Research Article Analytical Solution of the Schrödinger ...downloads.hindawi.com/journals/ahep/2014/952597.pdf · Analytical Solution of the Schrödinger Equation with Spatially

Research ArticleAnalytical Solution of the Schroumldinger Equation with SpatiallyVarying Effective Mass for Generalised Hylleraas Potential

Sanjib Meyur Smarajit Maji and S Debnath

Department of Mathematics Jadavpur University Kolkata 700032 India

Correspondence should be addressed to Sanjib Meyur sanjibmeyuryahoocoin

Received 30 May 2014 Revised 19 July 2014 Accepted 21 July 2014 Published 11 August 2014

Academic Editor Shi-Hai Dong

Copyright copy 2014 Sanjib Meyur et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We have obtained exact solution of the effectivemass Schrodinger equation for the generalisedHylleraas potentialThe exact boundstate energy eigenvalues and corresponding eigenfunctions are presented The bound state eigenfunctions are obtained in terms ofthe hypergeometric functions Results are also given for the special case of potential parameter

1 Introduction

The study of quantum mechanical systems within the frame-work of effective position dependent mass has been thesubject of much activity in recent years The Schrodingerequation with position-dependent (nonconstant) mass pro-vides an interesting and useful model for the description ofmany physical problems The most extensive use of such anequation is in the physics of semiconductor nanostructures[1 2] quantumdots [3] 3He clusters [4] quantum liquids [5]semiconductor heterostructures [6 7] and so forth

The solutions of nonrelativistic wave equations withconstant mass have been extended to the position dependentmass in recent studies [8ndash11] A general formalism for energyspectra andwave functionswas found in nonrelativistic prob-lems by using point canonical transformation [8] Comparedto the constant mass wave equation the position-dependentmass Schrodinger equation is more complex It is difficultto obtain its analytical solution as usual Several authorshave studied the effects of the position-dependent masson the solutions of the Schrodinger equation A position-dependent effective mass119898(119909) = 119898

1sdot 119898(119909) associated with

a quantummechanical particle constitutes a useful model forthe study of various potentials such as Morse potential [12ndash18] hard-core potential [18] Scarf potential [19ndash21] Poschl-Teller potential [22 23] spherically ring-shaped potential[24] Hulthen potential [25] Kratzer potential [26] and

Coulomb-like potential [27 28] Different techniques havebeen developed to obtain its exact solutions such as factor-ization methods [29] Nikiforov-Uvarov (NU) methods [30]and supersymmetric quantum mechanics [31] The position-dependent effective mass might have impact on high-energyphysics [31]

The objective of this paper is to investigate the position-dependent effective mass Schrodinger equation for the gen-eralised Hylleraas potential [32 33] by using the Nikiforov-Uvarov (NU) method (Figure 1) [30] Hylleraas potential isused to describe the interaction between two atoms in adiatomic molecule We have also investigated the solutionsof Hulthen potential and Woods-Saxon potential

The plan of the present paper is as follows In Section 2the Nikiforov-Uvarov method is summarized Section 3 isdevoted to the solution of the position-dependent effectivemass Schrodinger equation In Sections 4 and 5 the Hulthenpotential and Woods-Saxon potential are discussed respec-tively The paper is ended with a summary

2 Nikiforov-Uvarov Method

The NU method is a useful technique to solve the second-order linear differential equations with special orthogonalfunctions [34] In this method after employing an appropri-ate coordinate transformation 119904 = 119904(119909) the nonrelativistic

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2014 Article ID 952597 7 pageshttpdxdoiorg1011552014952597

2 Advances in High Energy Physics

minus6 minus4 minus2minus1000

minus800

minus600

minus400

minus200

x

Hyl

lera

as p

oten

tial

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4

0

Figure 1 Hylleraas potential for 1198811= 50 119881

2= 100 119886 = 10 119887 = 05

and 119889 = 20

Schrodinger equation 11988921205951198891199092+(119864minus119881(119909))120595 = 0 (ℏ = 2119898 =

1) can be written in the following form

12059510158401015840(119904) +

120591 (119904)

120590 (119904)

1205951015840(119904) +

(119904)

1205902(119904)

120595 (119904) = 0 (1)

where the prime denotes the differentiation with respect to 119904120590(119904) and (119904) are polynomials at most of second degree and120591(119904) is a polynomial at most of first degree In order to obtaina particular solution to (1) we set the followingwave functionas a multiple of two independent parts

