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Effect of Closed Classical Orbits on Quantum Spectra: Ionization Effect of Closed Classical Orbits on Quantum Spectra: Ionization
of Atoms in a Magnetic Field. II. Derivation of Formulas of Atoms in a Magnetic Field. II. Derivation of Formulas
M. L. Du
John B. Delos
William & Mary
, jbdelos@wm.edu
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Recommended Citation Recommended Citation
Du, M. L. and Delos, John B., Effect of Closed Classical Orbits on Quantum Spectra: Ionization of Atoms in
a Magnetic Field. II. Derivation of Formulas (1988).
Physical Review A
, 38(4), 1913-1930.
https://doi.org/10.1103/PhysRevA.38.1913
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PHYSICAL
REVIEW
A
VOLUME
38,
NUMBER 4
AUGUST
15,
1988
Effect of
closed classical orbits
on
quantum
spectra:
Ionization
of atoms
in
a magnetic
field. II. Derivation
of
formulas
M. L.
Du
and
J.B.
Delos
Joint Institute
for
Laboratory
Astrophysics,
Uni
uersity
of
Colorado and
National Bureau
of
Standards,
Boulder,
Colorado
80309-0440
and
Department
of
Physics,
College
of
William and
Mary,
Williamsburg, Virginia
23185
(Received
30
November
1987;
revised
manuscript
received 15
April
1988)
A
formula
is derived
for oscillations
in
the
near-threshold
absorption spectrum
of
an atom
in
a
magnetic
field. Three
approximations
are
used.
(1)
Near the
atomic
nucleus,
the
diamagnetic
field
is
negligible.
(2)
Far from
the
nucleus,
the waves
propagate
semiclassically.
(3)
Returning
waves
are
similar
to
(cylindrically
modified) Coulomb-scattering waves. With
use of these
approximations,
to-
gether
with the
physical
picture
described in
the
accompanying
paper,
an
algorithm
is
specified
for
calculation of
the
spectrum.
I.
INTRODUCTION
This is the second of two
papers
dealing
with the effect
of closed classical
trajectories
on
quantum spectra.
In
the first
paper,
we
explained
the
physical
picture,
and we
stated a
formula that connects closed classical orbits
with
oscillations in
the absorption
spectrum
of
a
hydrogen
atom
in
a
magnetic
field.
'
The
purpose
of this
paper
is
to
present
a
derivation of
that
formula.
The
derivation is
long,
but it is
quite
straightforward, provided
that
the
physical
picture
dis-
cussed
in
paper
I
is
kept
in mind. Let
us
recall that
phys-
ical
picture.
When a laser
is
applied
to an atom in
a
mag-
netic
field,
the atom
may
absorb
a
photon.
When
it
does,
the electron
goes
into a
near-zero-energy
outgoing
Coulomb
wave. This wave then
propagates
away
from
the
nucleus.
Sufficiently
far from
the nucleus,
the wave
propagates
according
to semiclassical
mechanics,
and it
is
correlated with classical
trajectories.
The wave fronts are
perpendicular
to
the
trajectories,
and the
waves
propa-
gate
along
the
trajectories. Eventually
the
trajectories
and the wave fronts are turned back
by
the
magnetic
field;
some of the orbits return
to
the
vicinity
of the
nu-
cleus,
and
the associated waves
(now incoming)
interfere
with the
outgoing
waves
to
produce
the observed
oscilla-
tions
in
the
absorption spectrum. (See
Fig.
1.
)
Since in
this
system
classical
trajectories
are
chaotic,
it
would be
impossible
to
find all of the trajectories
that
propagate
away
from and
return to the
vicinity
of the
nu-
cleus.
However,
since we
want to
calculate the absorp-
tion
spectrum
only
to
a
certain resolution
hE,
we
include
only
those
paths
which
return in
a
time
less than
T,
„=
2M/EE.
An additional complication
of
the
theory
arises from
the fact
that the
semiclassical
approximation
becomes
re-
liable
only
outside the
vicinity
of
the nucleus.
We
there-
fore
use
a
quantum
partial-wave
expansion
close to the
nucleus
and the
semiclassical
approximation
outside of
this
region.
The
two approximations
are
joined
on a
boundary sphere
of radius
rb
-50ao.
The
paper
is
organized
as follows.
