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ESSEÑA:\'ZA
REVISTA \IEXICA~A
Vibrational-rotational
v.e.
DE FISIC:\
-o l-l) 41.~-41l)
AGOSTO 199H
analysis of the HuIthén potencial using hydrogenic
eigenl'unction bases'
Aguilera-Navarro,
l,:!
E. Ley- Koo. 1,:~alle! s. f\latcos-COrlés:~
llnSTilllto tic FY\.ica Ji'úrica-UNESI}
tú/({ ¡'af1lplOfla
145, OI405-Y()() St70 Pl1//lo, S/~ Bmúl
:? DepartlUJlellfo
de F{sica.CCE, UlIh'('tsitlmll'
Estadltal de Londrilla.
Caixo Postal 60() 1, 8Ó05/-Y9() l..mulrillo. PI( IJrtr:.il
:{In sti tu 10 tic ¡:úicl1. 1/11 i rC'rsidad NaciOlItl I A lI/állOllla dc México
AIJ(Irtatio I}(JStaI lO- 3 fJ-1,()¡()()()
Al hico, D. ¡;:. Al nico
Recihido c1-l de diciemhre liL' 1997: iKcrlado el 20 de ahril de 1998
J lulthén's pOlenlial admils analylical solulions for its cncrgy eigcllvalucs and l'igl'nl"unclíons COlTl'sronding 10 ¡cro orhilal angular Illornentum
.~trlte.~.hui IL~non zero angular momcntUIll slales are not cqually kllOWll.Thís wOlk rrcscnls a vihralional-rotational analy sis (Jf Huhhén.s
potcnlialusíng hydrogenk cigcnfunction hases. whích may he 01" interesl amI usdul lo sltl(!l:IIISnf quantum mechanics al diffcrent stages.
K('ywonl.c QU:lntum Illechanics: Hulthén's potential: variational solU!iollS
El potencial de Ilullhén admitc soluciones :lllalílicas para sus eígenvalores de la energía y eigcntünciones correspondientes a estados con
momenlo angular orbital cero, pero sus estados con momento illlguJar diferenlc de crro no son i!-,u:llmcnte cOlHKídos. Este trahajo presenta UIl
an:í1isis vihracional-rotaci<mal del potencial de Ilullhén usando hases de eigentunciones hidrngénicas. el cual puede ser de interés y utilidad
para estudiantes de lllcc<lnica cu<Íntica en diferentes etaras.
/)I'SCI"i¡I!OI"l',I."
ivteC<Ínicacll:íntica; potencial dc Ilulthén: soluciones variacitlllalcs
PACS: O_l(l)
1. Intrnductinn
Lamek Hulthén inlroduced
Ihe potential
(1 )
in an article witll lhe tille" On the Eigcnsolution of Ihe
Schródinger Equation oflhe Deulcron" puhlished in Ihe "r!\ir
{¡;r A-lathclllarik, AsTmllomi och Fysik in 194211]. Thc shOrl
distancc behavior of Ihe polenlial is Iikc that of the Yllkawa
potential,
y
1° (,.)
and asymplOtically
=
-10-.
e ->.,
(2 )
A"
it hccoll1es the exponenlial
po!cntial.
i.l)
\Vhile Ihe Schrüdingcr equalion does not admit analytical
solutions for hoth potentials VY amI Vl-:. Hulthén showed
in Re!". I tha! F" has analylical exprcssions for the energy
cigcll\'allles ami eigcnfunctiolls of ils zero angular mOlllcnIUIll slales. in I""ac!,he used these eigensolutiolls of Ihe \'11
pOlcntial lo construct perturhalive solUlions 01"Ihe Yukawa
polcnlial in the case ol""the deuteron.
()lIllle other hand, the Schriidinger equalion does not admit analytical sollltions for Ihe non-zero angular mOlllentum
slales of Ihc Hulthén potcntial eilher. Correspondingly,
lhe
study nI' such stales has hecn lhe suhjecI of research anJ
Ihe Icsling grollnd nj" dilferenl thcorctical approaches in the
lasl fL~\'ldecalles. Refercnces 2-7. limiled lo lhe cighlics, illustrate Ihe mClhods ami results 01""different approxilllation
schellles and also contai n refcrenccs lo previoLls works.