120595 (119904) = 120601 (119904) 119910119899(119904) (2)

Equation (2) transformed (1) to a hypergeometric-type equa-tion

120590 (119904) 11991010158401015840

119899(119904) + 120591 (119904) 119910

1015840

119899(119904) + 120582119910

119899(119904) = 0 (3)

where first part of (2) 120601(119904) has a logarithmic derivative

1206011015840(119904)

120601 (119904)

=

120587 (119904)

120590 (119904)

(4)

and second part of (2) 119910119899(119904) is the hypergeometric-type

function whose polynomial solution satisfies the Rodriguesrelation

119910119899(119904) =

119862119899

120588 (119904)

119889119899

119889119904119899[120590119899(119904) 120588 (119904)] (5)

where 119862119899is normalization constant and the weight function

120588(119904) satisfies the relation as119889

119889119904

[120590 (119904) 120588 (119904)] = 120591 (119904) 120588 (119904) (6)

The function 120587(119904) and the eigenvalue 120582 required in thismethod are defined as

120587 (119904) = (

1205901015840minus 120591

2

) plusmn radic(

1205901015840minus 120591

2

)

2

minus + 119896120590 (7)

119896 = 120582 minus 1205871015840(119904) (8)

Hence the determination of 119896 is the essential point in thecalculation of 120587(119904) for which the discriminant of the squareroot in (7) is set to zero Also the eigenvalue equation definedin (8) takes the following new form

120582 = 120582119899= minus119899120591

1015840(119904) minus

119899 (119899 minus 1)

2

12059010158401015840(119904) 119899 = 0 1 2 (9)

120591 (119904) = 120591 (119904) + 2120587 (119904) 1205911015840(119904) lt 0 (10)

Since 120588(119904) gt 0 and 120590(119904) gt 0 the derivative of 120591(119904) should benegative [30] which helps to generate the essential conditionfor any choice of proper bound state solutions In additionthe energy eigenvalues are obtained from (8) and (9)

3 Position-Dependent Effective MassSchroumldinger Equation

In general working on position-dependent effective massHamiltonians is inspired by the von Roos Hamiltonian [35]proposal with ℏ = 2119898

0= 1

[minus

1

2

(119898120572(119909) 120597119909

119898120573(119909) 120597119909

119898120574(119909)

+ 119898120574(119909) 120597119909119898120573(119909) 120597119909119898120572(119909)) + 119881 (119909) ] 120593 (119909)

= 119864120593 (119909)

(11)

where ℏ = 21198980= 1 and 119898(119909) is the dimensionless form of

the function119898(119909) = 1198981sdot119898(119909)The ambiguity parameters are

constrained by the relation 120572 + 120573 + 120574 = minus1 and we have thefollowing time-independent Schrodinger equation from (11)

119867120593 (119909) equiv [minus120597119909(

1

119898 (119909)

) 120597119909+ 119881eff (119909) minus 119864]120593 (119909) = 0 (12)

where the effective potential is

119881eff (119909) = 119881 (119909) +

1

2

(120573 + 1)

11989810158401015840(119909)

1198982(119909)

minus [120572 (120572 + 120573 + 1) + (120573 + 1)]

11989810158402(119909)

1198983(119909)

(13)

where primes denote derivatives Thus Schrodinger equationtakes the form

(minus

1

119898 (119909)

1198892

1198891199092+

1198981015840(119909)

1198982(119909)

119889

119889119909

+ 119881eff (119909) minus 119864)120593 (119909) = 0 (14)

Advances in High Energy Physics 3

minus10 minus5

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 5 1000

02

04

06

08

10

Mas

s fun

ctio

nm(x)

Figure 2 Plot for mass function119898(119909) for 119887 = 05 and1198981= 1

Using the transformation [36] 120593(119909) = 119898](119909)120595(119909) in (14) we

have

minus

1198892

1198891199092minus (2] minus 1)

1198981015840(119909)

119898 (119909)

119889

119889119909

minus (] (] minus 2) + 120572 (120572 + 120573 + 1) + 120573 + 1)

11989810158402

1198982

+(

1

2

(120573 + 1) minus ])11989810158401015840(119909)

119898 (119909)

+ 119898 (119909) (119881 (119909) minus 119864)120595 (119909)

= 0

(15)

where 119881(119909) is the Hylleraas potential [29 30] given by

119881 (119909) = 1198811

119886 + 119890120582119909

119887 + 119890120582119909

minus 1198812

119889 + 119890120582119909

119887 + 119890120582119909

(16)

where 119886( = 119887) 119887 and 119889( = 119887) are the Hylleraas parameters 1198811

1198812are the potential depths and minusinfin lt 119909 lt infinHere we consider the following mass distribution

119898(119909) =

1198981

1 + 119887119890minus120582119909

(17)

The most extensive use of such kind of mass is in the physicsof semiconductor quantum well structures [11] The motionof electrons in them can often be described by the envelopefunction effective-mass Schrodinger equation where 119898

1is a

constant mass (Figure 2)Obviously it has the exponential form The above mass

function is convergent 119898(119909) rarr 1198981(finite) when 119909 rarr infin

In order to reduce the above equation (15) into Nikiforov-Uvarov equation we make the transformation 119904 = 1(1 +

119887119890minus120582119909

) (0 le 119904 le 1)

119860 = ] (] minus 2) + 120572 (120572 + 120573 + 1) + 120573 + 1

119861 = (

1

2

(120573 + 1) minus ])

120577 =

1

2

minus ]

119875 minus 1205772= minus 119860 + 2119861 minus

1198981(119886 minus 119887)119881

1

1198871205822

+

1198981(119889 minus 119887)119881

2

1198871205822


1198981(1198861198811minus 1198891198812minus 119887119864)

1198871205822

119877 minus 1205772= minus 119860 + 119861

119875 minus 119876 + 119877 =

1198981(1198811minus 1198812minus 119864)

1205822

= 1205762

(18)

also

1198981015840(119909)

119898 (119909)

= 120582 (1 minus 119904)

11989810158401015840(119909)

119898 (119909)

= 1205822(1 minus 119904) (1 minus 2119904)

(19)

Using (15)ndash(19) we have

1198892120595

1198891199042+

2] minus (2] + 1) 119904

119904 (1 minus 119904)

119889120595

119889119904

+

1

1199042(1 minus 119904)

2[(1205772minus 119875) 119904

2+ (119876 minus 2120577

2) 119904 + (120577

2minus 119877)]120595 = 0

(20)

Comparing (20) with (1) we have

120590 (119904) = 119904 (1 minus 119904)

(119904) = (1205772minus 119875) 119904

2+ (119876 minus 2120577

2) 119904 + (120577

2minus 119877)

120591 (119904) = 2] minus (2] + 1) 119904

(21)

Substituting these polynomials into (7) we have

120587 (119904) = (

1

2

minus ]) (1 minus 119904)

plusmn

(radic119877 minus 120576) 119904 minus radic119877 119896 = 119876 minus 2119877 + 2120576radic119877

(radic119877 + 120576) 119904 minus radic119877 119896 = 119876 minus 2119877 minus 2120576radic119877

(22)

For physical solutions it is necessary to choose

120587 (119904) = 120577 (1 minus 119904) minus (radic119877 + 120576) 119904 + radic119877

if 119896 = 119876 minus 2119877 minus 2120576radic119877

(23)


The origin of the Nikiforov-Uvarov method is negative signof derivative of 120591

1015840(119904) because the condition 120591

1015840(119904) lt 0

helps to generate energy eigenvalues and correspondingeigenfunctions Therefore 120591(119904) becomes

120591 (119904) = 1 + 2radic119877 minus 2 (2 + 2radic119877 + 2120576) 119904 (24)

Therefore from (8) and (9) we have

120582 = 119876 minus 2119877 minus 2120576radic119877 minus 120577 minus (radic119877 + 120576)

120582 = 120582119899= 2119899 (radic119877 + 120576) + 119899 (119899 + 1)

(25)

Comparing (25) we have

radic119877 + 120576 = minus(119899 +

1

2

) + radic119875 minus ] +1

4

radic119877 minus 120576 =

119876 minus 119875

minus (119899 + 12) + radic119875 minus ] + 14

(26)

Using (26) and (18) we have

1205762=

1

4

[2119899 + 1 + 2radic119877 minus radic119875 minus ] +1

4

]

2

(27)

