Spin (physics) Properties Of Spin Mathematical Formulation Of

As a quantum mechanical property, spin possesses a number of qualities that distinguish it from classical angular momentum. It is quantized, and can only take on discrete values. For instance, the spin angular momentum of an electron, measured along any particular direction, can only take on the values +ℏ/2 or -ℏ/2 (where ℏ is Planck's constant divided by 2π). Furthermore, the magnitude of the spin (a direction-independent quantity) is uniquely determined by the type of particle. Electrons are said to be "spin-half" particles, because the magnitude of every electron's spin is one half times ℏ. Other spin-half particles include neutrinos, protons, and neutrons. This has lent weight to the hypothetical particle group called Quarks, primarily because the Quark Theory predicts the proton and neutron being made up of three quarks (which are spin-half particles), one of whose spin is in the opposite direction as the other two. Photons are spin-one particles, and the hypothetical graviton is a spin-two particle. Certain exotic particles, such as pionIn particle physics, pion (short for 'pi meson') is the collective name for three subatomic particles discovered in 1947: π0, π+ and π−. Pions are the lightest mesons. Basic properties Pions have zero spin and are composed of first generatis, possess spin zero. The principles of quantum mechanics indicate that spin is restricted to integer or half-integer values, at least under normal conditions.

Mathematically, spin is not described by a vectorA vector in physics and engineering typically refers to a quantity that has close relationship to the spatial coordinates, informally described as an object with a "magnitude" and a "direction". The word vector is also now used for more general concepts (, unlike classical angular momentum. It is described by objects known as spinorIn mathematics and physics, in particular in the theory of the orthogonal groups, spinors are certain kinds of mathematical objects ( group representations of SO(N), roughly speaking) similar to vectors, but which change sign under a rotation of 2π rads, which act differently from vectors under coordinate rotationIn linear algebra and geometry, a coordinate rotation is a transformation from one system of coordinates to another system of coordinates, such that distance between any two points remains invariant under the transformation. In other words, it is an isomes.

It turns out that the spin of a particle is closely related to its properties in statistical mechanicsStatistical mechanics is the application of statistics, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. It provides a fr. Particles with half-integer spin obey Fermi-Dirac statisticsIn statistical thermodynamics, Fermi- Dirac statistics determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. Fermions are particles which are indistinguishable and obey the Pauli exclusion principl, and are known as fermionFermions named after Enrico Fermi, are particles which form totally-antisymmetric composite quantum states. As a result, they are subject to the Pauli exclusion principle and obey Fermi-Dirac statistics. The spin-statistics theorem states that fermions has. They are subject to the Pauli exclusion principleThe Pauli exclusion principle commonly referred to simply as the exclusion principle is a quantum mechanical principle formulated by Wolfgang Pauli in 1925, which states that no two identical fermions may occupy the same quantum state. It is one of the mo, 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.

Particles with spin possess a magnetic moment, just like a rotating electrically charged body in classical physics. However, this magnetic moment exists even for point particles like the electron, and for electrically neutral particles like the neutron. This magnetic moment can be experimentally observed, by the deflection of particles by inhomogenous magnetic fields (as in the Stern-Gerlach experiment) or by the magnetic fields generated by the particles themselves. In fact, ferromagnetism arises from the alignment of the spins of the atoms in a solid.

2 Mathematical formulation of Spin 1/2

The spin operator S behaves very much like L (angular momenum) where l = 1/2. The spin operator has two eigenvalues: , which corresponds for two eigenstates - spin up and spin down. It is common to measure the spin in one direction - and the corresponding operator will be where n is a unit vector in the desired direction and

is the vectorial spin operator and the σ-s are Pauli matrices.

For example: Lets measure the spin in the z direction (in cartesian coordinates). Then S_z has two eigenstates - spin up and spin down. If we assign coordinates vectors as follow

then the corresponding operator in that representation will be

In the matrix representation, the operator acts on the coordinate vectors, often called " Spinors".

3 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 L.H. 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.

Magnetic material may be modelled by a system of spins located at positions in a lattice, where the interaction of neighboring spins contributes to the total energy of the system and the states of the spins change according to some non-deterministic (probabilistic) rule (the dynamics of the system). In the Ising model spins have only two possible states (up and down), whereas in the Potts model they may have more than two possible states. This is discussed in detail in Spin Models (http://www.hermetic.ch/compsci/thesis/chap1.htm), particularly in the section Modelling Magnetic Material (http://www.hermetic.ch/compsci/thesis/chap1.htm#s1.3) and subsequent sections.

4 Applications

Well established applications of spin are Magnetic Resonance Imaging or MRI, and GMR drive head technology in modern hard drives

A possible application of spin is as a binary information carrier in spin transistor s. Electronics based on spin transistors is called spintronics.

5 See also

6 References

"Spintronics. Feature Article" in Scientific American, June 2002

Quantum mechanics

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1 Properties of spin