to k-th excited state accompanied by photoabsorption. The two transitions give the
same transition moment. Note that zeroth excited state means the ground state; see
(2.64) for basis vector representations.
We should be careful about “addresses” of the matrix accordingly. For example,
P 0, 1 in (4.27) represents a (1, 2) element of the matrix (4.28); P 2, 1 stands for a (3, 2)
element.
Suppose that we seek the transition dipole moments using coordinate representation. Then, we need to use (2.106) and perform definite integration. For instance,
we have
e
Z 1
À1
ψ 0 q
ð Þqψ 1 q
ð Þdq
that corresponds to (1, 2) element of (4.28). Indeed, the above integral gives e
ffiffiffiffiffiffi ffi
h
2mω
q
.
The confirmation is left for the readers. Nonetheless, to seek a definite integral of
product of higher excited-state wave functions becomes increasingly troublesome. In
this respect, the operator method described above provides us with a much better
insight into complicated calculations.
Equations (4.26)–(4.28) imply that the electric dipole transition is allowed to
occur only when the quantum number changes by one. Notice also that the transition
takes place between the even function and odd function; see Table 2.1 and (2.101).
Such a condition or restriction on the optical transition is called a selection rule. The
former equation of (4.27) shows that the transition takes place from the upper state to
the lower state accompanied by the photon emission. The latter equation, on the
other hand, shows that the transition takes place from the lower state to the upper
accompanied by the photon absorption.
4.3 Three-Dimensional System
The hydrogen-like atoms give us a typical example. Since we have fully investigated
the quantum states of those atoms, we make the most of the related results.
Example 4.3: An Electron in a Hydrogen Atom Unlike the one-dimensional
system, we have to take account of an angular momentum in the three-dimensional
system. We have already obtained explicit wave functions. Here we focus on 1s and
2p states of a hydrogen. For their normalized states we have
ϕ 1s
ð Þ ¼
ffiffiffiffiffiffiffi
1
πa 3
r
e
Àr=a ,
ð4:29Þ
132
4 Optical Transition and Selection Rules
same transition moment. Note that zeroth excited state means the ground state; see
(2.64) for basis vector representations.
We should be careful about “addresses” of the matrix accordingly. For example,
P 0, 1 in (4.27) represents a (1, 2) element of the matrix (4.28); P 2, 1 stands for a (3, 2)
element.
Suppose that we seek the transition dipole moments using coordinate representation. Then, we need to use (2.106) and perform definite integration. For instance,
we have
e
Z 1
À1
ψ 0 q
ð Þqψ 1 q
ð Þdq
that corresponds to (1, 2) element of (4.28). Indeed, the above integral gives e
ffiffiffiffiffiffi ffi
h
2mω
q
.
The confirmation is left for the readers. Nonetheless, to seek a definite integral of
product of higher excited-state wave functions becomes increasingly troublesome. In
this respect, the operator method described above provides us with a much better
insight into complicated calculations.
Equations (4.26)–(4.28) imply that the electric dipole transition is allowed to
occur only when the quantum number changes by one. Notice also that the transition
takes place between the even function and odd function; see Table 2.1 and (2.101).
Such a condition or restriction on the optical transition is called a selection rule. The
former equation of (4.27) shows that the transition takes place from the upper state to
the lower state accompanied by the photon emission. The latter equation, on the
other hand, shows that the transition takes place from the lower state to the upper
accompanied by the photon absorption.
4.3 Three-Dimensional System
The hydrogen-like atoms give us a typical example. Since we have fully investigated
the quantum states of those atoms, we make the most of the related results.
Example 4.3: An Electron in a Hydrogen Atom Unlike the one-dimensional
system, we have to take account of an angular momentum in the three-dimensional
system. We have already obtained explicit wave functions. Here we focus on 1s and
2p states of a hydrogen. For their normalized states we have
ϕ 1s
ð Þ ¼
ffiffiffiffiffiffiffi
1
πa 3
r
e
Àr=a ,
ð4:29Þ
132
4 Optical Transition and Selection Rules