This work is a didactic study of hOlh I.ero amI non-zcro
angular 1l10lllelltum slalcs of Hulthén's pOlcnlial. which may
he 01"inlerest amI uscful lo sludenls of quantulll lllechanic:s
al dilTerenl stagcs. For COlllplclcncss sake, our own versioll
01' thc exact eigensolution for s slates is incluJcd in SCCI. 2.
Variational solulions I""0l"
any angular Illomenlum I slalcs are
I"orlllulalcd Llsing hydrogenic eigenfuJ1ClioJ1s, and llulllcrical
reslllts are prcsenlcd in Sect. 3. The formulatioll ami nurnerical evaluation are carricu oul in two sLlccessivc slagcs. Section _,.1 is Jimited lo Ihe slales \Vilh no radial excitation ami
allows Ihe optilllizalioll 01"the nuclear charge varialional parameter in the hydrogcnic tri al functions for each value of 1
In ScCI. 3.2 Ihe Iincar varialional lllelhoe! is implelllented in
its lllalrix I"onn lIsing Ihe complele hydrogenic eigcnfunction
hases, obtaining al once Ihe statcs with successivc r:luial excilalions ror cach l. SectioI1.:1- includes a discussioll of lhe
rcsulls amlmelhods.
ve. AGUILERA-NAVARRO.
2. Exact eigensolution
for = o states
E LEY-KOO. ANO S. MATEOS.CORTÉS
of the Hulthén potential
e
Thc Schrodingcr
cqualion
lieles intcracting
via lhe Hullhén potcntial
for lhe rclativc
is wfilten as
<
inlO ()
in onler LOcnsure lhe correet hchavior
Then it is slraightfonvard
to eSlahlish
in terllls ol' lhe rcduccd l1lass Jl. Bccausc of the central naturc
pOlCntial the problclll
tcrms 01' lhe sphcrical
Xl
< 1,
.f
Herc we take
.'/(.,.)= .,.11(.,.)
lllotiOIl of {wo par-
(4)
orlhe
< ,. <
which maps O
additionally
adrnits Ihe separahle
solulions
(14 )
01" Eg. (9) for r ---+ O.
that Eq. (11) takcs the
form
¡{!.}¡
.,.(1 - ''')-1 .,
( ;1:-
in
+
[. (. 20)]
<lit
d."
2-3+-.,.A
harmonics
( 15)
=
\'(r.8.<;)
which are cigenfullclions
(5)
R(r)li",(H,<;),
01' lhe energy
mOll1cn~
and angular
The rcadcr can idcntify that Eq. (15) corresponds
canonical fOl"m01' lile ditlercntial equalion
10 the
tUIll. Thc radial fUllclion must satisfy lhe cquation
rfl/J
(hJ
"'(1-,,.)--'., +[,'-(a+h+
[_~:, (,1,,~.r' ,;:. _
_1'
01
Thc rcmaindcr
1))
l(l,~
e
for the hypcrgeollletric
-A'
]
_ e->.r
of this scclion
1)''')-1'
rf.1'-
=
R(r)
ER(r).
y
(6)
=
:!
(.1'
(16)
[81.
functioll
F'l((/,b;(':.r )
-ahy=O.
=
~ (a),(h),
LI
(e),s .
,
.r'.
( 17)
•
~=()
is rcstrictcd
lo s stalcs
with
f = () rOl' which we makc Ihe standard changc
whcre (a)., = a(a + 1) ... (a - 1 + .<) and (a)" = 1 are Ihe
Pochhallll1lcr sYlllhols. Also. the p;:¡rameters in Eq. (15) are
illllllediatcly
idenlillcd
as
(7)
(l
oblaining
Thc
squarc
houndary
0=1+--
lhe radial cquation.
intcgrahilil)'
conditions
A
condition
is translatcd
and
J(,.)
r~O
hchavior
Ilcntially decreasing
,,= 1 + -A +
lhe
(' = 2.
on the radial function
J(,.) -> O
Thc asymptotic
ioto
(R)
-> O.
