a more subtle form.
Quantum mechanics states that the component of angular momentum measured
along any direction (say along the z-axis) can only take on the values
where s is the principal spin quantum number discussed in the previous
section. One can see that there are 2s+1 possible values of sz. For
example, there are only two possible values for a spin-1/2 particle: sz
= +1/2 and sz = -1/2. These correspond to quantum states in which the
spin is pointing in the +z or -z directions respectively, and are often
referred to as "spin up" and "spin down". See spin-1/2.
For a given quantum state , it is possible to describe a spin vector
â?©Sâ?ª whose components are the expectation values of the
spin components along each axis, i.e., â?©Sâ?ª =
[â?©sxâ?ª, â?©syâ?ª, â?©szâ?ª]. This
vector describes the "direction" in which the spin is pointing,
corresponding to the classical concept of the axis of rotation. It turns
out that the spin vector is not very useful in actual quantum mechanical
calculations, because it cannot be measured directly â?" sx, sy and
sz cannot possess simultaneous definite values, because of a quantum
uncertainty relation between them. As a qualitative concept, however,
the spin vector is often handy because it is easy to picture
classically.
For instance, quantum mechanical spin can exhibit phenomena analogous to
classical gyroscopic effects. For example, one can exert a kind of
"torque" on an electron by putting it in a magnetic field (the field
acts upon the electron's intrinsic magnetic dipole moment â?" see
the following section). The result is that the spin vector undergoes
precession, just like a classical gyroscope.
Mathematically, quantum mechanical spin is not described by a vector as
in classical angular momentum. It is described using a family of objects
known as spinors. There are subtle differences between the behavior of
spinors and vectors under coordinate rotations. For example, rotating a
spin-1/2 particle by 360 degrees does not bring it back to the same
quantum state, but to the state with the opposite quantum phase; this is
detectable, in principle, with interference experiments. To return the
particle to its exact original state, one needs a 720 degree rotation!
[edit]
Spin and magnetic moment
Particles with spin can possess a magnetic dipole moment, just like a
rotating electrically charged body in classical electrodynamics. These
magnetic moments can be experimentally observed in several ways, e.g. by
the deflection of particles by inhomogeneous magnetic fields in a
Stern-Gerlach experiment, or by measuring the magnetic fields generated
by the particles themselves.
The intrinsic magnetic moment μ of a particle with charge q, mass
m, and spin S, is
where the dimensionless quantity g is called the gyromagnetic ratio or
g-factor.
The electron, despite being an elementary particle, possesses a nonzero
magnetic moment. One of the triumphs of the theory of quantum
electrodynamics is its accurate prediction of the electron g-factor,
which has been experimentally determined to have the value 2.002319...
The value of 2 arises from the Dirac equation, a fundamental equation
connecting the electron's spin with its electromagnetic properties, and
the correction of 0.002319... arises from the electron's interaction
with the surrounding electromagnetic field.
Composite particles also possess magnetic moments associated with their
spin. In particular, the neutron possesses a non-zero magnetic moment
despite being electrically neutral. This fact was an early indication
that the neutron is not an elementary particle. In fact, it is made up
of quarks, which are charged particles. The magnetic moment of the
neutron comes from the moments of the individual quarks and their
orbital motions.
The neutrinos are both elementary and electrically neutral, and theory
indicates that they have zero magnetic moment. The measurement of
neutrino magnetic moments is an active area of research. As of 2003, the
latest experimental results have put the neutrino magnetic moment at
less than 1.3 Ã? 10-10 times the electron's magnetic moment.
In ordinary materials, the magnetic dipole moments of individual atoms
produce magnetic fields that cancel one another, because each dipole
points in a random direction. In ferromagnetic materials, however, the
dipole moments are all lined up with one another, producing a
macroscopic, non-zero magnetic field. These are the ordinary "magnets"
with which we are all familiar.
