Journal of Modern Physics, 2010, 1, 399-404
doi:10.4236/jmp.2010.16057 Published Online December 2010 (http://www.SciRP.org/journal/jmp)
Calculation of the Effective G-Factor for the








222
12 3212
ns SnpPns S Transitions in
Hydrogen-Like Atoms and Its Application to the
Atomic Cesium
Ziya Saglam1, S. Burcin Bayram2, Mesude Saglam3
1Department of Physics, Faculty of Science, Aksaray University, Aksaray, Turkey
2Department of Physics, Miami University Ohio, USA
3Department of Physics, Ankara University, Ankara, Turkey
Received August 14, 2010; revised October 25, 2010; accepted October 27, 2010
Abstract
We have calculated the effective g-factor


*
g
for the






22 2
12 3212
ns Snp Pns S

 transitions in
hydrogen-like atoms and applied it to atomic cesium. We have identified that not only the g* factor in this
case is an integer number g* = 1, but also the existence of possible entangled states related to the above tran-
sitions. Furthermore we have used the above result to calculate the transition energies which are in complete
agreement (within the 1% margin error). Such results can give access to the production of new laser lights
from atomic cesium.
Keywords: Photonic Transitions, Hydrogen-Like (Hydrogenic) Atoms, Landé G-Factor, Quantum
Entanglement
1. Introduction
The study and accurate determination of the excited-state
properties of atomic and molecular systems, such as the
fine and hyperfine coupling constants and oscillator
strengths play an important role in testing high-precision
atomic theories and quantum mechanics.
In particular the investigation of hydrogen-like atoms
and its applications in modern technology go beyond the
hydrogen atom itself. Amongst hydrogenic atoms, ce-
sium with its low ionization potential and simplicity of
its outer shell structure has attracted a lot of attention. In
particular cesium has been involved in a number of sig-
nificant studies within the field of laser cooling as well
as in the development of atomic clocks. Furthermore
investigations of parity violation [1] in cesium have been
able to yield high precision results.
Recently it was also shown that cesium atoms are ideal
candidates for optical computers, since cesium vapor is
optically highly nonlinear, as well as possessing sensitiv-
tion a new series of scientific experiments on the proper-
ties and behavior of cesium atoms have been used to
prove fundamental connections between chaos theory [3]
and quantum entanglement.
The aim of the present st
ity much greater than most semiconductors [2]. In addi-
udy is to investigate the ef-
fective g-factor (
g

) for the






3 12
P transitions in hy-
dro-gen-like atoms and apply it to
ling a one-electron hydrogen-like
at
22 2
12 2
nsSnpn sS


atomic cesium. We
identify that the effective g factor has an integer value of
g*=1 as well as the existence of possible entangled states
related to the above transitions. We show that the al-
lowed transitions occur vertically from one crossing
point to another with respect to g* variable. Moreover
calculations of their corresponding energy values reveal
complete agreement with our previous results [4] (within
the 1% margin of error), giving access to new laser lights
from atomic cesium.
We start by mode
om in the presence of a magnetic field ˆ
BBz


. We
assume the sources of the magnetic field [5e the ,6] to b
proton’s magnetic moment,
p

and theag- electron’s m

Z. SAGLAM ET AL. 400
netic moment e

(or
j

) which has two components,
namely, the orbital part, l

and the spinning part
s

.
We note that thCoub potential in both cases re-
quires planar orbits for thlectron, therefore the dir-
tion of
e lom
e e ec
p

 is taken to be in z-direction.
Moreover it was recently shown by Saglam et al. [5,6]
that theantized magnetic flux through quthe electronic
orbits of the Dirac hydrogen atom corresponding to the
quantum state ,,
j
nlm is given by




0
,, jj nlmn lm where
72
hc
  is the flu
04.1410 Gc x quantum.
the aboationship is i
m
nd
e
ve rel
Since
ependent of


,
p
e

 , it can
gen-like atoms. Note that be easily gneralized to ehydro
in such a cases
p

would be replaced bagnetic
moment of the positively charged ion core. The above
calculated flux relts in very high magnetic field values
(such as 78
10 10GG) inside a hydrogen-like atom.
Therefore when such an atom is placed in an external
magnetic f

its effect would be to orient
y the m
su
B
ield ˆ
Bz
p

 in the z-direction and e

 in the opposite direction.
Nevertheless the agnetic field inside the atom will
be of the same orr with the initial value
78
10 10GG because of the fact that 7 10BG . Such
high magnetic field values have not been achieved ex-
so far.
The total magnetic field inside the atom will be called
the local magnetic f
total m
still de
, B
perime
ield . The investigation of th
ef
ntally
e
fective g-factor

*
g
requires to study the Zeeman
effect in detail, therefore theigenstates must be distin-
guished in the Ze sense including the quantum
number
e
eman
j
ls
mmm [7-8]. Using a non relativistic
Hamiltonian [9] we proceed to investigate the Zeeman
fine enere shown that for the above mag-
netic field values the spin-orbit coupling and the quad-
rupole moment energy can be neglected. Considering
diamagnetic and paramagnetic effects the energy eigen-
values become

gies. It can b

*
,, ,C Enlm BBmgm
j


  


2Bls
n



corresping to the ond,, j nlm  eigenstate. Note th
magnetic feld is replaced by the local magnetic f
factor is
at the
ieldiB
B and the Lande-g replaced by the effective
Lande-g factor,
g
 which is treated here as a varying
ameter. It should be noted that the same expression is
also applicable in the case of an atom subject to a laser
beam, where due to the photon’s magnetic moment [11]
and hence the large intrinsic magnetic field comparable
with the above large values [12], the same diamagnetic
and paramagnetic effects are present inside the atom.
To proceed further, by using the above energy relation
we define a new dimensionless function
par


*
2
,,,, l
ljc
m
24
g
C
fm gEnlmBn 



 



which allows us to investigate the effective-g-factor va
ues
l-


*
g
more easily. For and

0sl(1)pl

levels where we have 0, 1
l
m, the plotting of


fm,
lg

 as a function of
g
that c
of these lines correspond to either 1g,
shows rossings


s
p

r 2g crossiongs

, which corresponds to pp

crossings. As the subject of the present w t
tigate the effectctor 
*
ork iso inves-
ive g-fa
g
for the






22 2
12 3212
nsSnpPn sS

 transitions in hydro-
gen-like atoms, our graphic arly shows clethat for the
above transitions we have: g
ese cr
1. At these crossing
points, two states with opposite spin have the same en-
ergy value. We believe that thossing points corre-
spond also to the entangled states [12]. After finding the
effective g-factor value which is equal to one

 1g

,
we set 1g

in the energy expressions


,, ,Enlm B
j

. Substituting the value of 2
Be
mc

and

12
s
m

, we find:

**
22
l
m C
En n ,, 4
c c
j
m B



. Here the constant
is the characteristic of each hy
calculated from the ionization ene
ab
tiv-
ist
 
C drogen-like atom and
rgies. Applying the
ove results to atomic cesium and calculating the cor-
responding Zeeman fine energies at the s-p crossings,
reveals complete agreement with experiment [4,13]
(within the 1% margin of error). The above treatment can
give access to new laser lights from atomic cesium.
The outline of the present study is as follows: In Sec-
tion 2 we calculate the energy levels of the non-rela
ic hydrogen-like atom in the presence of an effec-
tive(local) magnetic field, *
B

. From the above energy
relation we calculate the effective g-factor for the






22
12 3212
nsSnpPn s

 transitions. In Sec-
tion 3 we apply the present energy results to atomic ce-
oncluding remarks.
2. Calculation of the Energy Levels of
2
S
sium. In Section 4 we present the c
LeHamiltonian [9] for a
ne-electron hydrogen-like atom in the presence of an
Hydrogen-Like Atoms in the Presence of a
Uniform Magnetic Field
t us consider the non-relativistic
o
external magnetic field, ˆ
BBz


:



2
2
11 .
24
Ze
0
22
2
8
B
H
prLS
m
 

r
e
LgSB Br
mc



  
  


(1)

Z. SAGLAM ET AL.401
The first term corresponds to the kinetic energy op-
erator, the second term is the Coulomb potential, the
third term is the spin-orbit coupling, the fo
dipole moment energy and the last term is the quadrupole
moment energy. As was discussed in [9] for fields up to
rth term is the
4T the spin-orbit coupling and the quadratic terms
can be neglected so the Hamiltonian is reduced to:
10

2
2
0
0
11
24
B
reduced B
Ze
H
pLgSBHH
mr


 
 

(2)
where 0
H
is the Hamiltonian for a free electron and

B
B
H
LgSB


 
 is the perturbing Hamiltonian
Using Dirac notation, we denote the eigenstates of 0
H
by ,,l
nlm thus writing:
02
,, ,,
l l
,, ln C
H
nlm Enlm nlm
n
 (3)
where 2
C
En  is the energy of the free hydrog
nen-
like atom (here the constant C is determined t
ionization energy).
If we denote the eigenstates of the Hamiltonian,
by
hrough the
reduced
H ,,
j
nl m where mj = ml + ms, the energy
eigenvalues