,.~=
may be cnsurcd
(9)
lhrough
an cxpo-
(\
(1 R)
For arhitrary values of thcsc paramctcrs,
series of Eq. ([ 7) is divergcnl 1'01';1"-t
l' --t 00, hcc<luse cacll nI' its asymplotic
lhe hypcrgcomctric
1 eorresponding
to
lcrms has the limit
vallle
.
(a),(h),
l 1111 ---
factor
.~~= (c) ....s!
->
1
( 19)
.
( 10)
The ooly way 10 avoid such a divergent hehavior fOl"Ihe solu(ion 01"Eq. ([5), in lhe rorm nI' Eq. (17) with lhe paralllelers
in which case Eq. (8) hecomes
(11 )
=
10 Ihe values
wilh
,,=
amI lhe reminder
statcs.
Hulthén
01' Eq. (1 X). is to ret.lucc the series with an inflnitc numhcr of
terms 10 a polynolllial of degrcc 11"
(l. 1.:2 .....
This can
he aecomplishcd
hy rcstricting lhe values of lhe parameter a
[hat lhe cnergy
E is ncgative
found Ihat Eq.( 11) is integrahle
-/1,.
( 12)
)-21IE/11'
fur bound
underthe
change
= 1
11
+ --
(20)
A
ensuring Ihal lhe cocrticieots
(-u,.).., with .•• > Itr. vanish.
According lo Eq. (12). this is cquivalent to lhe rcstrietion of
the eoergy lo ilS eigenvalues
nI' variahle
E"
(13)
-11,,:
fI1,\1[
= ---'2JI
-----:2JI\~)
,1 f¡l /\"1(n,. + 1)
Jin, Mex. Fís. 43 (4)( 199R)413-l1~
(11,.
+
1)
]'
(21 )
VIBRATIONAL.RarATIONAL
= "r
where "
+ 1
4uanlum number.
=
Thc corrcsponding
fonn
ANALYSIS OFTHE
1,2,3,
...
IIULTHÉN POTENCIAL USING IIYDROGENIC
EIGENFUNCTION
BASES
plays Ihe role 01' Ihe 10lal
radial eigenfunction
takes the cxplicit
= = =
fl(,-)
=
N_I_-_,,_-_Á_rexp
2F[
which involves
(2;'1:
[_~
,-
0
_ ,,)
+ 1;2;1-
r,-
,-]
e
-Á"]
(22)
E~' = -132/2112
functioll.
powcr series in the exponential
For a givcn pair 01' particlcs interacting via a Hulthén polential, Eqs. (21) and (22) show Ihat Ihe numher 01' hound .s
stalcs is lhe integce part of J2jlVo/t¡)...
anJ eigenfunclions
It is also instructivc (o view lhe Hulthén potcntial as a
type 01'scrccncd Coulomh pOlcntial, in which)' is lhe screcn-
fl n ,(,)-A"""
ing parameter.
Eqs. (1) and (2) show Ihat'if Ihe strength paralllctcr is chosen as Va
Z(;2)" then in lhe limit ), -t
=
Coulomh
and Yukawa
=
Prom here on we á~sume f¡,
JI
e
1 and Vo
Z ~\ in order to conform wilh lhe nOlalion of r2-i]. The eigcnsolutions
of Eq. (27) correspond
to the well-known
atomic hyorogen
Bohr cnergies
"A-"
2
2¡No
[ -(11-1), -'-.).:in
O. hOlh Ihe Hulthén
~15
pOlen ti al s reducc
lo Ihe
potcntial.
Ze2
IIr(,-) = --o
(23 )
r
It can he easily verified Ihal in su eh a limil. Eq. (21) is
rcduccd in lurn 10 lhe Bohe formula for Ihe atomic hydrogcll
cncrgics
e
.. _ . r
..1 -¡hin
[F[
r
•
(
(28)
~{3r)
..