Hence the energy becomes

119864119899

= minus

1205822

41198981

times[

[

2119899 + 1

minus 2radic

1198981(1198811minus 1198812)

1205822

minus

1198981(1198811119886 minus 1198812119889)

1198871205822

minus 120572 (120572 + 120573 + 1)

+ 2radicminus](] minus 1) minus 120572(120572 + 120573 + 1) minus

(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

+ (1198811minus 1198812) 0 le 119899 lt infin

(28)

The last term of the square bracket must be positive for Ben-Daniel and Dukersquos model [37] (120572 = ] = 0 120573 = minus1) (Figure 3)

From (6) (21) and (24) we obtain the weight function

120588 (119904) = 1199042radic119877

(1 minus 119904)2120576 (29)

and from (4) (21) and (23) we have

120601 (119904) = 119904radic119877+120577

(1 minus 119904)120576 (30)

minus3500

minus4000

minus4500

minus5000

minus5500

n

Hyl

lera

as en

ergy

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 1 2 3 4

Figure 3 Hylleraas energy 119864119899for 120572 = 0 120573 = minus1 119886 = 10 119887 = 05

119889 = 201198981= 1 119881

1= 50 and 119881

2= 100

Now we use the properties of Jacobi polynomial [35]

119875(120577120585)

119899(119909) =

(minus1)119899(1 minus 119909)

minus120577(1 + 119909)

minus120585

2119899119899

times

119889119899

119889119909119899[(1 minus 119909)

119899+120577(1 + 119909)

119899+120585]

119875(21205772120585)

119899(1 minus 2119904) =

(minus2)119899(119904)minus2120577

(1 minus 119904)minus2120585

2119899119899

times

119889119899

119889119909119899[119904119899+2120577

(1 minus 119904)119899+2120585

]

(31)

where 119875(119886119887)

119899(119909) (119886 gt minus1 119887 gt minus1) is the Jacobi polynomial

The wave functions (Figure 4) are obtained from (2) (5) and((29)ndash(31))

120595119899(119904) = 119873

119899119904radic119877+120577

(1 minus 119904)120576119875(2radic1198772120576)

119899(1 minus 2119904) (32)

where 119873119899is normalization constant to be determined from

the normalization condition

int

infin

minusinfin

1003816100381610038161003816120595119899 (

119909)1003816100381610038161003816

2119889119909 = 1 = int

1

0

1003816100381610038161003816120595119899 (

119904)1003816100381610038161003816

2119889119904 (33)

For acceptable solution it is required that |radic119877 + 120577| ge 120576 whenradic119877 + 120577 lt 0 120576 gt 0 andradic119877 + 120577 le |120576| whenradic119877 + 120577 gt 0 120576 lt 0

4 Hultheacuten Potential

We set the conditions 1198812

= 1198811 119886 = 1 + 119889 and 119887 = minus119902

the potential in (16) reduces to Hulthen potential (Figure 5)[38ndash41]

119881 (119909) = minus1198811

119890minus120582119909

1 minus 119902119890minus120582119909

(34)


n = 1

n = 4 n = 3n = 2

minus03

minus02

minus01

s

00 02 04 06 08 10

00

01

02

03

04

120595(s)

Figure 4 Wave functions 120595119899(119904) for 120572 = 0 120573 = minus1 ] = 12 and

120576 = 2

minus50

minus40

minus30

minus20

minus10

x

Hul

th en

pote

ntia

l

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4 6 8

0

Figure 5 Hulthen potential for 1198811= 1000 and 119902 = 05

Then the energy becomes

119864119899

= minus

1205822

41198981

times[

[

2119899 + 1 minus 2radic

11989811198811

1199021205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

(35)

with 0 le 119899 lt infin It is exactly the same result in the literature[38] for119898

1= 1

minus50

minus40

minus30

minus20

minus10

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4

0

minus4 minus2

Woo

ds-S

axon

pot

entia

l

Figure 6 Woods-Saxon potential for 1198810= 1000 and 119887 = 05

5 Woods-Saxon Potential

For the conditions 1198811= minus1198810 119886 = 0 and 119881

2= 0 the potential

given in (16) becomesWoods-Saxon potential (Figure 6) [40ndash45]

119881 (119909) = minus1198810

1

1 + 119887119890minus120582119909

(36)


119864119899

= minus

1205822

41198981

times[

[

2119899 + 1 minus 2radic1198981(1 minus 119887)119881

0

1198871205822

minus 120572 (120572 + 120573 + 1)

