discussed. Instead, the foundations of the theory of intrinsic angular
momentum will be presented as postulates whose validity is established
by comparison with experiment. NMR spectroscopy is a particularly
powerful demonstration of the concepts.
2.1.1 THESCHRo ̈DINGEREQUATION
The evolution in time of a quantum mechanical system is governed
by the Schro ̈dinger equation:
@ðÞt
@t
¼
i
h
HðÞt: ½ 2 : 1
The operatorHis termed the Hamiltonian of the system and incorporates
the essential physics determining the evolution of the system. The
Hamiltonian may be time dependent or time independent. Units in which
h¼1 will be assumed and factors ofhwill not be written explicitly; thus,
@ðÞt
@t
¼iHðÞt: ½ 2 : 2
When desired, necessary factors ofhcan be reintroduced by dimensional
analysis; equivalently, all energies are measured in angular frequency
units with dimensions of s–1. The solution of the Schro ̈dinger equation is
called thewavefunctionfor the system,(t). The wavefunction contains
all the knowable information about the state of the system and,
consequently, is a function of the variables appropriate to the system of
interest (e.g., spatial coordinates and spin coordinates). The probability
density that the system is in the state described by(t) at timetis given by
PðtÞ¼ðÞtðÞt, ½ 2 : 3 in which*(t) is the complex conjugate of(t). If the wavefunction is
known, then all the observable properties of the system can be deduced
by performing the appropriate mathematical operations upon the
wavefunction. Wavefunctions generally will be assumed to be normal-
ized such that
Z
ðÞtðÞt d ¼1, ½ 2 : 4
in which represents the generalized coordinates of the wavefunction
(and may include sums over spin states). If necessary, wavefunctions can
be normalized simply by defining
^0 ðtÞ¼ðÞt Z
ðÞtðÞt d 1 = 2
: ½ 2 : 5 30 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY