328
9 Semiclassical Theory: Path Integrals
Table 9.1 Comparison of exact eigenvalues of a quantum system and semiclassical eigenvalues
for silicon. Energies are given in milli-electron volts (from Gutzwiller 1982)
Exact quantum
Semiclassical
Exact quantum
Semiclassical
(even)
(even)
(odd)
(odd)
−31.27
−29.06
−10.48
−11.51
−8.83
−8.35
−5.10
−5.48
−4.75
−4.63
−3.12
−3.33
−3.75
−3.64
−2.30
−2.33
−2.85
−2.82
−2.15
−2.23
−2.11
−2.11
−1.64
−1.62
−1.87
−1.87
−1.49
−1.52
−1.52
−1.53
−1.32
−1.20
−1.38
−1.43
−1.09
−1.10
atom in the presence of a magnetic field have been used to measure magnetic field
strengths in stars (Wunner and Ruder 1987). From the point of view of chaos theory,
this system has also been important. It provided the first experimental observation
of the effect that a classical periodic orbit can have on quantum dynamics. The
connection was made by Edmonds (1970), who noted that large-scale structures
in the absorption spectrum measured by Garton and Tomkins (1969) could be
correlated with a periodic orbit that lies in the plane perpendicular to the magnetic
field.
More recent experiments (Holle et al. 1986; Main et al. 1986, 1994), carried out at
the ionization threshold using scaled variables, have provided the first experimental
observation of the effect of bifurcation of classical orbits on quantum dynamics. A
semiclassical theory connecting classical closed orbits, which begin and end at the
nucleus, to oscillations in the absorption spectrum was developed by Du and Delos
(1988a,b) (see also Mao and Delos 1992). The experiment of Main et al. (1994)
dramatically confirms the predictions of Du and Delos. Below we outline some of
the basic results of this work.
9.8.1 The Model
We consider the motion of the electron in a hydrogen atom that sits in a uniform
magnetic field, B = B 0 ˆ
z, which is directed along the z-axis. The vector potential is
A(r) =
1
2 B × r =
1
2 B 0 (x ˆ
y − y ˆ
x). The Hamiltonian for the electron dynamics can
be written
H 0 =
(p − eA) 2
2μ
−
κ 0 e 2
r
=
p 2
2μ
−
κ 0 e 2
r
−
1
2
ωL z +
μω 2
8
(x
2
+y
2 ) = E 0 ,
(9.149)
9 Semiclassical Theory: Path Integrals
Table 9.1 Comparison of exact eigenvalues of a quantum system and semiclassical eigenvalues
for silicon. Energies are given in milli-electron volts (from Gutzwiller 1982)
Exact quantum
Semiclassical
Exact quantum
Semiclassical
(even)
(even)
(odd)
(odd)
−31.27
−29.06
−10.48
−11.51
−8.83
−8.35
−5.10
−5.48
−4.75
−4.63
−3.12
−3.33
−3.75
−3.64
−2.30
−2.33
−2.85
−2.82
−2.15
−2.23
−2.11
−2.11
−1.64
−1.62
−1.87
−1.87
−1.49
−1.52
−1.52
−1.53
−1.32
−1.20
−1.38
−1.43
−1.09
−1.10
atom in the presence of a magnetic field have been used to measure magnetic field
strengths in stars (Wunner and Ruder 1987). From the point of view of chaos theory,
this system has also been important. It provided the first experimental observation
of the effect that a classical periodic orbit can have on quantum dynamics. The
connection was made by Edmonds (1970), who noted that large-scale structures
in the absorption spectrum measured by Garton and Tomkins (1969) could be
correlated with a periodic orbit that lies in the plane perpendicular to the magnetic
field.
More recent experiments (Holle et al. 1986; Main et al. 1986, 1994), carried out at
the ionization threshold using scaled variables, have provided the first experimental
observation of the effect of bifurcation of classical orbits on quantum dynamics. A
semiclassical theory connecting classical closed orbits, which begin and end at the
nucleus, to oscillations in the absorption spectrum was developed by Du and Delos
(1988a,b) (see also Mao and Delos 1992). The experiment of Main et al. (1994)
dramatically confirms the predictions of Du and Delos. Below we outline some of
the basic results of this work.
9.8.1 The Model
We consider the motion of the electron in a hydrogen atom that sits in a uniform
magnetic field, B = B 0 ˆ
z, which is directed along the z-axis. The vector potential is
A(r) =
1
2 B × r =
1
2 B 0 (x ˆ
y − y ˆ
x). The Hamiltonian for the electron dynamics can
be written
H 0 =
(p − eA) 2
2μ
−
κ 0 e 2
r
=
p 2
2μ
−
κ 0 e 2
r
−
1
2
ωL z +
μω 2
8
(x
2
+y
2 ) = E 0 ,
(9.149)