Spin (physics)

For other uses see Spin or Rotation.

In physics and chemistry, spin is the 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.

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 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. The concept of elementary particle spin was first proposed in 1925 by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit.

In quantum mechanics, spin is particularly 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") possesses several non-classical features and for such systems spin angular momentum cannot be associated with rotation but instead refers only to the presence of angular momentum. This article, to note, uses the term "particle" to refer to quantum mechanical systems, with the understanding that such actually exhibit wave-particle duality, and thus display both particle-like and wave-like behaviors.

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 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.

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

$S=\hbar \,{\sqrt {s(s+1)}},$

where $\hbar$ 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 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.

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

$\hbar s_{z},\qquad s_{z}=-s,-s+1,\cdots ,s-1,s$

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 $\langle S\rangle$ whose components are the expectation values of the spin components along each axis, i.e., $\langle S\rangle =[\langle s_{x}\rangle ,\langle s_{y}\rangle ,\langle s_{z}\rangle ]$. 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. 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!

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 $a_{\pm 1/2}$, giving amplitudes of finding it with projection of angular momentum equal to $\hbar /2$ and $-\hbar /2$, satisfying the requirement

$|a_{1/2}|^{2}+|a_{-1/2}|^{2}\,=1$

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:

$\sum _{m=-j}^{j}a_{m}^{*}b_{m}=\sum _{m=-j}^{j}(\sum _{n=-j}^{j}U_{nm}a_{n})^{*}(\sum _{k=-j}^{j}U_{km}b_{k})$

$\sum _{n=-j}^{j}\sum _{k=-j}^{j}U_{np}^{*}U_{kq}=\delta _{pq}$

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:

${\begin{pmatrix}a_{1/2}'\\a_{-1/2}'\end{pmatrix}}=\exp {(i\sigma _{z}\gamma /2)}\exp {(i\sigma _{x}\beta /2)}\exp {(i\sigma _{z}\alpha /2)}{\begin{pmatrix}a_{1/2}\\a_{-1/2}\end{pmatrix}}$

where $\alpha ,\beta ,\gamma$ 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 other than the trivial one.

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 $\psi$ with each particle. These spinors transform under Lorentz transformations according to the law

$\psi '=\exp {\left({\frac {1}{8}}\omega _{\mu \nu }[\gamma _{\mu },\gamma _{\nu }]\right)}\psi$

where $\gamma _{\mu }$ are gamma matrices and $\omega _{\mu \nu }$ is an antisymmetric 4x4 matrix parametrizing the transformation. It can be shown that the scalar product

$\langle \psi |\phi \rangle ={\bar {\psi }}\phi =\psi ^{\dagger }\gamma _{0}\phi$

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

$\mu =g\,{\frac {q}{2m}}\,S$

where the dimensionless quantity g is called the 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 0.00231456893... 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 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:

$S_{x}={\hbar \over 2}\sigma _{x}$

$S_{y}={\hbar \over 2}\sigma _{y}$

$S_{z}={\hbar \over 2}\sigma _{z}$

In the special case of spin-1/2 $\sigma _{x}$, $\sigma _{y}$ and $\sigma _{z}$ are the three Pauli Matrices, given by:

$\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}$

$\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}$

$\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$

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:

$\psi _{x+}={\begin{pmatrix}{1 \over {\sqrt {2}}}\\{1 \over {\sqrt {2}}}\end{pmatrix}}$, $\psi _{x-}={\begin{pmatrix}{1 \over {\sqrt {2}}}\\{-1 \over {\sqrt {2}}}\end{pmatrix}}$ $\psi _{y+}={\begin{pmatrix}{1 \over {\sqrt {2}}}\\{-i \over {\sqrt {2}}}\end{pmatrix}}$, $\psi _{y-}={\begin{pmatrix}{1 \over {\sqrt {2}}}\\{i \over {\sqrt {2}}}\end{pmatrix}}$, $\psi _{z+}={\begin{pmatrix}1\\0\end{pmatrix}}$, $\psi _{z-}={\begin{pmatrix}0\\1\end{pmatrix}}$.

