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6
Rotational Motion
So far we have used quantum mechanics to describe translation and vibration. In this
Fnal section on quantum mechanics, we will consider rotational motion. Our strategy
will be to use the separation of variables technique to separate rotation from translation
and vibration. In order to use separation of variables, we need to Fnd a coordinate
system in which the Schr¨odiner equation separates. ±or rotation in a plane, cylindrical
coordinates are appropriate, and for rotation in three dimensions, we will use spherical
polar coordinates.
6.1
Twodimensional rotation
We will start by considering the rotation of a diatomic molecule in the
x
−
y
plane.
Then, if we let
r
be the distance between the particles in the molecule, and
φ
be the
angle of the molecule as measured from the
x
axis, we can make the transformation to
cylindrical coordinates
x
=
r
cos
φr
=
p
x
2
+
y
2
(6.1)
y
=
r
sin
φφ
=atan(
y/x
)
.
The Laplacian,
∇
2
operator
∇
2
=
∂
2
∂x
2
+
∂
2
∂y
2
(6.2)
can be written as
∇
2
=
∂
2
∂r
2
+
1
r
∂
+
1
r
2
∂
2
∂φ
2
(6.3)
in cylindrical coordinates.
To separate rotation, which is described by the angle
φ
,
from vibration, described by the distance
r
, we can use the rigid rotor approximation.
A rigid rotor does not change shape. ±or a diatomic, this means that
r
is Fxed. Taking
r
=
r
0
, the Laplacian becomes
1
r
2
0
∂
2
2
.
(6.4)
±or a rigid rotor, the Laplacian is only a function of the angular variable
φ
.
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 Fall '06
 henkelman
 Physical chemistry, pH

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