,, ,
j
nZeeman
Enlm BEE


will contain
the Zeeman correction which is first order in B, i.e.:


0
,, ,,
,,
reduced jBj
,, ,
jj
H
nlmHHnlm
EBnlm

 (4)
Here the field dependent part
nlm

.
B
B
H
L BgS






corresponds to the Zeeman energy,
Z
eeman
E i.e.:

,, ,,
,,
B
jZeeman j
Bl sj
H
nlm Enlm
(5)
and (5quation
(4) gives the energy eigenvalue,
Bm gm nlm   
Substitution of the Equations (3)) in E




,, ,,,
jj
Enlm BEnm B
rection:
including the Zeeman cor-


2
,,
j
Bl s
C
EnmBBmgm
n
   (6)
where 2
B

ohr magneton and
e
mc
 is the B
g
is the
Lande-g factor which is equal to 2 for a free electron
an atom or ion in a free space the Lande-g fac
given by:
. For
tor [14] is




111JSS LL   (7)
121
J
gJJ  
agnetic field [12], we will have diamagnetic
and paramagnetic effects [10] inside the atom. Therefore
the magnetic field inside the atom must be
the local magnetic field
When an atom is subject to a laser beam because of
the photon’s magnetic moment [11] and hence the large
intrinsic m
replaced by
B

[15]. The Lande-g factor is
also replaced by the effective Lande-g factor,
g

which
is treated as a varying parameter [16,17]. Therefore the
magnetic field B and the Lande-g factor in Equation (6)
must be replaced by the effective values, B

and
g

respectively, such that:


2
,,
j
Bl s
C
EnmBBmgm
n


  (8)
Substituting the value of
2
Be
mc
  and 1
2
s
m


in Equation (8), we find:

*
2
,, 24
lc c
j
mg
C
EnmBn
*



 
 (9)
Where *
ceB

mc

 is the cyequenc- clotron fry corre
sponding to B

.
Let us proceed by defining a dimensionles
using the above energy expression given in E
such as:
s function
quation (9)


*
*l
m
2
,,, 24
j cl
g
C
fm gEnmBn 


 


 (10)
which does not have an explicit n dependence:
Plotting


,
l
f
mg

 for

0, 1m as a function
of
l
g

leawingure 1):
The h
ds to the follo
crossings of t
g results (Fi
ese ,
1)

l
f
mg

 lines co
spond to
rre-
1g

or 2g

.
2) Atts, with opposite
these
crints correspoangled st
F
these crossing pointwo states
spin have the same energy value. We believe that
ossing pond to entates [12].
3) or the crossings (entanglements) of the [


S
an
2
ns 12
d


2
32
np P] as well as [

2
12
ns S

and


2
12
np P]
states, ther is f effective g- factoound to be equal to one:
1g

.
4) The crossing for the 
pp states occur at
2g

.
In the present study since we concentrate on the






S n trans at th
for all energy calculations. Here we note the allowed
phically f
point tonother one.
22 2
12 3212
nspPn sS
sitione
crossing points, we substitute 1g in Equation (9)
otonic transitions occur vertrom one crossing
a
To find the energy eigenvalues, we start with


0sl

.
orientations we have
two
We note that depending on the spin


2
12
ns S states. In Dirac nn, these states are otatio

Z. SAGLAM ET AL. 402
Figure 1.  ,g

24 l
l
m
g
fm as a function of
g

for
. Identifying the effective g values (g*).The ,01
l
m
 s
p
 crossings occur at

1g; the 

pp
 crossings
occur at g
2
.
,0,0,n,0,0,1 2n and ,0,0,,0,0, 2n .
ation (9) the corresponding energy eigenvalues
ossing points are given by:
1n
From Equ
at the cr

*
2
,0,0,124
c
C
En n

  (11)
and

*
,0,0,12C En 24
c
n

  (12)
respectively.
Equations (11) and (12) we see that the state From
12
, 0,n0,,0,0,n has the lowest energy so it is
identified as the ground state.
Next we consider the

1pl

up states
states, w
states in total. The spin
hich are six
12
,1,1, , n12 ,1,0,n
and 12 ,1,1,n states are denoted by 2
32
e spin down states
np P while
th ,1,1,12n

