QMGreensite_merged

IfHis time independent, then [2.2] can be solved by the method
of separation of variables. Defining
(t)¼ ()’(t), in which ()
contains the time-independent spatial and spin variables [for simplicity,
() is frequently written as ] and’(t) contains time-dependent terms,
@
ðÞt
@t
¼
iH
ðÞt,

ðÞ

d’ðÞt

dt

¼
iH ðÞ’ðÞt,

Z

ðÞ ðÞd

d’ðÞt

dt

¼
i’ðÞt

Z

ðÞH ðÞd,

d’ðÞt

dt

¼
iE’ðÞt,

½ 2 : 6 Š

in which the energy of the system is defined by

E¼

Z

ðÞH ðÞd: ½ 2 : 7 Š

Solving [2.6] yields’(t)¼Cexp(–iEt). Using this result gives

ðtÞ¼ ðÞexpðÞ
iEt, ½ 2 : 8 Š

in which the integration constantChas been included in the normal-
ization of (). Ifhis reintroduced explicitly, then

ðtÞ¼ ðÞexpðÞ¼
iEt=h ðÞexpðÞ
i!t, ½ 2 : 9 Š

in whichE¼h!. As shown by [2.8] and [2.9], ifHis time independent,
then the time dependence of the wavefunctions is limited to a phase factor;
this factor cancels when calculating probability densities using [2.3].

2.1.2 EIGENVALUEEQUATIONS
The purpose of quantum mechanics, at least insofar as it is applied
to NMR spectroscopy, is to calculate the results expected from
experiments. In the language of quantum mechanics, every physically
observable quantity,A, has associated with it a Hermitian operatorA,
that satisfies the eigenvalue equation:

AfðÞ¼ fðÞ: ½ 2 : 10 Š

This equation defines a set of eigenfunctions,fi(), and eigenvalues,i,
fori¼1toN, that satisfy in turn

AfiðÞ¼ ifiðÞ: ½ 2 : 11 Š

2.1 POSTULATES OFQUANTUMMECHANICS 31