In Sec. II the
photon-absorption rate
is
expressed
by
the
average
oscillator-strength
density
Df
(E).
This
quantity
is
relat-
ed to
an
energy-averaged
Green's
function.
In Sec.
III
formulas are
presented
for
propagation
of
waves
using
a
semiclassical
approximation.
The formulas
are reduced
when the
cylindrical
symmetry
of the
system
is
con-
sidered. In Sec. IV the behavior of waves
in the
vicinity
of
the
nucleus is described. The initial
state
and
dipole
operator
are familiar.
The
outgoing
wave is
easy
to
de-
scribe. Detailed
analysis
is
required
for a
description
of
the
returning
waves. In
Sec.
V
the
results
of
Secs. II
—
IV
are
put
together
to
derive
the
formula
which describes
os-
cillations
in
the
spectrum.
In Sec.
VI
an
algorithm
for
computation
of the
spectrum
to a
specified
resolution is
described.
There is
nothing
difficult
in
this
paper,
but the full
analysis requires
a
lot of mathematical
details,
which
we
have
presented
as
compactly
as
possible.
The title
and
the
first sentence
of each
subsection
tell what
happens
therein.
We
emphasize
the
importance
of
keeping
the
physical
picture
in
mind,
to avoid
getting
lost in the
for-
mulas. Constant reference
to
Fig.
1 should be
helpful.
II. BASIC
FORMULAS
FOR
SPKCTRA
Fundamental
quantities needed for
quantitative
calcu-
lation
of
spectra
are
defined, and
relationships
between
them
are derived.
The
photon-absorption rate is
related
to an
average oscillator-strength
density,
and
the latter
quantity
is
related
to a matrix
element
of an
energy-
averaged
Green's
function.
A. The
photon-absorption rate
is related
to
the
average
oscillator-strength
density
D
f
(
E)
Given
a collection of
N;
one-electron
atoms in an
ini-
tial
quantum
state
lb;
of
energy
E;,
if
a
radiation
field
is
applied
to the
atoms,
then
the rate
of
absorption
of
pho-
tons,
or
the rate of
production
of
atoms
in
an
excited
state
g&
(energy
E&),
is
38 1913
1988
The
American
Physical Society
1914
M.
L.
DU AND
J.B.
DELOS
38
06,
"dipole-coordinate"
operator, etiual
to the
projection
of
the electron coordinate
r
=ix
+
jy+kz
onto the
direction
of
polarization
of the
field,
D
=r'
Aol
~
Ao
~
(2.
2b)
where the vector
potential
for the
electromagnetic
wave
is
given
by
A(r,
t)=
A
exp[i(k
r
cot)—
)+
A
'exp[
i—
(k
r
.
tot—
)]
.
(2.
2c)
The
operator
D can
be
written in
the
form
D
=a+(x
+iy)+a
(x iy)+—
a z
=r(a
+sin8e'~+a
sin8e
'4'+a
cos8),
(2.
2d)
where
a+,
a,
and a are
polarization
coeScients for
the
radiation
field,
a+=
—,
'(A„i
A~—
)l
~
A
u
=
—,
'(so+iso)r~
Ao~
a
=A,
/(A
(2.
2e)
FIG. 1.
Physical picture
of the
absorption
process.
(1)
The
atom is
initially
in the
2p,
state,
with the
oscillating field due
to
the laser
present.
(2)
The
oscillating
field
produces zero-energy
Coulomb
waves,
which
propagate
outward in all directions.
(3)
For
distances
greater
than
about
50ao,
a
semiclassical
approxi-
mation is
appropriate,
and we can
propagate
the wave
outward
by
following
classical
trajectories.
(4)
A
pencil
of
trajectories
propagates
outward,
encounters
a
caustic
(5),
a focus
(6),
and
another
caustic
(7).
This
group
of
trajectories
started out in
such a direction that it turned around
and returned toward the
atom
(8).
Around
50ao,
we describe it as an
incoming
zero-
energy
Coulomb wave
(9),
which
continues
to
propagate
inward
(10),
until it
overlaps
with the initial
2p,
state
(11).
Interference
between
steadily
produced
outgoing
and
incoming
waves
leads
to
oscillations in
the
absorption spectrum. (The
sizes of the first
and last
parts
of
the
figure
are
about
10ao,
the sizes of the
second and fourth
are
about
60ao,
and the size of
the third is
about
3000ao.
