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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. 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.
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Repeat this sequence at least four times, but preferably for a few minutes. This practice is often used in Cognitive Behavioral Therapy sessions and can be quite effective in disrupting anxiety lingering in your subconscious.
If your thoughts are going a mile a minute, try jotting some realistic thinking aka evidence-based realities into your journal to calm your mind and extinguish the fear.
Finally, consider how you can make your life kinder for your mind, body, and soul moving forward. The goal is to find what is sustainable and works for you.
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Meltzer Marilyn Manson William S. The Verge. Retrieved February 14, International Business Times. Retrieved November 8, Retrieved May 29, San Francisco Chronicle.
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