_-_
-",,21+2,
11
(29)
=
whcrc n
nI. + 1 + 1 [9j. The reaJcr rnay recognize
thal
Eqs. (28) a",1 (29) reduce lo Eqs. (24) and (25) for the case
01' 1
IJ sta les.
=
Thc varialional analysis nf Eq. (6) lO he carried OUI nex! is
hased on Irial funclions conslrucleo with lhe basis of El], (29),
and taking ¡i as a varialionai parameter.
[n SecL 3.1 the analysis is reslricted lo the slates wilhout
any raJial excitation. rOl' which it is sufficicnl to staft with the
Irial functions 01' the Iype 01' Eq. (29) wilh /l,
O. The analysis 01' Sec!. 3.2 incluJes radial excilalions anJ requires trial
functions conslructed as linear combinations
of Ihe complete
hasis 01' Eq. (29) wilh 11.,.
n, 1,2,.
=
=
(24 )
which are illfinilc in number,
Eq. (27) also become hydrogenic
fl
(. (,-) = Ne-"W:;"
z.' •
[F[
and
lhe eigcnfunclions
( -(11
-
1); 2;
22e2¡t1')
2
11,
recalling
1
thal
the
contluenl
hypergeomctric
F[ ("; c; x. ) is the limil 01' lhe hypergeomelric
lim
1'-+00
2F[ (a,b;c;=-b')
3, Variational solutions
tial for any f. sta tes
In this seclion
we considcr
in
19J
=
[F[(a;c;:r).
n
Wc choose
'
Calculatiol1
the normaJizcd
for States
hydrogenic
witltOUI Radial
Irial functions
(25)
function
funclion [8]:
(26)
of the Hulthén Poten-
the solulion
.1.1. Varialioll:ll
Exdtation
of Eq. (6) for statcs
wilh any angular momcntum l. Since no analytical solutions
are known in this case, we propose a variational analysis. Thc
conneclion
hetween lhe Hulthén pOlential and the Coulomb
potenlial ami their eigenfunclions,
eSlablished in Ihe previous
Seclion, suggests thal the atomic hydrogen eigenfunctions
may be approprialc
as a basis to conslruct variational solulions 01' Eq. (6). Here we consider the equivalent 01' Eq. (6)
for a Coulomh potent with a nuclear charge parameler /3
fl7'(,.)
=
~
(-iJ)
2'+'
[ ~
1
(21 + 2)!
] [/2
.'
l'
e
-¡Jr/"
(30)
rol' the varialional ca!culalioll 01' lhe energy cigenvalucs of
the slales 01' Ihe Hullhén potcntial without radial excilation
=
=
for which nI'
O ami n
1 + 1. The calculation involvcs
lhe cvaluation 01' lhe cxpectation
value of the cnergy using
El]. (6), ami its lllinimization
wilh rcspecl lo lhe varialional
paramclcr ¡:J. BOlh sleps are carried out as described ncxt.
The evalualioll 01' Ihe cxpectation
value al' the kinetic energy lerm in El). (6) Iin' Ihe Irial function 01' Eq. (30). is the
same as Ihe corresponding
evalualion in Eq. (27). In the lal.
ter wc use lhe virial Ihcorcm for the Coulomh potential 19)
lo concJuJe lhal thc cxpeclalion
value 01' the kinelic cncrgy
is the negativc 01' the expectation
valuc al' lhe total cnergy
nI' Eq. (28). The evalualion 01' the expeclalion
value 01' Ihe
Hulthén po(enti;¡[ is slraightforward
making use of ils geo~
melric series rcprescnlalion:
ReI'. Me". Fis. ~3 (4) (1~98) 413-11 ~
v.e.
