Elements of Modern Physics
94
Apart from playing an important role in the discussion of the kinetic energy
in spherical coordinates [Eq. (3.135)], the angular momentum operator plays a
significant role in determining the rotational energy levels of a rigid rotator. For
example, the rotational energy levels of a di-atomic molecule are given by the
Hamiltonian
H =
2
1 L
2I
(3.155)
where l is the moment of inertia, and the corresponding energy levels are given
by
E =
2
1 ( 1)
2
+
l l
I
, l = 0, 1, ...
(3.156)
3.12 EXAMPLES
Here some important properties of quantum mechanical system and their
applications are discussed.
Example 1
Since indeterminacy is not an essential part of classical mechanics, it is suggetive
that classical measurements may be related to the averages of quantum
mechanical measurements. This is illustrated by Ehrenfest’s theorem.
Consider the time-derivative of the average position given by
r
〈 〉
d
dt
=
*
3
∫ ψ ψ
d
d r
dt
r
=
*
*
3
3
∂ψ
∂ψ
ψ
+
ψ
∂
∂
∫
∫
d r
d r
t
t
r
r
(3.157)
Using the Schrödinger equation, and cancelling the potential energy terms
(potential is real),
〈 〉
d
dt
r =
*
2
3
2
*
3
(
)
2
∫ ψ ∇ ψ
− ∫ ∇ ψ
ψ
i
d r
d r
m
r
r
(3.158)
Integrating by parts gives
〈 〉
d
dt
r =
*
3
1
(
)
i
d r
m
ψ − ∇ ψ
∫
=
〈 〉
m
p
(3.159)
94
Apart from playing an important role in the discussion of the kinetic energy
in spherical coordinates [Eq. (3.135)], the angular momentum operator plays a
significant role in determining the rotational energy levels of a rigid rotator. For
example, the rotational energy levels of a di-atomic molecule are given by the
Hamiltonian
H =
2
1 L
2I
(3.155)
where l is the moment of inertia, and the corresponding energy levels are given
by
E =
2
1 ( 1)
2
+
l l
I
, l = 0, 1, ...
(3.156)
3.12 EXAMPLES
Here some important properties of quantum mechanical system and their
applications are discussed.
Example 1
Since indeterminacy is not an essential part of classical mechanics, it is suggetive
that classical measurements may be related to the averages of quantum
mechanical measurements. This is illustrated by Ehrenfest’s theorem.
Consider the time-derivative of the average position given by
r
〈 〉
d
dt
=
*
3
∫ ψ ψ
d
d r
dt
r
=
*
*
3
3
∂ψ
∂ψ
ψ
+
ψ
∂
∂
∫
∫
d r
d r
t
t
r
r
(3.157)
Using the Schrödinger equation, and cancelling the potential energy terms
(potential is real),
〈 〉
d
dt
r =
*
2
3
2
*
3
(
)
2
∫ ψ ∇ ψ
− ∫ ∇ ψ
ψ
i
d r
d r
m
r
r
(3.158)
Integrating by parts gives
〈 〉
d
dt
r =
*
3
1
(
)
i
d r
m
ψ − ∇ ψ
∫
=
〈 〉
m
p
(3.159)