Spin (physics)

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In physics and chemistry, spin has a special meaning, representing a non-classical kind of angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. Although this special property is only explained in the relativistic quantum mechanics of Paul Dirac, it plays a most-important role already in non-relativistic quantum mechanics, e.g., it essentially determines the structure of atoms.

In classical mechanics, any spin angular momentum of a body is associated with self rotation, e.g., the rotation of the body around its own center of mass. For example, the spin of the Earth is associated with its daily rotation about the polar axis. On the other hand, the orbital angular momentum of the Earth is associated with its annual motion around the Sun.

In fact, in classical theories there is no analogue to the quantum mechanical property meant by the name spin. The concept of this nonclassical property of elementary particles was first proposed in 1925 by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit; but the name related to the phenomenon of spin in physics is Wolfgang Pauli.

Spin in quantum mechanics

In quantum mechanics, spin is an intrinsic property of all elementary particles related to angular momentum.

Spin obeys commutation relations analogous to those of the orbital angular momentum:


It follows (as with angular momentum) that the eigenvectors of and (expressed as kets in the total basis) are:


The raising and lowering spin operators acting on these eigenvectors gives:

, where

But unlike orbital angular momentum the eigenvectors are not spherical harmonics. They are not functions of and . There is also no reason to exclude half integer values of s and m.

In quantum mechanics, the non-classical property spin is especially important for systems at atomic length scales, such as individual atoms, protons, or electrons. Such particles and the spin of quantum mechanical systems ("particle spin") possess several non-classical features and for such systems spin angular momentum cannot be associated with rotation but instead refers only to the presence of an 'angular momentum-like' property. (Note that particles are quantum mechanical entities, which also exhibit wave-like behavior due to the so-called wave-particle duality.)

Precisely, in addition to their other properties, all quantum mechanical particles possess the above-mentioned non-classical kind of intrinsic "spin". This is quantized in units of the reduced action constant , such that the state function of the particle is, e.g., not , but where is out of the following discrete set of values:     . One distinguishes bosons (S=0 or 1 or 2 or ...) and fermions (S=1/2 or 3/2 or 5/2 or ...). The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.

Spin and the Pauli exclusion principle

For systems of N identical particles this is related to the Pauli exclusion principle, which states that by interchanges of any two of the N particles one must have

Thus, for bosons the prefactor will reduce to +1, for fermions, in contrast, to (-1). In quantum mechanics all particles are either bosons or fermions. In relativistic quantum field theories also "supersymmetric" particles exist, where linear combinations of bosonic and fermionic components appear. Only in two dimensions you are allowed to replace the prefactor by any complex number of magnitude 1
(-> anyons).

Electrons are fermions with S=1/2; quanta of light ("photons") are bosons with S=1. This shows also explicitly that the property spin cannot be fully explained as a classical intrinsic orbital angular momentum, e.g., similar to that of a "spinning top", since orbital angular rotations would lead to integer values of s. Instead one is dealing with an essential legacy of relativity. The photon, in contrast, is always relativistic (velocity , and the corresponding classical theory, that of Maxwell, is also relativistic.

The above permutation postulate for N-particle state functions has most-important consequences in daily life, e.g. the already mentioned periodic table of the chemists or biologists.

After this condensed presentation of some essentials, a broad overview is given:

Broad overview

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 can be determined, 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.

According to quantum mechanics, the angular momentum of any system is quantized. The magnitude of angular momentum, , can only take on the values according to this relation:

where is the reduced Planck'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 on 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 hypothetical Higgs boson is unique among elementary particles in having a spin of zero.

The spin of composite particles, such as protons, neutrons, atomic nuclei, and atoms, is made up of the spins of the constituent particles, and their total angular momentum is the sum of their spin and 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 valence quarks and the surrounding sea quarks and gluons is an active area of research.

History

Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang 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 state at the same time.

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 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit. Under the advice of Paul Ehrenfest, they published their results. 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 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; that is, 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, 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.

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

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 whose components are the expectation values of the spin components along each axis, i.e., . 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 — sx, sy and sz cannot possess simultaneous definite values, because of a quantum uncertainty relation between them. However, for statistically large collections of particles that have been placed in the same pure quantum state, such as through the use of a Stern-Gerlach apparatus, the spin vector does have a well-defined experimental meaning: It specifies the direction in ordinary space in which a subsequent detector must be oriented in order to achieve the maximum possible probability (100%) of detecting every particle in the collection. For spin-1/2 particles, this maximum probability drops off smoothly as the angle between the spin vector and the detector increases, until at an angle of 180 degrees —that is, for detectors oriented in the opposite direction to the spin vector—the expectation of detecting particles from the collection reaches a minimum of 0%.

As a qualitative concept, 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. A spin zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle 180 degree can bring it back to the same quantum state and a spin-4 particle should be rotated 90 degrees to bring it back to the same quantum state. The spin 2 particle can be analogous to a straight stick that looks the same even after it is rotated 180 degrees and a spin 0 particle can be imagined as sphere which looks the same after whatever angle it is turned through.

Spin and rotations

As described above, quantum mechanics states that component of angular momentum measured along any direction can only take a number of discrete values. The most convenient quantum mechanical description of particle's spin is therefore with a set of complex numbers corresponding to amplitudes of finding a given value of projection of its intrinsic angular momentum on a given axis. For instance, for a spin 1/2 particle, we would need two numbers , giving amplitudes of finding it with projection of angular momentum equal to and , satisfying the requirement

Since these numbers depend on the choice of the axis, they transform into each other non-trivially when this axis is rotated. It's clear that the transformation law must be linear, so we can represent it by associating a matrix with each rotation, and the product of two transformation matrices corresponding to rotations A and B must be equal (up to phase) to the matrix representing rotation AB. Further, rotations preserve quantum mechanical inner product, and so should our transformation matrices:

Mathematically speaking, these matrices furnish a unitary projective representation of the rotation group SO(3). Each such representation corresponds to a representation of the covering group of SO(3), which is SU(2). There is one irreducible representation of SU(2) for each dimension. For example, spin 1/2 particles transform under rotations according to a 2-dimensional representation, which is generated by Pauli matrices:

where are Euler angles.