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 ($S_{x}$, $S_{y}$, $S_{z}$) on that axis, ${\hbar \over 2}$ and ${-\hbar \over 2}$. The quantum state of a particle (with respect to spin), can be represented by a two component spinor:

$\psi ={\begin{pmatrix}{a+bi}\\{c+di}\end{pmatrix}}$

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 ${\hbar \over 2}$ is just $\mid \langle \psi \mid \psi _{x+}\rangle \mid ^{2}$. Correspondingly, the probability is will be measured as ${-\hbar \over 2}$ is just $\mid \langle \psi \mid \psi _{x-}\rangle \mid ^{2}$. 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 $\mid \langle \psi _{x+}\mid \psi _{x+}\rangle \mid ^{2}=1$, etc), provided that no measurements of the spin are made along other axes (see section compatibility below).

Measurement of the Spin along an Arbitrary Axis

If we measure the spin along an arbitrary axis, at arbitrary angles $\phi$ and $\theta$ (see spherical coordinates), then we can construct an arbitrary spin operator:

$S_{arb}=\cos \theta \cos \phi S_{x}+\sin \theta \cos \phi S_{y}+\sin \phi S_{z}={\hbar \over 2}(\cos \theta \cos \phi \sigma _{x}+\sin \theta \cos \phi \sigma _{y}+\sin \phi \sigma _{z})$

Since the sum of hermitian matrices is itself hermitian, $S_{arb}$ will have only real eigenvalues. But what are these eigenvalues? To determine this, we must note two properties of the Pauli matrices:

• $\sigma _{x}^{2}=\sigma _{y}^{2}=\sigma _{z}^{2}=I$
• Any pair of distinct Pauli matrices anticommute, e.g. $\sigma _{x}\sigma _{y}=-\sigma _{y}\sigma _{x}$

We now construct the operator corresponding to the square of the spin measured along our arbitrary axis:

$S_{arb}^{2}={\frac {\hbar ^{2}}{4}}(\cos \theta \cos \phi \sigma _{x}+\sin \theta \cos \phi \sigma _{y}+\sin \phi \sigma _{z})(\cos \theta \cos \phi \sigma _{x}+\sin \theta \cos \phi \sigma _{y}+\sin \phi \sigma _{z})={\frac {\hbar ^{2}}{4}}I$

Where the cross terms vanish because of anticommutation, and the square terms add up to identity. It follows that the magnitude of any eigenvalues of $S_{arb}$ must be either ${\hbar \over 2}$ or ${-\hbar \over 2}$. Therefore, the spin can only take these two allowed values when measured over any axis.

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:

$\mid \langle \psi _{x+/-}\mid \psi _{y+/-}\rangle \mid ^{2}=\mid \langle \psi _{x+/-}\mid \psi _{z+/-}\rangle \mid ^{2}=\mid \langle \psi _{y+/-}\mid \psi _{z+/-}\rangle \mid ^{2}={\frac {1}{2}}$

So when we measure the spin of a particle along the x-axis as, for example, ${\hbar \over 2}$, the particle's spin state collapses into the eigenstate $\mid \psi _{x+}\rangle$. When we then subsequently measure the particle's spin along the y-axis, the spin state will now collapse into either $\mid \psi _{y+}\rangle$ or $\mid \psi _{y-}\rangle$, each with probability ${\frac {1}{2}}$. Let us say, in our example, that we measure ${-\hbar \over 2}$. When we now return to measure the particle's spin along the x-axis again, the probabilities that we will measure ${\hbar \over 2}$ or ${-\hbar \over 2}$ are each ${\frac {1}{2}}$ (i.e. they are $\mid \langle \psi _{x+}\mid \psi _{y-}\rangle \mid ^{2}$ and $\mid \langle \psi _{x-}\mid \psi _{y-}\rangle \mid ^{2}$). 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.

Applications

Well established applications of spin are nuclear magnetic resonance spectroscopy in chemistry, electron spin resonance spectroscopy in chemistry and physics, 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.

See also

• Angular momentum
• Helicity
• Pauli matrices
• Rarita-Schwinger equation
• Representation theory of SU(2)
• Spin quantum number
• Spin-1/2
• Spin tensor
• Spinor
• Yrast

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.

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

External links

• Goudsmit on the discovery of electron spin