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r
r [(j σ ⋅φ ê )] m
exp(jσ ⋅ êφ ) = ∑ (8.56)
m m !

Now separating the even power and odd power terms, using the just derived
r
result for the odd and even powers of (σ⋅ ê ), and Taylor expansions of the
cosine and sine functions, we obtain the desired result.

Example 8.15
Find the time development of the spin state of an electron in a constant mag-
netic flux density.

Solution: [For readers not interested in the physical background of this prob-
lem, they can immediately jump to the paragraph following Eq. (8.59).]

Physical Background: In addition to the spatio-temporal dynamics, the elec-
tron and all other elementary particles of nature also have internal degrees of
freedom; which means that even if the particle has no translational motion,
its state may still evolve in time. The spin of a particle is such an internal
degree of freedom. The electron spin internal degree of freedom requires for
its representation a two-dimensional vector, that is, two fundamental states
are possible. As may be familiar to you from your elementary chemistry
courses, the up and down states of the electron are required to satisfactorily
describe the number of electrons in the different orbitals of the atoms. For the
up state, the eigenvalue of the spin matrix is positive; while for the down
state, the eigenvalue is negative (respectively h/2 and –h/2, where h = 1.0546
–34
× 10 J.s = h/(2π), and h is Planck’s constant).
Due to spin, the quantum mechanical dynamics of an electron in a mag-
netic flux density does not only include quantum mechanically the time
development equivalent to the classical motion that we described in Exam-
ples 8.13 and 8.14; it also includes precession of the spin around the external
magnetic flux density, similar to that experienced by a small magnet dipole
in the presence of a magnetic flux density.
The magnetic dipole moment due to the spin internal degree of freedom of
an electron is proportional to the Pauli’s spin matrix; specifically:

r r
µ =−µ σ (8.57)
B
where µ = 0.927 × 10 J/Tesla.
–23
B
In the same notation, the electron spin angular momentum is given by:
r h r
S = σ (8.58)
2

© 2001 by CRC Press LLC