Particles with higher spin transform in a similar way using higher-dimensional representations; see the article on Pauli matrices for matrices generating rotations for spin 1 and 3/2.

Spin and Lorentz transformations

We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we'd immediately discover a major obstacle. Unlike SO(3), the group of Lorentz transformations SO(3,1) is non-compact and therefore does not have any faithful unitary finite-dimensional representations.

In case of spin 1/2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product that is preserved by this representation. We associate a 4-component Dirac spinor with each particle. These spinors transform under Lorentz transformations according to the law

where are gamma matrices and is an antisymmetric 4x4 matrix parametrizing the transformation. It can be shown that the scalar product

is preserved. (It is not, however, positive definite, so the representation is not unitary.)

Spin and magnetic moments

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 g-factor. For exclusively orbital rotations it would be 1.

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 0.00231456893... arises from the electron's interaction with the surrounding electromagnetic field, including its own field.[citation needed]

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 electrically 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 2001, the latest experimental results have put the neutrino magnetic moment at less than 1.2 × 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. Ferromagnetic materials below their Curie temperature, however, exhibit magnetic domains in which the atomic dipole moments are locally aligned, producing a macroscopic, non-zero magnetic field from the domain. These are the ordinary "magnets" with which we are all familiar.

The study of the behavior of such "spin models" is a thriving area of research 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 interesting results in the theory of phase transitions.

The spin-statistics connection

The spin of a particle has crucial consequences for its properties in statistical mechanics. Particles with half-integer spin obey Fermi-Dirac statistics, and are known as fermions. They are required to occupy antisymmetric quantum states (see the article on identical particles.) This property forbids fermions from sharing quantum states - a restriction known as the Pauli exclusion principle. Particles with integer spin, on the other hand, obey Bose-Einstein statistics, and are known as bosons. These particles occupy "symmetric states", and can therefore share quantum 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.

Mathematical formulation of spin in quantum mechanics

Pauli matrices and spin operators

The quantum mechanical operators associated with spin observables are:

In the special case of spin-1/2 , and are the three Pauli matrices, given by:

Measurement of the spin along the x, y, and z axes

Each of the (hermitian) Pauli matrices has two eigenvalues, +1 and -1. The corresponding normalized eigenvectors are:

, , , , , .

By the postulates of quantum mechanics, an experiment designed to measure the electron spin on the x, y or z axis can only yield an eigenvalue of the spin operator (, , ) on that axis, and . The quantum state of a particle (with respect to spin), can be represented by a two component spinor:

When the spin of this particle is measured with respect to a given axis (in this example, the x-axis), the probability that its spin will be measured as is just . Correspondingly, the probability that its spin will be measured as is just . Following the measurement, the spin state of the particle will collapse into the corresponding eigenstate. As a result, if the particle's spin along a given axis has been measured to have a given eigenvalue, all measurements will yield the same eigenvalue (since , etc), provided that no measurements of the spin are made along other axes (see compatibility section below).

Measurement of the spin along an arbitrary axis

The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Let be an arbitrary unit vector. Then the operator for spin in this direction is simply . The operator has eigenvalues of , just like the usual Pauli spin matrices. This method of finding the operator for spin in an arbitrary direction generalizes to higher spin states, one takes the dot product of the direction with a vector of the three operators for the three x,y,z axis directions.

A normalized spinor for spin-1/2 in the direction (which works for all spin states except spin down where it will give 0/0), is:

The above spinor is obtained by normalizing the left column of the matrix where "1" is the 2x2 unit matrix. This trick of writing the eigenvectors of the Pauli matrices depends on certain details of the density matrix representation of quantum states.

Compatibility of spin measurements

Since the Pauli matrices anticommute, measurements of spin along the different axes are incompatible. This means that if, for example, we know the spin along the x-axis, and we then measure the spin along the y-axis, we have invalidated our previous knowledge of the x-axis spin. This can be seen from the property of the eigenvectors (i.e. eigenstates) of the Pauli matrices that:

So when we measure the spin of a particle along the x-axis as, for example, , the particle's spin state collapses into the eigenstate . When we then subsequently measure the particle's spin along the y-axis, the spin state will now collapse into either or , each with probability . Let us say, in our example, that we measure . When we now return to measure the particle's spin along the x-axis again, the probabilities that we will measure or are each (i.e. they are and ). This implies that our original measurement of the spin along the x-axis is no longer valid, since the spin along the x-axis will now be measured to have either eigenvalue with equal probability.

Direct and indirect applications

Well established direct applications of spin are nuclear magnetic resonance spectroscopy in chemistry; electron spin resonance spectroscopy in chemistry and physics; proton spin density with magnetic resonance imaging (MRI) in medicine; 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.

But finally we remind to the many indirect applications based on spin and the Pauli principle, e.g. the periodic table of Dmitri Mendeleev.

See also

References

  • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-111892-7.
  • Shankar, R. (1994). "chapter 14-Spin". Principles of Quantum Mechanics (2nd ed.). Springer. ISBN 0-306-44790-8.
  • Lawden, Derek (2005). The Mathematical Principles of Quantum Mechanics. Dover. ISBN 0-486-44223-3.

External links

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