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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. This phenomenon is known as electron spin resonance (ESR).
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. When the spin of the electron is + + 1 / 2 the atom moves away from the stronger field, and when the spin is − + 1 / 2 the atom moves toward it. Thus the beam of silver atoms is ...
The spin magnetic moment of the electron is =, where is the spin (or intrinsic angular-momentum) vector, is the Bohr magneton, and = is the electron-spin g-factor. Here μ {\displaystyle {\boldsymbol {\mu }}} is a negative constant multiplied by the spin , so the spin magnetic moment is antiparallel to the spin.
c is the speed of light, and ħ is the reduced Planck constant; these fundamental physical constants reflect special relativity and quantum mechanics, respectively. α and β are 4x4 Gamma matrices encoding the spin properties of relativistic particles and ensuring the equation's correct transformation under Lorentz transformations.
Spin and magnetic moment are in the same direction for γ > 0 (as for protons). Protons, neutrons, and many nuclei carry nuclear spin, which gives rise to a gyromagnetic ratio as above. The ratio is conventionally written in terms of the proton mass and charge, even for neutrons and for other nuclei, for the sake of simplicity and consistency.
A fundamental physical constant occurring in quantum mechanics is the Planck constant, h. A common abbreviation is ħ = h /2 π , also known as the reduced Planck constant or Dirac constant . Quantity (common name/s)
The spin–statistics theorem proves that the observed relationship between the intrinsic spin of a particle (angular momentum not due to the orbital motion) and the quantum particle statistics of collections of such particles is a consequence of the mathematics of quantum mechanics.
Note that μ is a negative constant multiplied by the spin, so the magnetic moment is antiparallel to the spin angular momentum. The spin g-factor g s = 2 comes from the Dirac equation, a fundamental equation connecting the electron's spin with its electromagnetic properties. Reduction of the Dirac equation for an electron in a magnetic field ...