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Physica Scripta. Vol. 46,354-356, 1992.
The Hydrogen Atom in Crossed Static Electromagnetic and
Non-Resonant Laser Fields
Trygve Helgaker and Igor Tomashevsky*
Department of Chemistry, University of Oslo, Blindem, N-0315Oslo 3, Norway
Received February 26,1992; accepted May 20,1992
Abstract
The energy splittings and wave function of a hydrogen atom in crossed
uniform static electromagnetic and non-resonant monochromatic electric
fields of arbitrary mutual orientation are obtained within the "one-shell"
approximation. The intensities of the Lyman lines are also obtained. A
special analytical method is used. Relativistic corrections and spin-orbit
interactions are not considered.
1. Introduction
represents the interaction with the monochromatic electric
field. The wave function 4 is the time-dependent part of the
Floquet (quasi-energetic)function
Y = 4 exp [ -i(Eio)
I
+ E)t]
(2)
of period T = 2n/w. The Floquet eigenvalue or quasi-energy
E is measured relative to the unperturbed value E!,').
We now write eq. (1) in the representation of the interaction with the time-dependent field
We consider the interaction of a hydrogen atom with a nona $ = (Hint- E)$
resonant electric field F cos cot in the presence of uniform i at
static electric and magnetic fields S and 8' assuming that
these fields are too weak for ionization. This situation may
4 = exp (-i V sin cot)+
(3)
exist in plasmas and has been investigated analytically by
Gavrilenko [l] when the frequency o is in resonance with
the static field splitting. This paper examines the non- where $ is periodic in t in the same way as 4 and
resonant effect of this interaction analytically. In Section 2
we derive simple approximate analytical formulas for the
energy levels and the wave function in the presence of external fields. The region of validity is discussed in Section 3.
x exp (-i V sin wt)
Finally in Section 4 we use these results to calculate the
(4)
intensities of the Lyman spectral lines.
We consider Hintas a perturbation and find the solution to
eq. (3)in the form of a series
2. Spectral line splitting and wave functions
0
0
We consider the influence of external fields on the atomic
shell of principal quantum number n. We assume that the
frequency co of the variable electric field as well as the interaction with all external fields are small compared with the
separation between level n of energy ELo) and the neighboring levels. Therefore we do not consider the admixture of
other states to the states in shell n. We also neglect spinorbit interactions and other relativistic corrections,
assuming that the splitting induced by external fields is large
compared with the fine structure effects. We therefore
assume that the Hamiltonian acts only in the subspace of
the unperturbed states of shell n.
The Schrodinger equation for the hydrogen atom in an
external time-periodical field may be written in the form
(using atomic units)
a
iat
=(
d S + pB18'
+ V cos a t - E)4
(1)
in the one-shell approximation. Here 1 is the angular
momentum operator, pB is the Bohr magneton, and V = dF
* Permanent
address: Department of Mathematics, ALTI, Nab. Lenina
17A, 163061 Arkhangelsk, U.S.S.R.
Physica Scripta 46
I(/ = $ ( O ) + $(U + . . .
E = E(') + E'2' + . . .
Substituting eqs (5)into eq. (3)we find that
pendent
$(O)
E
(5)
is time inde-
$(yr)
and that
$(')
(6)
is periodic in t and fulfils the equation
(7)
Therefore, integrating eq. (7) over one period T we find that
the first order quasi-energy E(') and the zero order function
$(') satisfy the time-independent equation
(8)
where
U =1
T
pint
dt
(9)
If we direct F along the z-axis, U may be written in the form
(see Appendix)
The Hydrogen Atom in Crossed Static Electromagnetic and Non-Resonant Laser Fields
355
3. Validity region
in terms of the effective static electric and magnetic fields
We used time-dependent perturbation theory for eq. (3)
treating Hint(ot)as a perturbation. The small parameter is
s e = C H xJob), H yJob), s z l
(11b)
proportional to A/o, where A is the energy splitting in the
where J o ( p ) is the Bessel function and
static electric and magnetic fields. It may be shown that the
perturbation
series for the quasi-energy eq. ( 5 ) contains odd
3nF
p=(12)
powers
only.
Therefore, the first order approximation E(’)
20
should be good to second order and indeed Cohn et al. [4]
The eigenvalues of eq. (8) when the operator is written in the find that E“) already coincides with the exact value for
form eq. (10) have been determined by Demkov et al. [3] o x 2A. The function
has the first correction $(I) which
and are given by
satisfies eq. (7). Consequently, the function eq. (17) is a good
approximation to the quasi-energetic wave function Y at
E!,!,!,,,. = lg, I n’ Ig, In”
(13)
higher frequencies.
where
We have also neglected the mixing of the states in shell n
with
all other states. This restricts the electric fields to the
(14)
g1 = P B s e - i n s , , g2 = P B H e + %.*e
F 4 n - 4 and 2 4 lo3 n-4 in atomic units.
values 9,
and the quantum numbers may take the following values
s
e
=[9xJ0@),
9yJ0@),
(1la)
9zI
+
If spin is taken into account, one must add 2 p B s H to eq.
(10) and _ + p B H (sign depending on relative orientation) to
eq. (13). However, these shifts are not observed since the
spin quantum number does not change in optical transitions.
