Mathematical Methods for Physicists: A concise introduction - Site Map

COMMUTING OPERATORS
Multiplying both sides from the left by the bra vector h' i j we ®nd
b i ˆ Xn
jˆ1
a j h' i
X
jA
n
' j ˆ a j A ij :
~
jˆ1
…5:31†
Referred to the basis j 1 i; j 2 i; ...; j n i the same vectors jXi and jYi are
jXi ˆPn
iˆ1 a i 0 j i i, and jYi ˆPn
iˆ1 b i 0 j i i, and Eqs. (5.31) are replaced by
bi 0 ˆ Xn
aj
0 X
h i jA
n
j ˆ aj 0 Aij;
0
~
jˆ1
where Aij 0 ˆh i jA j i, which is related to A ij by the following relation:
~ Aij 0
ˆ h i jA j ˆ h U'i jA U' j ˆ h 'i jU*A U ' j ˆ…U*A~ U† ij
~ ~ ~
or using the rule for matrix multiplication
jˆ1
Aij 0
ˆ h i jA j ˆ…U*A~ U† ij ˆ Xn
~
X n
rˆ1 sˆ1
U ir *A rs U sj :
…5:32†
From Eqs. (5.32) we can ®nd the matrix representation of an operator with
respect to a new basis.
If the operator A
~
transforms vector jXi into vector jYi which is vector jXi itself
multiplied by a scalar : jYi ˆjXi, then Eq. (5.30) becomes an eigenvalue
equation:
A
~
j Xi ˆ j Xi:
Commuting operators
In general, operators do not commute. But commuting operators do exist and
they are of importance in quantum mechanics. As Hermitian operators play a
dominant role in quantum mechanics, and the eigenvalues and the eigenvectors of
a Hermitian operator are real and form a complete set, respectively, we shall
concentrate on Hermitian operators. It is straightforward to prove that
Two commuting Hermitian operators possess a complete orthonormal
set of common eigenvectors, and vice versa.
If A
~
and A
~
jvi ˆjvi are two commuting Hermitian operators, and if
A
~
jvi ˆjvi;
…5:33†
then we have to show that
B
~
jvi ˆjvi:
…5:34†
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