~16
AGUILERA.NAV.ARRO,
E. LEY.KOO, ANIl S. MATEOS.CORTÉS
Tt\ BLL l. Sur<:essivc cntrics corrcspond lo statcs n1 without radial cxcitatioll for which 1 = 11- 1. scrccning para meter ..\. scalcd variational
paramctcr ¡3/ 11. minimizcd cncrgy Emin• ¡¡ud cxael cncrgy E""art. (mm Eq. (21) fUf .<; slalcs and f["1n [2] for olher slalcs.
/3/11
E",,,,
01
,\
Is
0.025
0.999921874999206
-o. 18757812,1
Euact
-0.487578125
0,050
0.999687499949197
-OA ,5312483
-0..t75312500
0,100
O.99H7499%759909
-O.4S 12497 4U
-O ..tG12G()(JOO
IU50
O.9971H7463302950
-0.,12,811189
-0.427812500
0.200
O.99-ttJ9979540.U7..f
-(1.-1019958,4
- O.105000000
0,3(X)
O,9HH747717623 176
-O.:~(j 1229.1G:1
-1136125011110
0.025
0.4994787642H4H 1O
-0112,60438
(1.050
04979101 H46H4437
-0101114201,
-0,101043
0.100
0.49 156005 1950H61
-0.079172425
-0.0,9179
11.150
0.4H0683851~)35646
-O.05!HO;J;JOJ
0.200
0464744947741120
-0.0.1176951J
-0.041886
IUOO
0.4125146623(~)447
-0.013136,3,
-O. (113,90
0,025
O.3316847791~)624
-(1.0.13602856
-11.0436113
0.050
0326639664406316
- O.0;12, 50023
-0.032753
0.304639712141441
-{),O\.1428372
-0.014484
11.150
0.256 U 197235H986
-(1.0111111;1290
-1I.1J1I1;191
0.025
0,24618690901 H531
- (1.0l9G9029;1
-0.019691
0,050
0.233H65HH253761 H
-O.lll()047!J.'3G
-11.1110062
0.075
0,20HHOH334469317
-O.OO2.1G7821
-0.002556
2p
O.! 00
41'
(31)
whcrc ((X, (1) is the gcneralizcd Ricmann zeta function [IOJ.
Tite lirst step is complcted by writing the expectation valuc
of the total energy.
E(l3)
=
32
.!.--, - Z.\
211 -
(~{3)2'+3 (
- ,
(21
'11,/\
+ 3,
0{3)+
",
1
,
(2)
H/\
Tite sccond step consists in obtaining the dcrivativc of this
cncrgy with rcspcct lo Ihe variational parameter and finding
ilS zeros.
dE =0,
d(3
The implementation of this step was carricd out in a personal compulcr using the Mathematica program l11].
Illustrativc numerical rcsults are presentetl in Table 1,
whcrc the successivc columns corrcspond to the spectroscopic dcsignation 01'the statcs without radial excitation. fOI"
which 1 ::::n-l.
In the chosen valucs of Ihe screening paramcler .\ 01'the Hulthén potential. to the scalcd variational
parameter (3/11 cvalualcd via Eqs. (32) ano (33), to the respective minilllized energies Elllill fmm Eq. (32), and 10 Ihe
exact cncrgics from Eq. (21) for.'i states and from [21 for the
other slatcs. The readcr Illay notice that the variational parameter ¡-J lakes valucs c10sc to Z :::: 1. with systcmatically
increasing dcpartures from such a value as the screcning parameter inCITases ami as we move to statcs wilh increasing
rotational cxcitatioll. Corrcspondingly, the variational ellefgies uf Ihe Hullhén pOLential, starting from the values of Lhe
hydrogcn aLomic cncrgics Eq. (28) ror.\ :::: 0, show similar
dcpartures fmm Ihese valucs as .\ and l wke on largcr values;
such a trcnd can hc understood on the basis of Eq. (32). The
exact encrgics are included as points of rcfcrence for numerical cumparison. The ovcrall conclusion is lhat the variational
cnergics are very c10se to the exact encrgies wilh syslematically ilH.:rcasingdcparturcs for larger values of.\ and l.
