Page 78 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics
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3.3 Hermitian operators 69
hö j øi hø j öi
hö j cøi chö j øi
hcö j øi c hö j øi
where c is an arbitrary complex constant. Since the integral hø j øi equals
hø j øi, the scalar product hø j øi is real.
If the scalar product of ø and ö vanishes, i.e., if hö j øi 0, then ø and ö
^
are said to be orthogonal. If the eigenfunctions ø i of an operator A obey the
expressions
hø j j ø i i 0 all i, j with i 6 j
the functions ø i form an orthogonal set. Furthermore, if the scalar product of
ø i with itself is unity, the function ø i is said to be normalized. A set of
functions which are both orthogonal to one another and normalized are said to
be orthonormal
hø j j ø i i ä ij (3:6)
where ä ij is the Kronecker delta function,
ä ij 1, i j
(3:7)
0, i 6 j
3.3 Hermitian operators
^
The linear operator A is hermitian with respect to the set of functions ø i of the
variables q 1 , q 2 , ... if it possesses the property that
^
^
ø Aø i dô ø i (Aø j ) dô (3:8)
j
The integration is over the entire range of all the variables. The differential dô
has the form
dô w(q 1 , q 2 , ...)dq 1 dq 2 ...
where w(q 1 , q 2 , ...)isa weighting function that depends on the choice of the
coordinates q 1 , q 2 , ... For cartesian coordinates the weighting function
2
w(x, y, z) equals unity; for spherical coordinates, w(r, è, j) equals r sin è.
Special variables introduced to simplify speci®c problems have their own
weighting functions, which may differ from unity (see for example Section
6.3). Equation (3.8) may also be expressed in Dirac notation
^ ^
hø j j Aø i ihAø j j ø i i (3:9)
in which the brackets indicate integration over all the variables using their
weighting function.
