The number of eigenvalues and eigenfunctions,N, is determined by
the system of interest and may be finite or infinite. The adjoint of
an operator is defined byAy¼AT*, in which T indicates transposition
andindicates complex conjugation. The adjoint operator satisfies the
eigenvalue equation,
fðÞ Ay¼fðÞ : ½ 2 : 12
Hermitian operators are self-adjoint, A¼Ay, and satisfy the
relationship
Z
fð ÞAgð Þd ¼
Z
gð ÞAfð Þd
½ 2 : 13 for well-behaved functions f( ) and g( ). If f( ) is a normalized
eigenfuntion of the operatorAwith eigenvalue, then the following
relationships are obtained from [2.10] and [2.12]:
Z
fðÞ AfðÞ d ¼
Z
fðÞ fðÞ d ¼, ½ 2 : 14
Z
fðÞ AyfðÞ d ¼Z
fðÞ fðÞ d ¼: ½ 2 : 15 If the operatorA corresponds to an observable quantity, then the
eigenvalues ofAmust be real numbers. Thus,¼and equating [2.14]
and [2.15] proves that A¼Ay and A is Hermitian. Consequently,
operators corresponding to observable quantities in quantum mechanics
must be Hermitian.
The eigenfunctions of a Hermitian operator form a complete
orthonormal set. The orthonormality condition is
Z
fið Þfjð Þd ¼i,j, ½ 2 : 16
in whichi, jis the Kronecker delta with values
i,j¼ 0 for1 for ii¼^6 ¼jj
: ½ 2 : 17 Unnormalized eigenfunctions can be normalized as in [2.5]; if necessary,
the wavefunctions can be orthogonalized using a procedure known as
the Gram–Schmidt process ( 5 ). Acompleteset of orthonormal functions,
(^) n, constitutes a set of basis functions for a vector space of dimensionN,
called the Hilbert space. Therefore, an arbitrary function defined in the
32 CHAPTER 2 THEORETICALDESCRIPTION OFNMR SPECTROSCOPY