When S = 0 we obtain an expression for the linear Stark
effect in the presence of a monochromatic external field
F cos ot
E::?,,,, = jn(n’
+ n”) I
1 Jcos’
a
+ sin2 a J i @ )
4. Intensities of the Lyman spectral lines
The periodic function 4 may be expanded in a Fourier
series in t. Therefore, the quasi-energetic state may be
written as a superposition of stationary states with energies
E & ko, k = 0, 1, 2, .... From a physical point of view
E & kw is the energy of the system “atom + field”, i.e., the
quasi-energy of the atom and the energy of the k quanta [SI.
It leads to a splitting of each line into a few components
with separation U.
We consider the transitions from the quasi-energetic state
Y,,,,.,,..[eq. (17)] to the ground state $loo. The frequencies of
these transitions are
(15)
where a is the angle between F and S.A similar result for
F I F and n = 2 was obtained numerically by Cohn,
Bakshi and Kalman [4] and analytically by Gavrilenko and
+ E!,’) - E‘’’1 + ko, k = 0, _+ 1, _+ 2, .. . (18)
Oks 151. When the time-dependent field F vanishes the func- Qk z
tion Jo(p) becomes unity. Equation (15) then reduces to the The intensity of light polarized along e emitted into the
well-known formula for the linear Stark effect in a static solid angle dO is given by
field and eq. (13) reduces to the formula for shifts in static
a: I er””‘””
electric and magnetic fields [3].
J do =2nc3 100,k 1’ do
Following Demkov et al. [3] we write the eigenfunctions
of U in eq. (10) in terms of the wave functions in parabolic Using in eq. (17) the Wigner function in the form [2]
coordinates
[2] with quantization along the z-axis or
F:
Dj”,(a, p, 0) = dim,(/?)exp (im’a)
(20)
$!,:’n,
=
f:
iI,i2=-j
DiSil(41, 0 1 ,
OP$i2(42
9
0 2
O)$nili2
(16)
The coefficients o’,i are the Wigner functions and $niliz
equals $nnln2m with i, = (m n, - nJ2 and i, = (m - n2
nJ2. The ($k, 0,) are the spherical angular coordinates
of g k (0 < 4 k < 2n and 0 < @k < n, 0, is the angle between
g k and F). Then using the representation diagonal in I/ we
obtain from eqs (2) and (3) an expression for the zero order
approximation to the quasi-energetic wave functions :
+
+
Y!,:!,..(r,t) = exp [ -i(E!,’’
x
il.
iz=
-j
+ E!,!,!,,,,)t]
Dii1(a1, ~
x exp [ip(i,
‘
1 o,~ ! , , i ~ ( a9 2~2
and the values of the matrix elements of the operator r
between the states
and
in parabolic coordinates
[7], we obtain the following expressions for the coordinates
of the vector
k:
j
znn‘n*’
100,k = 2cn
xnn’n’’
100,k
1rJk(2rp) exp ( i r A a ) d ~ ‘ r ( p 1 ) d ! ‘ ’ - r ( p 2 )
r=
-j
1
2 cn(xl
=-
+
i
x2)9
YyO$: k = -2 cn(xl -
(21)
Here
9
0)
- i2) sin ~ t ] $ , , ~ ~ ~ ~ ( r ) (17)
Higher approximations to the quasi-energy E and the quasienergetic function Y cannot be written in a simple analytical
way. We therefore restrict ourselves to the above results and
consider the validity region.
x1= exp (-ia2)
j- 1
Jn’
r=-j
- (2r + 1)2Jk[(2r + 1)pl
x exp (irAa)d’,.r(fll)di.,,
-
r=l-j
4,)
(224
356
T . Helgaker and 1. Tomashevsky
where
It is easy to see that
+
C, = 24n3(l - r ~ ) ” - ~ / ( l
- a2
(23)
contain only n or n - 1 terms and
Aa = al
The formulas [eq. (21)]
are convenient for calculations. The validity region is discussed in Section 3.
= (dF
In the basis of the states in parabolic coordinates
with quantization axis along the z-axis [2] the operator U
may be written in the form
Uki
= (dF+ P B w)ki
JO@ki)
uki
= (ds -k
- X ~
PB w)ki
X- Y f y
JO@ki)
PB1,
*x
+ PB1, * y ) k i
JO@ki)
(A6)
for off-diagonal elements. The only non-vanishing offdiagonal elements of the operators I,, l, and d,, d, are
between the states $nnlnZm
and JIn. n l * 1, n2. m T 1,
$, ,1 n2 1, mT 1. Therefore, all p k i are equal to & 3nF/2w and
we obtain eq. (10).
(AI)
where
References
and
is the
function’ To Obtain
we have used the identity
m
exp (ip sin at) =
~ , ( p exp
)
(inwt)
n=--m
Physica Scripta 46
(A5)
*z)kk
for diagonal elements and
=(
Appendix
+ PB w ) k k = (-z9z + PB
this expression
(A4)
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2. Landau, L. D.and Lifshitz, E. M., “Quantum Mechanics (Non-Relativistic Theory)” (Pergamon,Oxford 1977).
3. Demkov, Yu. N., Monozon, B. S. and Ostrovsky, V. N., Sov. Phys.
JETP 30,775 (1970).
4. Cohn, A., Bakshi, P. and Kalman, G.,Phys. Rev. Lett. 29,324 (1972).
5. Gavrilenko,V. P. and Oks, E. A., Sov. Phys. JETP 53,1122 (1981).
6. Delone, N. B. and Krainov, V. P., “Atom in Strong Light Field”
(Springer, Berlin 1985).
7. Bethe, H.A. and Salpeter, E. E., “Quantum Mechanics of One- and
Twoelectron Atoms” (Plenum, New York 1957).