!I"". M"x. Fú. 43 (4) (199H) 413-419
VIBRATlONAL.ROTATlONAL ANALYSIS OF THE HULTHÉN POTENCIAL USING HYlJkOGENIC EIGENFUNCTION BASES
3.2. Linear variational
excitation
method
fOf
states witlt an)' radial
for Ihe expansion coeffkients
417
a'/1lr:
,v
In this section wc forrnulate and implement lhe varialionai 50lulion 01'lhe Schrodingcr cquation ror states with any radial
cxc;tation of the Hullhén potent;a!. Eq. (6), using the hasis of
hydrogenic functions, Eq. (29). We consider a trial function
in lhe form of lhe linear superposition:
(34)
L (¡"",.(n,.fl(H
II
-
E)I",I')
= O,
n,.=::O
'llr
=
0.1, 2, ... "V.
(35)
Thc solulion 01' these cquations requires Ihe vanishing
01' Ihe determinanl of the matrix (HII - El), which determines Ihe variational cnergy eigenvalucs Ev and the aptimizcd eigenvcctors
By writing lhe Hulthén Harniltor'
nian in tcnns of Ihe Coulomb Hamiltanian. Eq. (27), the secular equatioll hccolllcs
a¡!Il
in which lhe nuclear charge parameter 13 in lhe hydrogcnic
functiol1s is givcn lhe optimized values obtaincd in lhe previotlS section for choscn values DI' A and l.
Thc substitution of lhe trial fUlletinn, Eq. (34), in lhe
SchroJinger Eq. (6), and lhe subsequcnt and succcssive scalar
Jl1ulliplication by c3ch one of lhe hydrogenic eigenfuctions
R;;'rf(r). Icads lo lhe se! of linear homogeneous equatians
I
(n'el~ln f)=N,
,
=
L
t"
(1I'.el '" e->(p+I)'ln
,
p:=O
N
",'
E) = N ,N
''',1
~~
",1
The matrix clcmenls of the Cou!omh and Hulthén pOlential are constructed lIsing the explicit forms al' the hydrogenic
funclions 01'Eq. (29):
(-n;).,(-n,.),
~:Si;(2E + 2),s'(2C
+ 2),1'
(2(1/n')"(2iJ/n)'(2E+I+s+I)'
(2/3/n' + 211/11.)21+2+'+'
", ",
(')
(
2(3)'
'" '"
-n, , -11,),
",1L L (2E
+ 2),s'(2C + 2) l' ( ni
.~::::O
1::::0
.
1
(2(+2+8+1)'
,\u+,+,+t
Thc nurnerical construction and diaganalization of the matrices was implemented in a personal compulcr wilh the Mathcmalica Program [11].
Table II contains the numerical rcsults l'or lhe linear varational cncrgies of the eigenstates of the Hulthén polential
wilh successivcly incrcasing radial amI rotational excitations.
The enlries in the two first columns, corresponding to the
screening parameter and Ihe states without radial excitations,
coincide with their counterparts in Table 1; Ihe entries in the
following (;olumns correspond to the states with nr = 1,2,3.
Thc energies rcported in cach row were obtained from lhe
diagonalization al' 20x20 matrices eonslructed using Eqs.
(37) and (38), and the corresponding values ofthc variational
paramcter (3 l'rom Tablc L Again, Ihe exact energies l'rom
Eq. (21) for s states and from [2] for states with rotalional
cxcilation are included in parenthesis for numerical comparison. For the states without radial exeitation the irnprovcrnent
of lhe linear variational energies 01' Table 11 rclativc lO the
simple variationul energies of Table 1 is very small rol' the
Jower values of'\ and C, bUI it becornes quantitatively significant rOl' the higher values ol' A and [as in the specific cases 01"
E(3d) for'\ = 0.150, and E( 4f) for ,\ = 0.050 and 0.075.
Por lhe states with radial exeitation the linear variational cncrgics show convergence towards lhe cxact energies with a
dccreasing number of digits as the values of "\, U1' and E get
Iarger. This lrend ol' the convergence and accuracy of the lin-
(37)
(2/1)'
'/1
[,
,
.