The study of the behavior of such "spin models" is a thriving cottage
industry in condensed matter physics. For instance, the Ising model
describes spins (dipoles) that have only two possible states, up and
down, whereas in the Heisenberg model the spin vector is allowed to
point in any direction. These models have many interesting properties,
which have led to many interesting results in the theory of phase
transitions. [2] [3]
[edit]
The spin-statistics connection
It turns out that the spin of a particle is closely related to its
properties in statistical mechanics. Particles with half-integer spin
obey Fermi-Dirac statistics, and are known as fermions. They are subject
to the Pauli exclusion principle, which forbids them from sharing
quantum states, and are described in quantum theory by "antisymmetric
states" (see the article on identical particles.) Particles with integer
spin, on the other hand, obey Bose-Einstein statistics, and are known as
bosons. These particles can share quantum states, and are described
using "symmetric states". The proof of this is known as the
spin-statistics theorem, which relies on both quantum mechanics and the
theory of special relativity. In fact, the connection between spin and
statistics is one of the most important and remarkable consequences of
special relativity.
[edit]
Applications
Well established applications of spin are Magnetic Resonance Imaging or
MRI, and GMR drive head technology in modern hard disks.
A possible application of spin is as a binary information carrier in
spin transistors. Electronics based on spin transistors is called
spintronics.
[edit]
History
Wolfgang Pauli was possibly the most influential physicist in the theory
of spin. Spin was first discovered in the context of the emission
spectrum of alkali metals. In 1924 Pauli introduced what he called a
"two-valued quantum degree of freedom" associated with the electron in
the outermost shell. This allowed him to formulate the Pauli exclusion
principle, stating that no two electrons can share the same quantum
numbers.
The physical interpretation of Pauli's "degree of freedom" was initially
unknown. Ralph Kronig, one of Landé's assistants, suggested in
early 1925 that it was produced by the self-rotation of the electron.
When Pauli heard about the idea, he criticized it severely, noting that
the electron's hypothetical surface would have to be moving faster than
the speed of light in order for it to rotate quickly enough to produce
the necessary angular momentum. This would violate the theory of
relativity. Largely due to Pauli's criticism, Kronig decided not to
publish his idea.
In the fall of that year, the same thought came to two young Dutch
physicists, George Uhlenbeck and Samuel Goudsmit. Under the advice of
Paul Ehrenfest, they published their results in a small paper. It met a
favorable response, especially after Llewellyn Thomas managed to resolve
a factor of two discrepancy between experimental results and Uhlenbeck
and Goudsmit's calculations (and Kronig's unpublished ones). This
discrepancy was due to the necessity to take into account the
orientation of the electron's tangent frame, in addition to its
position; mathematically speaking, a fiber bundle description is needed.
The tangent bundle effect is additive and relativistic (i.e. it vanishes
if c goes to infinity); it is one half of the value obtained without
regard for the tangent space orientation, but with opposite sign. Thus
the combined effect differs from the latter by a factor two (Thomas
precession).
Despite his initial objections to the idea, Pauli formalized the theory
of spin in 1927, using the modern theory of quantum mechanics discovered
by Schrödinger and Heisenberg. He pioneered the use of Pauli
matrices as a representation of the spin operators, and introduced a
two-component spinor wave-function.
Pauli's theory of spin was non-relativistic. However, in 1928, Paul
Dirac published the Dirac equation, which described the relativistic
electron. In the Dirac equation, a four-component spinor (known as a
"Dirac spinor") was used for the electron wave-function.
In 1940, Pauli proved the spin-statistics theorem, which states that
fermions have half-integer spin and bosons integer spin.
[edit]
See also
Angular momentum
Helicity
Pauli matrices
Representation theory of SU(2)
Spin tensor
Spinor
[edit]
References
Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.),
Prentice Hall. ISBN 013805326X.
"Spintronics. Feature Article" in Scientific American, June 2002
[edit]
External links
Goudsmit on the discovery of electron spin
Retrieved from "http://en.wikipedia.org/wiki/Spin_%28physics%29";
Categories: Rotational symmetry | Quantum field theory
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