, ,1,0,12nand
,1,1,12n  are written as 2
12
np P. Using Equation
(9) the corresponding energy eigenvalues are:

*
2
4
* *
2
3
,1,1,1 224
cc c
CC
En nn


 
 )

 (13

** *
22
,1,1,1 2244
C
En nn
cc c
C

  
  (14)

*
c
C
2
,1,0,1 24
En n

  (15)


*
2
,1,0,124
c
C
En n


 (16)

**
22
,1,1,1 224
*
cc c
CC
En nn

4


 
(17)

**
22
3
,1,1,1 224
*
cc c
CC
En nn

4

 
 
(18)
3. Application to Atomic Cesium
The ionization energy for cesium is , while the
smallest amount of energy th
the ground state to the neares
herefore in Equation (11) setting th
3.89 eV
at allows a transition
t excited state is
e lowest en
from
1.38 eV .
ergy to T
3.89 eV

we can write:

2
6, 0, 0,124
6
c
EV   (19)
From Equation (12) and Equation (17) the nearest
excited state energy is given by
3.89
Ce



24
6
c
C




. Since

the smallest amount of energy that allows a transition
from the ground is then the energ
level will be
1.38 eV ,y of this

23.89 1.38
4 6
  (20) 2.51
ceV

C
 
From Equations (19) and (20), we find that the un-
knowns: 115 CeV

and
*
1.38
ceV

. Thus for the
2

cesium atom Equation (9) is written as:

2
1115
, ,,, ,,1.380.69
2
jl l
E
nlmBE nlmBmg
n



 
ange of es the values (
lot the energies
bers:

(21)
where the rl tak-1,0,1) for the m
present transitions. In Figure 2 we p
given in Equation (21) for quantum num6n

and
10n

.
Plotting the energies corresponding to Equation (21)
against
g

for 6,7,8, 9,10n

gives crossings at
1g

as ex
which are denoted
pected0 main crossing points . We observe 1
by the letters:
A
, '
A
B, ' B, C,

Z. SAGLAM ET AL.403
Figure 2. The plots of 

,,
l
E
nm g

with respect to
g

for
the


6,s
p

and

,s10
p
 stat
occur at g
respect to
es for at
it iscrossand transitios take
plly with
omic cesium. As
n seen the
ace vertical
ings 1
g
.
and
'
C, D, ' D
F
, '
F
. Their energies and related
n in
from
states are give Table 1.
To excite electron
A
to '
A
,which orre-
sponds to a
c

22
32
66
12
s
S
ener
pP transition, the re-
quiredgy is

'2.513.891.38 AA eV .
To excite electron from '
EE
A
to
F
, which corresponds
to 

22
12
632 10pP sS tsiran
2.51 
e giv
44
erim
ti

en
e
oe required en-
rgy is When
electron rela
nts involving cesium
clusions
n, th
2.06 
by:
Th
e'
xes bac toe ground state the energy of
the corresponding laser will
0.45
FA
EE eV .
k
ent with

th
b
exp
the

0.453.89 3.
FA
E eV .ese values are
in good agreem
lasers [4].
4. Con
We have calculated the effective g-factor 
E
g
 for the

22 2
ns Snp P  transitions in hydro-
gen-like atoms and applied it to atomic cesium. We have
found that the value of
12 32 12
n sS
g
 is exactly equal to 1 for the
bove mena
p
tioned photonan
resent results to atomic cesium gives complete agree-
nd the raleted states at the crossing
,10.
ic trsitions. Application of the
Table 1. Energies a
oints for n = 6,7,8,9p
Energies Mixture of states
3.89
A
EeV 6,0,0, and 61,1,
'2.51 A
EeV 6, 0, 0, and 6,1, 1,


3.04
B
EeV 7, 0, 0, and 7 , 1, 1,
'1.66 B
EeV 7,0,0,  and 7,1,1,


2.49
C
Ee V8, 0, 0,  and 8,1,1, 
'1.11 C
Ee V8, 0, 0, and 8,1,1,


2.09
D
Ee V9,0,0, and 9 , 1, 1,
'0.71 D
Ee V9, 0,0, and 9,1, 1,


0.45
F
Ee V10,0,0, and 10,1,1,
'0.93 F
Ee V10,0,0, and 10,1, 1,


ment (within the margin of error) with the previous
experimental results. We have also s the extence
of possible entangled states related to the above transi-
tions. Application the above tre ac-
cess to the production of new laser from atomic
cesium.
. References
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s ofatment can give
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