+ 2radicminus] (] minus 1) minus 120572 (120572 + 120573 + 1)minus

(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

minus 1198810

(37)

where 0 le 119899 lt infin

6 Conclusion

We have applied the NUmethod derived for the exponential-type potentials to obtain the bound state solutions of the effec-tive Schrodinger equation with position-dependent mass forthe Hylleraas potential Furthermore a suitable choice ofa position mass function of the exponential-like form hasalso been devised Also we have shown that our results areconsistent with ones obtained before


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors wish to deliver their sincere gratitude to thereferee for constructive suggestions and technical commentson the paper

References

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[2] P Harrison Quantum Wells Wires and Dots John Wiley ampSons New York NY USA 2000

[3] L Serra and E Lipparini ldquoSpin response of unpolarizedquantum dotsrdquo Europhysics Letters vol 40 no 6 pp 667ndash6721997

[4] M Barranco M Pi S M Gatica E S Hernandez and JNavarro ldquoStructure and energetics of mixed 4He-3He dropsrdquoPhysical Review B vol 56 no 14 pp 8997ndash9003 1997

[5] F Arias De Saavedra J Boronat A Polls and A FabrocinildquoEffective mass of one He4 atom in liquid He3rdquo Physical ReviewB vol 50 no 6 pp 4248ndash4251 1994

[6] I Galbraith and G Duggan ldquoEnvelope-functionmatching con-ditions forGaAs(AlGa)As heterojunctionsrdquoPhysical ReviewBvol 38 no 14 pp 10057ndash10059 1988

[7] C Weisbuch and B Vinter Quantum Semiconductor Het-erostructure Academic Press New York NY USA 1993

[8] L Dekar L Chetouani and T F Hammann ldquoWave function forsmooth potential and mass steprdquo Physical Review A vol 59 no1 pp 107ndash112 1999

[9] A D Alhaidari ldquoSolution of the Dirac equation with position-dependent mass in the Coulomb fieldrdquo Physics Letters A vol322 no 1-2 pp 72ndash77 2004

[10] B Roy and P Roy ldquoA Lie algebraic approach to effective massSchrodinger equationsrdquo Journal of Physics AMathematical andGeneral vol 35 no 17 pp 3961ndash3969 2002

[11] B Gonul D Tutcu and O Ozer ldquoSupersymmetric approachto exactly solvable systems with position-dependent effectivemassesrdquoModern Physics Letters A vol 17 no 31 pp 2057ndash20662002

[12] S M Ikhdair ldquoRotation and vibration of diatomic moleculein the spatially-dependent mass Schrodinger equation withgeneralized q-deformedMorse potentialrdquoChemical Physics vol361 no 1-2 pp 9ndash17 2009

[13] C Tezcan R Sever and O Yesiltas ldquoA new approach to theexact solutions of the effective mass Schrodinger equationrdquoInternational Journal of Theoretical Physics vol 47 no 6 pp1713ndash1721 2008

[14] A Arda and R Sever ldquoBound state solutions of the Schrodingerfor generalizedMorse potential with position-dependentmassrdquoCommunications in Theoretical Physics vol 56 no 1 pp 51ndash542011

[15] G Levai andO Ozer ldquoAn exactly solvable Schrodinger equationwith finite positive position-dependent effective massrdquo Journalof Mathematical Physics vol 51 no 9 Article ID 092103 2010

[16] B Bagchi C Quesne and R Roychoudhury ldquoPseudo-Hermiticity and some consequences of a generalized quantumconditionrdquo Journal of Physics A vol 38 no 40 pp L647ndashL6522005

[17] J Yu S-H Dong and G-H Sun ldquoSeries solutions of theSchrodinger equation with position-dependent mass for theMorse potentialrdquo Physics Letters A vol 322 no 5-6 pp 290ndash297 2004

[18] S-H Dong and M Lozada-Cassou ldquoExact solutions of theSchrodinger equation with the position-dependent mass fora hard-core potentialrdquo Physics Letters A General Atomic andSolid State Physics vol 337 no 4mdash6 pp 313ndash320 2005

[19] O Mustafa and S H Mazharimousavi ldquoFirst-order inter-twining operators with position dependent mass and 120578-weak-pseudo-Hermiticity generatorsrdquo International Journal of Theo-retical Physics vol 47 no 2 pp 446ndash454 2008

