QMGreensite_merged

Next, (^) iis expanded in the eigenfuctions ofB:
AB (^) i¼aiB (^) i,
AB
X
j
cijj¼aiB
X
j
cijj,
X
j
cijbjAj¼ai
X
j
cijbjj,
X
j
cijbjAj
aij

¼ 0 :
½ 2 : 24 Š
By definition,cij 6 ¼0 for at least one value ofj¼k. Thus, the bracketed
term in the last equation is zero for somek:
Ak
aik¼0, Ak¼aik: ½ 2 : 25 Š
Thus, k is an eigenfunction of Awith eigenvalue ai and must be
identical to (^) i(to within a constant of proportionality). This equality is
satisfied for all members of the set of eigenfunctions, and; therefore,
the general theorem must hold.
The commutator ofAandBis defined as
½Š¼A,B AB
BA: ½ 2 : 26 Š
The earlier result can be restated: if the commutator of two operators
vanishes, then the operators have the same eigenfunctions. If the
commutator does not vanish, then a Heisenberg uncertainty relationship
can be established for the two operators ( 5 ).
2.1.4 EXPECTATIONVALUE OF THEMAGNETICMOMENT
As should now be clear, each operator for an observable quantity
defines a set of basis vectors. Any complete orthonormal set can be used
to expand an arbitrary wavefunction; consequently, a basis set can be
chosen for computational convenience. In no case can the expectation
value of an operator depend upon the choice of the basis functions.
As an example of these ideas, the time-dependent expectation value
of the magnetic moment l¼ I of a single spin (I¼1/2) will be
calculated. Using [2.9] and [2.18], the wavefunction for the spin in the
static magnetic field can be written as

¼c þc ¼aexp½Š
i! t þbexp½
i! tŠ , ½ 2 : 27 Š
2.1 POSTULATES OFQUANTUMMECHANICS 35