Chemistry Reference
In-Depth Information
By analogy with the classical analysis result, Equation (A9.39), we can also obtain an
operator for the square of magnitude of the angular momentum:
l
(
l
L
2
Y
lm
l
=
2
I T
θφ
Y
lm
l
=
2
l
(
l
+
1)
Y
lm
l
and
L
=
+
1)
(A9.42)
So the magnitude of the angular momentum is quantized and set by the quantum number
l
.
This result also confirms that the natural unit for angular momentum is the Planck constant
over 2
.
In the spherical polar coordinate system the angle
π
sets the position away from the
X
-axis measured parallel to the
XY
plane (Figure A9.1). The normal to this plane is clearly
the
Z
-axis. So the
φ
angle can be used to describe that part of the rotation which is about
the
Z
-axis, i.e. the
Z
-component of the angular momentum
L
z
. The angular wavefunction
must also obey the differential equation
φ
∂
∂φ
−
i
Y
lm
l
=
L
z
Y
lm
l
(A9.43)
The left-hand side is the operator for the
L
z
component of the angular momentum acting
on the spherical harmonic function (see Further Reading section of this appendix). We can
apply this operator to the general form of the spherical harmonic solutions:
√
2
∂
∂φ
m
l
√
2
−
i
exp( i
m
l
φ
)
=
exp( i
m
l
φ
)
=
m
l
Y
lm
l
(A9.44)
π
π
so that
L
z
=
m
l
(A9.45)
In the quantum description we have found the magnitude of the angular momentum and
its
Z
-component. This gives a set of planes on which the electron can be thought to move,
as illustrated in Figure A9.5a. The direction of the angular momentum can never be pinned
down precisely; this is one manifestation of the uncertainty principle. The relationship
Z
(a)
(b)
Z
m
l
= 1
l
= 1
L
L
z
m
l
= 0
m
l
= -1
Figure A9.5
(a) In quantum mechanics the angular momentum vector can be orientated
anywhere on a cone around the Z-axis, so that the plane in which the electron moves is
not fixed. (b) The magnitude of the angular momentum is controlled by the quantum num-
ber l, while the magnetic quantum number m
l
determines the Z-component of the angular
momentum.