[20] H Motavali ldquoBound state solutions of the Dirac equationfor the Scarf-type potential using Nikiforov-Uvarov methodrdquoModern Physics Letters A vol 24 no 15 pp 1227ndash1236 2009

[21] A P Zhang P Shi Y W Ling and Z W Hua ldquoSolutions ofone-dimensional effective mass Schrodinger equation for PT-symmetric Scarf potentialrdquo Acta Physica Polonica A vol 120no 6 pp 987ndash991 2011

[22] M G Miranda G Sun and S-H Dong ldquoThe solution of thesecond Poschl-Teller-like potential by Nikiforov-Uvarovmethodrdquo International Journal of Modern Physics E vol 129 no1 pp 123ndash129 2010

[23] H Hassanabadi E Maghsoodi S Zarrinkamar and H Rahi-mov ldquoDirac equation for generalized Poschl-Teller scalarand vector potentials and a Coulomb tensor interaction byNikiforov-Uvarov methodrdquo Journal of Mathematical Physicsvol 53 no 2 Article ID 022104 2012

[24] M C Zhang G H Sun and S-H Dong ldquoExactly completesolutions of the Schrodinger equation with a spherically har-monic oscillatory ring-shaped potentialrdquo Physics Letters A vol374 no 5 pp 704ndash708 2010

[25] F Yasuk and M K Bahar ldquoApproximate solutions of the Diracequation with position-dependent mass for the Hulthen poten-tial by the asymptotic iteration methodrdquo Physica Scripta vol85 no 4 Article ID 045004 2012

[26] S M Ikhdair and R Sever ldquoExactly solvable effective mass119863-dimensional Schrodinger equation for pseudoharmonic andmodified Kratzer problemsrdquo International Journal of ModernPhysics C vol 20 no 3 pp 361ndash372 2009

[27] R Koc andMKoca ldquoA systematic study on the exact solution ofthe position dependent mass Schrodinger equationrdquo Journal ofPhysics A Mathematical and General vol 36 no 29 pp 8105ndash8112 2003

[28] S Ortakaya ldquoPseudospin symmetry in position-dependentmass dirac-coulomb problem by using laplace transform andconvolution integralrdquo Few-Body Systems vol 54 no 11 pp2073ndash2080 2013

[29] S-H Dong Factorization Method in Quantum MechanicsSpringer New York NY USA 2007

[30] A F Nikiforov andV B Uvarov Special Functions ofMathemat-ical Physics Birkhaauser Basel Switzerland 1988

[31] B Midya B Roy and T Tanaka ldquoEffect of position-dependentmass on dynamical breaking of type B and type 119883

2N-fold

supersymmetryrdquo Journal of Physics A Mathematical and The-oretical vol 45 no 20 Article ID 205303 22 pages 2012


[32] E A Hylleraas ldquoEnergy formula and potential distribution ofdiatomic moleculesrdquo Journal of Chemical Physics vol 3 article595 no 9 1935

[33] A N Ikot ldquoSolution of Dirac equation with generalized hyller-aas potentialrdquo Communications in Theoretical Physics vol 59no 3 pp 268ndash272 2013

[34] M Abramowitz and I StegunHandbook ofMathematical Func-tion with Formulas Graphs and Mathematical Tables DoverNew York NY USA 1964

[35] O von Roos ldquoPosition-dependent effective masses in semicon-ductor theoryrdquo Physical Review B vol 27 no 12 pp 7547ndash75521983

[36] R Sever C Tezcan O Yesiltas and M Bucurgat ldquoExactsolution of effective mass Schrodinger equation for the Hulthenpotentialrdquo International Journal of Theoretical Physics vol 47no 9 pp 2243ndash2248 2008

[37] D J BenDaniel and C B Duke ldquoSpace-charge effects on ele-ctron tunnelingrdquo Physical Review vol 152 no 2 pp 683ndash6921966

[38] LHulthen ldquoUber die eigenlosungender schrodinger-gleichungder deuteronsrdquo Arkiv for Matematik A vol 28 no 5 pp 1ndash121942

[39] S Meyur and S Debnath ldquoReal spectrum of non-HermitianHamiltonians for hulthen potentialrdquo Modern Physics Letters Avol 23 no 25 pp 2077ndash2084 2008

