Spin (physics)

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This article is about intrinsic angular momentum. For a related concept see rotation. For other uses see spin.

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, such as individual atoms, protons, or electrons. The effects of quantum mechanics are important when describing such particles and the spin of quantum mechanical systems possesses several unusual features, which will be described in the remainder of this article. For such systems, associating the spin angular momentum with rotation is not possible, so the word "spin" does not connote rotation but rather only the presence of angular momentum.

(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.)

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

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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, akin to a particle's electric charge and mass.

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 <math>\hbar</math> is Planck's constant divided by 2π (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]

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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, but 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

<math>\hbar s_z, \qquad s_z = - s, - s + 1, \cdots, s</math>

where s is the principal spin quantum number discussed in the previous section. One can see that there are 2s+1 possible values of s_z. For example, there are only two possible values for a spin-1/2 particle: s_z = +1/2 and s_z = -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 ⟨S⟩ whose components are the expectation values of the spin components along each axis, i.e., ⟨S⟩ = [⟨s_x⟩, ⟨s_y⟩, ⟨s_z⟩]. 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 — s_x, s_y and s_z 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 — 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!

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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 2.0023193043768(86), with the first 12 figures certain. 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 0.00231930437... arises from the electron's interaction with the surrounding electromagnetic field, including its own 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 × 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]

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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.

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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.

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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.

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