(3
(21 + ,1+" + 1,>:
(1-;;; 1) ]
+;;:
+ I
(38)
I
ear varialional energies l'olJows the expeclcd hehavior; more
accurate nurnerical results can be obtained by enlarging the
size ol' the matrices, which we could not do in our personal
computer due to its lirnitations in preeision and mernory capacity. In any case the rcported values are good cnough l'or
our didactic purposcs, and comparable with thosc of [2-7].
Also in our diagonalization procedure we obtain Ihe energics
01'stales with higher radial excitations, and lhe aVnr expansion cocfficients 01'Eq. (34) for the linear variational eigenfunctions.
4. Discussion
The sludy 01' lhe Hulthén potential prcsented in Ihis papel'
covers tapies of intcrest for students of quantum mcchanics.
Scction 2 illustrales <Inexamplc of an exact analytical solulion 01'lhe Schrodinger equation yielding the energy eigen.
values Eq. (21) and cigenfunctions Eq. (22) of Hulthén's potC!llial
o stalCs, including their limiting atomic hydrogenic fonns Eqs. (24) and Eq. (25) whcn the screening parametcr beco mes vanishingly smal!. Section 3.1 presents the
simple varational calculation ofthe cncrgies ofthe statcs with
any angular Illomentum and no radial excilalions using the
corresponding atomic hydrogenic functions. Eq. (30), as trial
e=
ve. AGUILERA-NAVARRO.
418
TABl.E 11. Successive
rolational cxcitations.
>.
E. LEY. KOO. AND S. MATEOS-CORTÉS
enlrics correspond lo screcning paramctcr ,,\, linear variational cncrgics £(n1) ol' statcs with aod without radial alld
For comparison exacl energies from Eq.(z 1) for ..•slatcs <lnd from [2J fUf othcr statcs are included in parenthesis.
E(ls)
E(2s)
E(3,)
E(4s)
-0.01999981
-0.4875781245
-0.1128124890
-0.043758616
( -0.4875781250)
(-0.11281250)
( -0.043758681)
(-0.02000)
0.050
-0.47531249
-0.1012498299
-0.033367177
-0.01124820682
(-0.47531250)
(-O 101250)
(-0(33368056)
(-0011250)
0.100
-0.4512498668
-0.0799975629
-0.016797794
-0.001246256
( -0.4512500)
( - O0800000)
(-0.(16805556)
(-0001250)
01125
0.150
0.200
0.300
-0.4278118279
-0.061239729
-0.00,)8044125
( -0.4278125)
( -006112500)
(-00(5868056)
-0.40499788
-0.04497502
-0.0005490027
( -0.4050000)
( -004500(0)
( -0.(005556)
-0.3612394377
( -0.3612500)
>.
E(2p)
-0.0199.1,]4
( -0.0200(00)
E(4p)
E(3p)
0.025
-0.1127604501
-004370687
(-0.043707)
(-0.019949)
0.050
-0.1010422
-003316.]28
-O.011OCJ728
(-0.101043)
(-0.(33165)
(-0011058)
0.100
-0.07917547
-0.01605176
-O.OOO'i5387
( -0.079179)
(-0016(54)
(-0.00(754)
-0.059421418
-0.00,]462177
O. 150
-0019948&
(-0.0(4466)
0.200
-0.0418222
( -0041886)
0.300
-0.0134634
(-0.013790)
>.
E(3d)
E(4d)
0.025
-0.04360294
-0.01984620
0.050
(1.100
(-0.043603)
(-0.(19846)
-0.0327515
-0.010666642
(-0032753)
(-0.01(667)
-0.01445634
(-0.014484)
(U 50
-0.00124455
( -0001391)
>.
