in which! 0 ¼! –! ¼ B 0 is the Larmor frequency and ¼ b ais
a phase angle. These results utilize the following equations for the
angular momentum operators (note that only the equations forIzare
eigenvalue equations):
Ix ¼
h
2
, Ix ¼
h
2
,Iy ¼
ih
2
, Iy ¼
ih
2
,Iz ¼
h
2
, Iz ¼
h
2
,½ 2 : 31 together with the orthonormality of the wavefunctions. Equations [2.31]
are derived from the Pauli spin matrices as shown in Section 2.2.5. The
three equations, [2.29]–[2.30], represent a vector of constant magnitude
precessing about thez-axis with an angular velocity! 0. This result is
identical to the predicted motion of the magnetic moment obtained from
the Bloch model.
2.2 The Density Matrix
Calculations of scalar products and expectation values are frequent
operations in quantum mechanics. Such calculations are facilitated by a
formulation of quantum mechanics that focuses on thedensity matrix
rather than on the wavefunction for a system. Additionally, the symbolic
manipulations required are simplified by using a notational system
introduced into quantum mechanics by Dirac ( 6 ).
2.2.1 DIRACNOTATION
The Dirac notation is a compact formalism for representing
the scalar product. In this notation, a wavefunction, , is represented
by the ket function, j i, and the conjugate wavefunction, *, is
represented by thebrafunction,h j. In the Dirac notation, the scalar
product of and’is written as the contraction of the brah jand the
ketj’i,
(^)
’
Z
’d : ½ 2 : 32
2.2 THEDENSITYMATRIX 37