Chemistry Reference
In-Depth Information
1.3 Operations and Operators
Besides functions, we must also consider the action of operations on operators. In
quantum chemistry, operators, such as the Hamiltonian,
, are usually spatial func-
tions and, as such, are transformed in the same way as ordinary functions, e.g.,
H
(P
1
)
H
( R
−
1
P
1
)
. So why devote a special section to this? Well, operators are
different from functions in the sense that they also operate on a subsequent argu-
ment, which is itself usually a function. Hence, when symmetry is applied to an
operator, it will also affect whatever follows the operator. Symmetry operations act
on the entire expression at once. This can be stated for a general operator
=
H
O
as
follows:
R
=
O
Rf
O
f
(1.17)
O
by smuggling
R
−
1
R
(
=
E
)
From this we can identify the transformed operator
into the left-hand side of the equation:
R
O
R
−
1
Rf
=
O
Rf
(1.18)
This equality is true for any function
f
and thus implies
4
that the operators preced-
ing
Rf
on both sides must be the same:
O
=
R
O
R
−
1
(1.19)
This equation provides the algebraic formalism for the transformation of an op-
erator. In the case where
O
=
O
, we say that the operator is
invariant
under the
symmetry operation. Equation (
1.19
) then reduces to
R
O
−
O
R
=[
R,
O
]=
0
(1.20)
This bracket is know as the
commutator
of
R
and
O
. If the commutator vanishes, we
say that
R
and
commute. This is typically the case for the Hamiltonian. As an ap-
plication, we shall study the functional transformations of the differential operators
∂
O
∂x
,
∂y
under a rotation around the positive
z
-axis. To find the transformed opera-
tors, we have to work out expressions such as
x( R
−
1
P
1
)
. Hence, we
are confronted with a functional form, viz., the derivative operator, of a transformed
argument,
x
, but this is precisely where classical analysis comes to our rescue be-
cause it provides the chain rule needed to work out the coordinate change. We have:
∂x
where
x
=
∂
∂
∂x
=
∂x
∂x
∂
∂x
+
∂y
∂x
∂
∂y
(1.21)
In order to evaluate this equation, we have to determine the partial derivatives in the
transformed coordinates. Using the result in Sect.
1.1
but keeping in mind that the
4
The fact that two operators transform a given function in the same way does not automatically
imply that those operators are the same. Operators will be the same if this relationship extends over
the entire Hilbert space.
