Spin (physics)
Spinors and bispinors behave similarly to vectors: they have definite magnitudes and change under rotations; however, they use an unconventional "direction".
The earliest models for electron spin imagined a rotating charged mass, but this model fails when examined in detail: the required space distribution does not match limits on the electron radius: the required rotation speed exceeds the speed of light.
Wolfgang Pauli, a central figure in the history of quantum spin, initially rejected any idea that the "degree of freedom" he introduced to explain experimental observations was related to rotation.
[6] The first classical model for spin proposed a small rigid particle rotating about an axis, as ordinary use of the word may suggest.
Therefore it is a matter of interpretation whether the Hamiltonian must include such a term, and whether this aspect of classical mechanics extends into quantum mechanics (any particle's intrinsic spin angular momentum, S, is a quantum number arising from a "spinor" in the mathematical solution to the Dirac equation, rather than being a more nearly physical quantity, like orbital angular momentum L).
Pauli described this connection between spin and statistics as "one of the most important applications of the special relativity theory".
Particles with spin can possess a magnetic dipole moment, just like a rotating electrically charged body in classical electrodynamics.
[16] The value of 2 arises from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties; and the deviation from −2 arises from the electron's interaction with the surrounding quantum fields, including its own electromagnetic field and virtual particles.
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.
The equivalent behaviour of protons in atomic nuclei is used in nuclear magnetic resonance (NMR) spectroscopy and imaging.
This permutation postulate for N-particle state functions has most important consequences in daily life, e.g. the periodic table of the chemical elements.
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.
For instance, for a spin-1/2 particle, we would need two numbers a±1/2, giving amplitudes of finding it with projection of angular momentum equal to +ħ/2 and −ħ/2, satisfying the requirement
We could try the same approach to determine the behavior of spin under general Lorentz transformations, but we would immediately discover a major obstacle.
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.)
The above spinor is obtained in the usual way by diagonalizing the σu matrix and finding the eigenstates corresponding to the eigenvalues.
Precise measurements of the g-factor of the electron have played an important role in the development and verification of quantum electrodynamics.
The manipulation of spin in dilute magnetic semiconductor materials, such as metal-doped ZnO or TiO2 imparts a further degree of freedom and has the potential to facilitate the fabrication of more efficient electronics.
[32] There are many indirect applications and manifestations of spin and the associated Pauli exclusion principle, starting with the periodic table of chemistry.
Additional information was known from changes to atomic spectra observed in strong magnetic fields, known as the Zeeman effect.
In 1924, Wolfgang Pauli used this large collection of empirical observations to propose a new degree of freedom,[7] introducing what he called a "two-valuedness not describable classically"[34] associated with the electron in the outermost shell.
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.
[36] The young physicists immediately regretted the publication: Hendrik Lorentz and Werner Heisenberg both pointed out problems with the concept of a spinning electron.
Fortunately, by February 1926, Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results for the fine structure in the hydrogen spectrum and calculations based on Uhlenbeck and Goudsmit's (and Kronig's unpublished) model.
[7] Thomas' result convinced Pauli that electron spin was the correct interpretation of his two-valued degree of freedom, while he continued to insist that the classical rotating charge model is invalid.
[38] The original interpretation assumed the two spots observed in the experiment were due to quantized orbital angular momentum.
However, in 1927 Ronald Fraser showed that Sodium atoms are isotropic with no orbital angular momentum and suggested that the observed magnetic properties were due to electron spin.
[40] Once the quantum theory became established, it became clear that the original interpretation could not have been correct: the possible values of orbital angular momentum along one axis is always an odd number, unlike the observations.