)
For
example,
if
the
light
is
linearly
polarized
with
electric
field
along
the
z
axis,
then
a
=1
and
a+=a
=0,
while
if the
light
is
circularly
polarized
and
traveling
in the z
direction, then a =0,
and either
a+
=1
and
a
=0
(giv-
ing
transitions
having
b,m
+1)
or
a+
=0
and
a
=1
(giv-
ing
transitions
having
b,
m
=
—
1);
mfi is the
component
of electronic
angular
momentum
on the z axis.
In the
experiments
of interest
to
us,
the
energy
width
of the
photon
beam is
large compared
with the
spacing
between
the
energy
levels,
and
so
transitions
occur from
a
given
initial
state
to
many
final
states. As a
consequence,
the measured
absorption
rate
is
dNf
=
J
p(Ef
)dEI,
(2.
3)
where
p(Ef
)
is the
density
of
states
of
the
system
at
ener-
gy
Ef.
For
bound states with
discrete
energy
levels,
p(Ef
)
is a sum
of
5
functions
peaked
at each
discrete
en-
ergy
level
E„,
dNf
=Bf;N;I(co)
.
dt
(2.
1)
Here
I(co)des
is
the
energy
flux
density
(energy
per
unit
area
per
unit
time) in
the
frequency
range
dc@
(It
is
as-.
sumed
that
the
range
of
energies
in the
photon
beam is
large compared to the
natural
linewidth
for the
transi-
tion.
)
Bf;
is
the
induced
absorption coefficient
between
initial
and final
states.
In
many
textbooks
on
quantum
mechanics it
is
shown
that
8f;
is
given
by
p(Ef
)=
g
5(Ef E„),
—
(2.
4)
while
for free
states with
a
continuous
range
of
levels,
by
enclosing the
whole
system
in
a
box of
finite
volume
V,
one finds
that
p(Ef
) goes
to
infinity
and
tPf
goes
to
zero
in
such
a
way
that
~
tgf
~
p(Ef
)
approaches
a finite
limit
as
V~
~.
It
is
convenient
to write
the
photon intensity in
the
form
4
2
2
(2.2a)
I
(a)
)da)
=Iog,
„,
(Ef
E)dEf,
—
where
(2.
5a)
where
f;
and
gf
are the
initial and
final
quantum
states
of the
atom,
—
e
is the
electron
charge,
c
is
the
speed
of
light,
and
A is
the
Planck
constant
over
2~. D
is
the
Io
—
—
I
I(co)des
is
the
integrated
intensity
of
the
laser
beam
and
(2.
5b)
38
EFFECT
OF CLOSED
CLASSICAL ORBITS. . . . II. 1915
(2.
5c)
=ION;
f
BI;
p(EI)g,
„p,
(E~
E)d—
EI
.
dt
(2.
6)
Obviously
the
observed
average absorption spectrum
de-
pends
upon
the width of
the
convolution
function, but
normally
it should not be sensitive
to
the detailed
shape
of
this
function.
In theoretical
calculations,
the oscillator
strength
is
often
preferred
over the
induced
absorption
coeScient.
For
a
discrete transition, the oscillator
strength
is
defined
as
2m,
(EI E;
)—
fp=
+2
I
i@f
ID
I@;&
I'
(2.7a)
is the
energy
of the
initial
state
plus
the
average
energy
A'co
of
the
laser
photons.
The function
g,
„,
(EI
E—
)
is a
convolution function
representing
the
line
shape
of the
laser beam
—
the
intensity
at
energy
EI
when the laser is
tuned so that its
maximum
intensity
occurs
at
energy
E.
The measured
absorption
rate is
obtained
by
combin-
ing
(2.
1),
(2.
3),
and
(2.
5),
B.
The
propagator
and
Green's
function are
defined
K(r,
r')
=exp[
—
i(r
r'—
)HI%)=[~(t',
t))t,
K(q,
t;q't')=(q
~
K(t,
t')
~
q')
E"
q
s"
q
pE
—
iE"(t
—
t'jib
~~«
we
(2.
11a)
(2.11b)
(2.
11c)
It
is
also
convenient
to
define the
"forward
propagator"
E+,
such that
K(q,
t;
qt')
for t
&t',
K+(q,
r;q'r')=
'0
f
(2.11d)
The
outgoing-wave
Green's
function is
G+(q,
q',
E)
=
(q
~
0
+
~
q')
(2.
