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It follows as with angular momentum that the eigenvectors of S 2 and S z expressed as kets in the total S basis are:. The spin raising and lowering operators acting on these eigenvectors give:.
But unlike orbital angular momentum the eigenvectors are not spherical harmonics. There is also no reason to exclude half-integer values of s and m s.
In addition to their other properties, all quantum mechanical particles possess an intrinsic spin though this value may be equal to zero.
One distinguishes bosons integer spin and fermions half-integer spin. The total angular momentum conserved in interaction processes is then the sum of the orbital angular momentum and the spin.
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.
In quantum mechanics all particles are either bosons or fermions. In some speculative relativistic quantum field theories " supersymmetric " particles also exist, where linear combinations of bosonic and fermionic components appear.
The above permutation postulate for N -particle state functions has most-important consequences in daily life, e. As described above, quantum mechanics states that components 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.
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 the 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.
Starting with S x. Using the spin operator commutation relations , we see that the commutators evaluate to i S y for the odd terms in the series, and to S x for all of the even terms.
Note that since we only relied on the spin operator commutation relations, this proof holds for any dimension i. A generic rotation in 3-dimensional space can be built by compounding operators of this type using Euler angles :.
An irreducible representation of this group of operators is furnished by the Wigner D-matrix :. Recalling that a generic spin state can be written as a superposition of states with definite m , we see that if s is an integer, the values of m are all integers, and this matrix corresponds to the identity operator.
This fact is a crucial element of the proof of the spin-statistics theorem. We could try the same approach to determine the behavior of spin under general Lorentz transformations , but we would 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.
These spinors transform under Lorentz transformations according to the law. It can be shown that the scalar product.
The corresponding normalized eigenvectors are:. Because any eigenvector multiplied by a constant is still an eigenvector, there is ambiguity about the overall sign.
In this article, the convention is chosen to make the first element imaginary and negative if there is a sign ambiguity. The present convention is used by software such as sympy; while many physics textbooks, such as Sakurai and Griffiths, prefer to make it real and positive.
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 corresponding spin operator S x , S y or S z on that axis, i.
The quantum state of a particle with respect to spin , can be represented by a two component spinor :. Following the measurement, the spin state of the particle will collapse into the corresponding eigenstate.
The operator to measure spin along an arbitrary axis direction is easily obtained from the Pauli spin matrices. Then the operator for spin in this direction is simply.
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.
In quantum mechanics, vectors are termed "normalized" when multiplied by a normalizing factor, which results in the vector having a length of unity.
Since the Pauli matrices do not commute , 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. This implies that the 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.
By taking Kronecker products of this representation with itself repeatedly, one may construct all higher irreducible representations.
That is, the resulting spin operators for higher spin systems in three spatial dimensions, for arbitrarily large s , can be calculated using this spin operator and ladder operators.
The resulting irreducible representations yield the following spin matrices and eigenvalues in the z-basis. Also useful in the quantum mechanics of multiparticle systems, the general Pauli group G n is defined to consist of all n -fold tensor products of Pauli matrices.
The analog formula of Euler's formula in terms of the Pauli matrices :. For example, see the isotopes of bismuth in which the List of isotopes includes the column Nuclear spin and parity.
Spin has important theoretical implications and practical applications. Well-established direct applications of spin include:. Electron spin plays an important role in magnetism , with applications for instance in computer memories.
The manipulation of nuclear spin by radiofrequency waves nuclear magnetic resonance is important in chemical spectroscopy and medical imaging.
Spin-orbit coupling leads to the fine structure of atomic spectra, which is used in atomic clocks and in the modern definition of the second.
Precise measurements of the g -factor of the electron have played an important role in the development and verification of quantum electrodynamics.
Photon spin is associated with the polarization of light photon polarization. An emerging application of spin is as a binary information carrier in spin transistors.
The original concept, proposed in , is known as Datta-Das spin transistor. The manipulation of spin in dilute magnetic semiconductor materials , such as metal-doped ZnO or TiO 2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.
There are many indirect applications and manifestations of spin and the associated Pauli exclusion principle , starting with the periodic table of chemistry.
Spin was first discovered in the context of the emission spectrum of alkali metals. In , Wolfgang Pauli introduced what he called a "two-valuedness not describable classically"  associated with the electron in the outermost shell.
This allowed him to formulate the Pauli exclusion principle , stating that no two electrons can have the same quantum state in the same quantum system.
The physical interpretation of Pauli's "degree of freedom" was initially unknown. 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.
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 results.
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 , known to Ludwik Silberstein in Despite his initial objections, Pauli formalized the theory of spin in , using the modern theory of quantum mechanics invented 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.
But in the valence shell of silver atoms, there is a single electron whose spin remains unbalanced. The unbalanced spin creates spin magnetic moment , making the electron act like a very small magnet.
As the atoms pass through the in-homogeneous magnetic field, the force moment in the magnetic field influences the electron's dipole until its position matches the direction of the stronger field.
The atom would then be pulled toward or away from the stronger magnetic field a specific amount, depending on the value of the valence electron's spin.
Thus the beam of silver atoms is split while traveling through the in-homogeneous magnetic field, according to the spin of each atom's valence electron.
In Phipps and Taylor conducted a similar experiment, using atoms of hydrogen with similar results. Later scientists conducted experiments using other atoms that have only one electron in their valence shell: copper , gold , sodium , potassium.
Every time there were two lines formed on the metallic plate. The atomic nucleus also may have spin, but protons and neutrons are much heavier than electrons about times , and the magnetic dipole moment is inversely proportional to the mass.
So the nuclear magnetic dipole momentum is much smaller than that of the whole atom. This small magnetic dipole was later measured by Stern, Frisch and Easterman.
In , Paul Dirac developed a relativistic wave equation , now termed the Dirac equation , which predicted the spin magnetic moment correctly, and at the same time treated the electron as a point-like particle.
Solving the Dirac equation for the energy levels of an electron in the hydrogen atom, all four quantum numbers including s occurred naturally and agreed well with experiment.
For some atoms the spins of several unpaired electrons s 1 , s 2 , Here L is the total orbital angular momentum quantum number.
From Wikipedia, the free encyclopedia. Quantum number parameterizing spin and angular momentum. Starts With A Bang.
Retrieved The History of the Laser. CRC Press. Retrieved 22 March Physical Chemistry 8th ed. Freeman , p. Electron configuration. Electron shell Atomic orbital Quantum mechanics Introduction to quantum mechanics.
Periodic table electron configurations Electron configurations of the elements data page. Pauli exclusion principle Hund's rule Aufbau principle.