E( 4f)
0.025
-0.01969075
0050
-0.01005618
0.075
-0.00252351
(-0019691)
( -0.010062)
( -0.002556)
R"v. Mex. FiJ. 43 (4) (1998) 413-419
VIHRATIONAL.ROTATIONALANALYSIS
OFTHE HULTHÉN POTENCIAL USING HYI)ROGENIC EIGENFUNCTION BASES
functions. The nurncrical values of the variational cncrgies
are quite accurate, comparing favorably with the cxact cncl"gics ror the lowcr values of A.and R, and showing ¡ncrcasing
deviations as these parameters become larger. In Sect. 3.2 the
linear variational rncthod is formulated for states with both
radial and rotaliana! excitations, using thc matrix formulation wilh the complete atomic hydrogenic cigenfunction basis, El]. (29). For eaeh value 01'
e, the
energy
eigenvalues
01'
the states with successively ¡ncrcasing radial cxcitations are
obtained simultaneously in the same numerical diagonalizalion proccss. The energies of the states wilh no radial excitation show an improvement when going fram their values
o( Tahle 1 to those of Table 11,the improvement hcing more
apprcciahle rol' highcr values of A and e. Thc linear variat¡onal cnergies of the states with hoth radial and rotalional
exeilations show reasonable eonvergenec and aeeuraey when
eompared with lheir values rcporled in the rcseareh Iiteraturc [2-7]. Students of quantum meehanics may eomplement
theír sludy of tapies on prohlems with analylical solutions,
simple variational ealculations and linear variational calculalions hy working
the details
of See!s. 2, 3.! and 3.2. While
419
lhe firsl lwo tapies are covercd und illustraled in standard
eourses, the last olle is lIsually trcatcd only fonnally. The
availabilily 01' eomputers makes lhe latter workable and inslruelive for stuJenls.
Apart
from the speeifie
study
01' Ihe Hulthén
potentia!
of SeeL 3, it muy he pointcd out that lhe simple variational
methoJ anJ the linear variational method as formulatcJ in
Secls. 3.1 und 3.2 can he systematically applieJ to other polentials. For instance, the Yukawa and exponential potentials, Eqs. (2) and (3), can be invesligated using lhe same
atomic hydrogcnic cigenfunetions basis, Eqs. (30) and (29);
in hoth cases the evalualion of the matrix clements is straight4
forward, involving faetoriaIs instcad of gencralizeJ Riemann
reta funetions
in the counlerparts
01' Eqs. (32) and (38). 01'
eourse, other basis 01' funetions may be used for these and
other potentiais trying to improve the convergenee and aeellracy of the rcsults in eaeh specific situation. Also, the reader
muy become awarc 01'the approximate perturbativc methods
of [1-7], and he interested in comparing thcm among lhemselves' anJ with the variationalmelhods.
Conacyt scientillc cx- 8. I.N. Sneddon ~/}ecial Fl/llctiallS oI MarhemalicaJ Physics and
Clwmistry. 2nd cdition (Edinburgh: Oliver and Boyd. Ltd .. 1961),
Chap.l!.
1. L Hullhén, Ark. Mat. Astron. Fys. 28A (1942) 1.
*. Work done under the Brazil CNPq-Mexico
changc programo
2. CS. Lai, and W.C Lín Phys. Lett. 7M (1980) 335.
3. R. DUIl. and U. Mukherji Phys. Len 90A (1982) 395.
4. S.H. Palil J. Phys. Al7 (1984) 575.
5. V.S. Popnv, and VM. W.einbcrg Phys. Len 107A (1985) 371.
G. B. Roy, ,nd R. Royehoudury J. Phys. A2U (1987) 3051.
,. P. Mauhys, and H. DcMeyer Phys. Re\'. A38 (1988) 1168.
9. E. Merzhachcr Quantum Mechanics. (Wiley and Sons, New York.
19(1) Chaps. 8 and 10.
10. E. T. Whiuaker. and G. N. Watson A Course vI Mm/ern Alla/ysis
Cambridge: University Press 4th edition rcprinted (1952) P 265.
11. S. Wolfram Marhemmica. A S~'r'stemlar Doing Mathematic by
Compu/er. (Addison-Wcsley Publishing. CO.. Redwood City. California, (19X8).
Re\'. Me.\'. Fís. 43 (4) (1998) 413--419
1/--pages
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