Page 80 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

3.3 Hermitian operators 71
^ ^ 2 ^ ^
any integral power of A. Since jAøj is always positive, the integral hAø j Aøi
is positive and consequently
^ 2
hø j A øi > 0 (3:12)

Eigenvalues
The eigenvalues of a hermitian operator are real. To prove this statement, we
consider the eigenvalue equation
^
Aø ˆ áø (3:13)
^
^
where A is hermitian, ø is an eigenfunction of A, and á is the corresponding

eigenvalue. Multiplying by ø and integrating give
^
hø j Aøiˆ áhø j øi (3:14)
Multiplication of the complex conjugate of equation (3.13) by ø and integrat-
ing give
^
hAø j øiˆ á hø j øi (3:15)
^
Because A is hermitian, the left-hand sides of equations (3.14) and (3.15) are
equal, so that

(á ÿ á )hø j øiˆ 0 (3:16)
Since the integral in equation (3.16) is not equal to zero, we conclude that

á ˆ á and thus á is real.

Orthogonality theorem
^
If ø 1 and ø 2 are eigenfunctions of a hermitian operator A with different
eigenvalues á 1 and á 2 , then ø 1 and ø 2 are orthogonal. To prove this theorem,
we begin with the integral
^
hø 2 j Aø 1 iˆ á 1 hø 2 j ø 1 i (3:17)
^
Since A is hermitian and á 2 is real, the left-hand side may be written as
^ ^
hø 2 j Aø 1 iˆhAø 2 j ø 1 iˆ á 2 hø 2 j ø 1 i
Thus, equation (3.17) becomes
(á 2 ÿ á 1 )hø 2 j ø 1 iˆ 0
Since á 1 6ˆ á 2 , the functions ø 1 and ø 2 are orthogonal.
Since the Dirac notation suppresses the variables involved in the integration,
we re-express the orthogonality relation in integral notation

ø (q 1 , q 2 , ...)ø 1 (q 1 , q 2 , ...)w(q 1 , q 2 , ...)dq 1 dq 2 ... ˆ 0
2
This expression serves as a reminder that, in general, the eigenfunctions of a