12a)
Given a
time-independent
quantum
Hamiltonian
operator H(
—
iAV,
q),
the
propagator
K(q,
t;q',
r')
is
defined as the
coordinate
representation
of the
evolution
operator
m,
c
(E&
E;
)—
(2.7b)
oo
iEt
dt K+(q,
t;q',
O)
exp
SA
0
where
m,
is
the mass of
the electron. When
transitions
occur to
a
group
of unresolved final
states,
it is
appropri-
ate
to
define
the oscillator-strength
density (the
oscillator
strength
per
unit increment
of
energy)
Df
(EI
)
as
(2.12b)
where E
=E+c.
i
and
c.
~+0.
All
quantum
dynamical
properties
of
a
system
can be calculated
if the
propagator
or
Green's
function is
known.
Df
(EI)=
fI;
p(EI)
(2.
8)
and to define
the
experimentally
averaged
oscillator-
strength density
by
the
formula
C.
An
energy-averaged
Green's
function
is related
to the
finite-time
propagator
DI,
„,
(E)
=
(E
E;)
f
(E—
I
E—
;
)
'Df
(EI—
)
Xg
pg(Ef
E)dEf
(2.9a)
If
the
propagator
is
known
for
times
t
up
to
some
max-
imum,
0
(
t
(
T,
then we can
calculate
a
"finite-
resolution"
or
"energy-averaged"
Green's
function. The
range
of
energy
averaging
hE is related to
the maximum
time
Thy
the formula
TEE-2vrR.
Let
us
define
Normally
the laser beam
is
narrow
enough
that
(EI
E;
)
=
(E
E;
)
—
over the
si—
gnificant
range
of
EI,
and
Df,
„,
(E)
is
a
simple average
of
Df
(EI
),
Df,
„p,
(E)-
f
Df
(E~)g,
„p,
(Ey
E)de
.
(2.
9b—
)
G
(q,
q',
E)—
:
(imari)
'
f
K+(q, t;q',
0)g(t)
exp(iEt/A)dt,
0
(2.13a)
where
g(t)
is
a
general
cutoff function. We define
g
(t)
to
be
symmetric
in
time, so
g(
t)
=g(t).
It
is
ea—
sy
to
prove
that 6
is
an
energy-averaged
Green's
function,
specifically,
G~(q,
q';E)=
f
G+(q,
q';E')g(E
E')dE',
(2—
.
13b)
Combining
(2.7a)
with
(2.
9a)
we
find
(2.9c)
where
m,
c(E
E;
)—
Df,
„p,
(E)=
f
BI;
p(EI
)g,
„,
(E~
E)dE~
.
—
2'
e
Hence
the
measured
absorption
rate (2.
6)
can
be
ex-
pressed
in terms of
Df,
„,
(E)
as
222
=ION,
Df,
„,
(E)
.
dt
'
m,
c(E
—
E,
)
g
(E)
=
f
g
(t)e'
'~"
dt
2M
g
t cos
Et/A
dt .
1
7TR
0
(2.13c)
The
goal
of the
theory
developed
below
is
the theoretical
calculation of the
average
oscillator-strength
density.
Proof
From (2.
12b),
the
.
inverse Fourier
transform
gives
1916
M.
L.
DU
AND J. B.
DELOS
38
K+(q, t;q',
0)
=
f
G+(q,
q',
E)e
' '~"
dE
.
2'
(2.14)
2m,
(E
E—
;
)
Dfg(E)=
3
Re
f
(Dp;
A(t,
O)
I
Dp;
)
M3
0
Substitute
this
formula for
K+
into
Eq.
(2.
13a),
reverse
the order of
integration,
and
use the
6-function
formula
Xg(i)e'
""«
.
(2.
20)
f
exp[i
(EI
E}t
—
/fi]dt
=5(EI
E)
—
(2.15)
to
arrive
at
(2.
13b).
D. The
oscillator-strength
density
is
expressed
in
terms
of
the
propagator
and the
Green's
function
The
oscillator-strength
density
is
related to matrix
ele-
ments
of the
propagator
and
to those
of the
Green's
func-
tion
by
the
formulas
Proof.
Equation
(2.19)
follows
trivially
from
(2.
