Chapter 13
Spin Operators
Abstract Spin operators are introduced in this chapter. The spin 1/2 and 1 are
looked upon explicitly. Projectors into magnetic sub-states and irreducible spin ten-
sors are defined. Spin traces of multiple products of these tensors and their role for
the expansion of density operators and the evaluation of averages are elucidated.
The last section deals with the rotational angular momenta of linear molecules, in
particular with tensor operators. One application is the anisotropic dielectric tensor
of a gas of rotating molecules.
The orbital angular momentum of particles is linked with their linear momentum.
Most elementary particles, like electrons, protons, neutrons, neutrinos posses an
intrinsic angular momentum, conventionally called ‘spin’, which is not caused by
their translational motion. Here properties of spin operators are discussed and rules
are presented for tensors constructed from the Cartesian components of the spin
operators. Furthermore, tensor operators associated with the rotational motion of
linear molecules are treated.
13.1 Spin Commutation Relations
13.1.1 Spin Operators and Spin Matrices.
The spin operator of a particle with spins, in units of, is denoted bys.The
components of this operator can be represented by hermitian( 2 s+ 1 )×( 2 s+ 1 )
matrices. Fors=^12 , e.g. these are the two-by-two Pauli matrices. The quantitysis
a positive integer or halve-integer number, viz.s=^12 ,ors=1, ors=^32 ,oretc.
The Cartesian components of the spin operator obey the angular momentum com-
mutation relations, analogous to those of the dimensionless orbital angular momen-
tum, cf. Sect.7.6.2. The spin commutation relations
sμsν−sνsμ≡[sμ,sν]−=iεμνλsλ. (13.1)© Springer International Publishing Switzerland 2015
S. Hess,Tensors for Physics, Undergraduate Lecture Notes in Physics,
DOI 10.1007/978-3-319-12787-3_13
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