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This phenomenon is known as electron spin resonance ESR. The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance NMR spectroscopy and imaging.

Mathematically, quantum-mechanical spin states are described by vector-like objects known as spinors. There are subtle differences between the behavior of spinors and vectors under coordinate rotations.

To return the particle to its exact original state, one needs a degree rotation. The Plate trick and Möbius strip give non-quantum analogies.

A spin-zero particle can only have a single quantum state, even after torque is applied. Rotating a spin-2 particle degrees 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 degrees and a spin-0 particle can be imagined as sphere, which looks the same after whatever angle it is turned through.

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

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