18),
(2.
16b),
and
(2.
13b).
Then
Eq.
(2.20) follows from
(2.19)
and
(2.13a).
Ideally,
we should
take
g
(E'
E)
i—
n
(2.18)
in the
same
form
as that
for
the laser
profile,
as in
Eq.
(2.
5a).
Howev-
er,
while the
resulting
theoretical
average
oscillator-
strength density
should
depend
upon
the width hE of the
convolution
function,
it should not be
sensitive
to the
de-
tailed form
of
this
function. Therefore
we
consider the
special
case that
g(t)
corresponds
to
a
sharp
cutoff,
2m,
(EI E;
)—
Df(EI)=
3
Ref
(Dp;
I
K+(t,
O)
ID/,
)
g'=
0
IrI
T
(2.
21a)
iEIt/fi
&(e
dt,
2m,
(E/
E,
)—
Df(EI)=
—
Im(DQ,
I
G
+
I
Dg;
)
.
(2.
16a)
In this case we use
only
a
finite-time
propagator
E(t,
O),
for 0
&
t
(
T,
and the
resulting
oscillator-strength
density
is
averaged
over
energy
with
the
convolution function
(2.
16b)
Proof.
First
establish
the
relationship
f
(D0;
I
«& 0)
I
DW
&
Xe
I
dt .
(2.17)
To
prove
this,
use
(2.
1lc)
on
the right-hand
side of
(2.
17),
reverse
the order
of
integration,
and use
(2.
15).
To
prove
(2.
16a)
from
(2.
17),
use
time-reversal
symmetry
of the
propagator,
(2.
11a),
and the
definition
of
Df,
(2.
7)
and
(2.8).
Then
(2.12b) leads from
(2.
16a) to
(2.16b).
E.
An
energy-averaged
oscillator-strength
density
is related
to the
Snite-time
propagator
and to the
energy-averaged
Green's
function
It is then
easy
to
prove
that
(2.
18)
2m,
(E
E,
)—
Df
(E)=
—
Im(Dip;
I
G
(E)
I
Dg,
)
M2
If
the
propagator k(t,
O)
is
calculated
for
only
a
finite-
time interval
0
(
t
(
T,
then
an
averaged
oscillator-
strength
density
is
determined, and so the
spectrum
can
be calculated
to a
corresponding
resolution.
The same
"low-resolution"
spectrum
can be calculated from
the
energy-averaged
Green's
function.
Let
us define
Dfg(E)
=(E
E,
)
f
(E'
E;
)
'—
Df
(E')g (E—
'
E)dE'
.
—
1
sin[(E
E')T/A—
]
(E
E')—
(2.21b)
In this
way,
we obtain
a theoretically
averaged
oscillator-strength
density,
Df,
h,
«(E).
This
quantity
will
be compared
to the
experimentally averaged
measure-
ments
Df,
„,
(E).
We
take
the width
(in
energy)
of
the
theoretical convolution function
comparable
to the
ener-
gy
width
of
the laser beam.
Equivalently,
we evaluate the
propagator
up
to a
maximum
time
T which is
compara-
ble
to 2irR/(experimental
energy
resolution).
Henceforth
we
no
longer
distinguish between
Df,
h«,
(E)
and
Df,
„p,
(E).
III. SEMICLASSICAL PROPAGATION OF WAVES
A.
Applicability
of
the semiclassical
approximation
is examined
Perfectly rigorous (necessary
and
sufficient) conditions
for the
validity
of the semiclassical
approximation
are
not
known,
particularly
for
multidimensional
problems.
However,
for motion
along
a
line,
a
set
of
physically
reasonable criteria is
generally
accepted.
We
shall
show
that
for
our
system,
these criteria are satisfied for
motion
along
the
p
axis if
p
is
not
too
small.
The
potential
ener-
gy
is
After
the
outgoing
waves are
produced
by
the laser
from
the
initial
state
P;,
these
waves
propagate
forward
in the combined Coulomb and
magnetic
fields.
Sufficiently
far from
the
nucleus,
this
propagation
can
be
described
using
the semiclassical
approximation.
Here
we
describe the semiclassical
method of
propagation
of
waves from
an
initial surface.
and that
(2.
19)
2
1
&(q)=-
{
p2+zz}i/2
8m
+
2
e8
P
(3.
1)
