334
9 Semiclassical Theory: Path Integrals
9.8.4 Semiclassical Cross Section
From Eq. (9.79), the matrix element of the Green’s function in configuration space,
in the semiclassical approximation, can be written
r| ˆ
G + (E ν )|r
= ¯
G(r, r
; E ν ) + G osc (r, r
; E ν ).
(9.163)
In Eq. (9.163), ¯
G(r, r ; E ν ) is a slowly varying background term and G osc (r, r ; E ν )
is an oscillatory term that can be written
G osc (r, r
; E ν ) =
β
A β (r, r
) exp
i
¯
h
S β (r, r
; E ν ) −
iπ
2
n β
, (9.164)
where
β is the sum over all classical orbits with energy E ν that travel from point r
to point r , A β (r, r ) is an amplitude whose value depends on the stability of the βth
orbit, n β is the Maslov index, and S β (r, r ; E ν ) =
r
r dr ·p is the action integral for
the β th orbit. The imaginary part of the Green’s function is given by
Im[[r| ˆ
G + (E ν )|r
=
β
A β (r, r
) sin
1
¯
h
S β (r, r
; E ν ) −
π
2
n β
.
(9.165)
It is this imaginary part that appears in the absorption cross section in Eq. (9.160).
There are two important effects that we now discuss. First, although the offdiagonal matrix elements of the Green’s function appear in the absorption cross
section, the dominant contribution comes from closed orbits that correspond to the
diagonal matrix elements, r| ˆ
G + (E ν )|r. Closed orbits are orbits that begin at point
r and return to the same point r, but the momenta of the outgoing and incoming
orbits may point in different directions. Waves that leave the neighborhood of the
nucleus along such an orbit will return to the same point and may constructively
interfere with the outgoing orbit. This assumes, of course, that the time of the round
trip is short enough that the wave remains fairly coherent during its propagation.
This then brings us to the second important effect, which has to do with the finite
resolution of the laser used to measure the absorption spectrum. Because of the
finite resolution of the laser, the incident radiation can only excite the hydrogen
atom from the ground state to an energy interval of final states with width E. This
means that only contributions from orbits with fairly short periods can be resolved.
The long-period orbits that give rise to the fine-scale energy resolution are washed
out. Thus, the dominant features in the experimental absorption cross section should
be determined by the short-period closed orbits.
For orbits that return to the same point, the Green’s function (propagator) may
be written
Im[[r| ˆ
G + (E ν )|r] =
β
l β
A β,l β (r) sin
l β S β (E ν ) −
π
2
n β
,
(9.166)
9 Semiclassical Theory: Path Integrals
9.8.4 Semiclassical Cross Section
From Eq. (9.79), the matrix element of the Green’s function in configuration space,
in the semiclassical approximation, can be written
r| ˆ
G + (E ν )|r
= ¯
G(r, r
; E ν ) + G osc (r, r
; E ν ).
(9.163)
In Eq. (9.163), ¯
G(r, r ; E ν ) is a slowly varying background term and G osc (r, r ; E ν )
is an oscillatory term that can be written
G osc (r, r
; E ν ) =
β
A β (r, r
) exp
i
¯
h
S β (r, r
; E ν ) −
iπ
2
n β
, (9.164)
where
β is the sum over all classical orbits with energy E ν that travel from point r
to point r , A β (r, r ) is an amplitude whose value depends on the stability of the βth
orbit, n β is the Maslov index, and S β (r, r ; E ν ) =
r
r dr ·p is the action integral for
the β th orbit. The imaginary part of the Green’s function is given by
Im[[r| ˆ
G + (E ν )|r
=
β
A β (r, r
) sin
1
¯
h
S β (r, r
; E ν ) −
π
2
n β
.
(9.165)
It is this imaginary part that appears in the absorption cross section in Eq. (9.160).
There are two important effects that we now discuss. First, although the offdiagonal matrix elements of the Green’s function appear in the absorption cross
section, the dominant contribution comes from closed orbits that correspond to the
diagonal matrix elements, r| ˆ
G + (E ν )|r. Closed orbits are orbits that begin at point
r and return to the same point r, but the momenta of the outgoing and incoming
orbits may point in different directions. Waves that leave the neighborhood of the
nucleus along such an orbit will return to the same point and may constructively
interfere with the outgoing orbit. This assumes, of course, that the time of the round
trip is short enough that the wave remains fairly coherent during its propagation.
This then brings us to the second important effect, which has to do with the finite
resolution of the laser used to measure the absorption spectrum. Because of the
finite resolution of the laser, the incident radiation can only excite the hydrogen
atom from the ground state to an energy interval of final states with width E. This
means that only contributions from orbits with fairly short periods can be resolved.
The long-period orbits that give rise to the fine-scale energy resolution are washed
out. Thus, the dominant features in the experimental absorption cross section should
be determined by the short-period closed orbits.
For orbits that return to the same point, the Green’s function (propagator) may
be written
Im[[r| ˆ
G + (E ν )|r] =
β
l β
A β,l β (r) sin
l β S β (E ν ) −
π
2
n β
,
(9.166)