[40] O Aydogdu A Arda and R Sever ldquoScattering of a spinlessparticle by an asymmetricHulthen potential within the effectivemass formalismrdquo Journal ofMathematical Physics vol 53 no 10Article ID 102111 2012

[41] D S Saxon and R DWoods ldquoDiffuse surface optical model fornucleon-nuclei scatteringrdquo Physical Review vol 95 no 2 pp577ndash578 1954

[42] A Arda and R Sever ldquoBound states of the Klein-Gordonequation for Woods-Saxon potential with position dependentmassrdquo International Journal of Modern Physics C vol 19 no 5pp 763ndash773 2008

[43] C Berkdemir A Berkdemir andR Sever ldquoPolynomial Solutionof the Schrodinger equation for the generalized Woods-Saxonpotentialrdquo Physical Review C vol 74 no 3 Article ID 0399022006

[44] O Aydogdu A Arda and R Sever ldquoEffective-mass Diracequation for Woods-Saxon potential scattering bound statesand resonancesrdquo Journal ofMathematical Physics vol 53 ArticleID 042106 2012

[45] O Aydogdu and R Sever ldquoPseudospin and spin symmetry inthe Dirac equation with Woods-Saxon potential and tensorpotentialrdquo European Physical Journal A vol 43 no 1 pp 73ndash812009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



AstronomyAdvances in

International Journal of


Superconductivity


Statistical MechanicsInternational Journal of


GravityJournal of


AstrophysicsJournal of


Physics Research International


Solid State PhysicsJournal of

Computational Methods in Physics

Journal of



Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014


PhotonicsJournal of


Journal of

Biophysics


ThermodynamicsJournal of


minus6 minus4 minus2minus1000

minus800

minus600

minus400

minus200

x

Hyl

lera

as p

oten

tial

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4

0

Figure 1 Hylleraas potential for 1198811= 50 119881

2= 100 119886 = 10 119887 = 05

and 119889 = 20

Schrodinger equation 11988921205951198891199092+(119864minus119881(119909))120595 = 0 (ℏ = 2119898 =

1) can be written in the following form

12059510158401015840(119904) +

120591 (119904)

120590 (119904)

1205951015840(119904) +

(119904)

1205902(119904)

120595 (119904) = 0 (1)

where the prime denotes the differentiation with respect to 119904120590(119904) and (119904) are polynomials at most of second degree and120591(119904) is a polynomial at most of first degree In order to obtaina particular solution to (1) we set the followingwave functionas a multiple of two independent parts

120595 (119904) = 120601 (119904) 119910119899(119904) (2)

Equation (2) transformed (1) to a hypergeometric-type equa-tion

120590 (119904) 11991010158401015840

119899(119904) + 120591 (119904) 119910

1015840

119899(119904) + 120582119910

119899(119904) = 0 (3)

where first part of (2) 120601(119904) has a logarithmic derivative

1206011015840(119904)

120601 (119904)

=

120587 (119904)

120590 (119904)

(4)

and second part of (2) 119910119899(119904) is the hypergeometric-type

function whose polynomial solution satisfies the Rodriguesrelation

119910119899(119904) =

119862119899

120588 (119904)

119889119899

119889119904119899[120590119899(119904) 120588 (119904)] (5)

where 119862119899is normalization constant and the weight function

120588(119904) satisfies the relation as119889

119889119904

[120590 (119904) 120588 (119904)] = 120591 (119904) 120588 (119904) (6)

The function 120587(119904) and the eigenvalue 120582 required in thismethod are defined as

120587 (119904) = (

1205901015840minus 120591

2

) plusmn radic(

1205901015840minus 120591

2

)

2

minus + 119896120590 (7)

119896 = 120582 minus 1205871015840(119904) (8)

Hence the determination of 119896 is the essential point in thecalculation of 120587(119904) for which the discriminant of the squareroot in (7) is set to zero Also the eigenvalue equation definedin (8) takes the following new form

120582 = 120582119899= minus119899120591

1015840(119904) minus

119899 (119899 minus 1)

2

12059010158401015840(119904) 119899 = 0 1 2 (9)

120591 (119904) = 120591 (119904) + 2120587 (119904) 1205911015840(119904) lt 0 (10)

Since 120588(119904) gt 0 and 120590(119904) gt 0 the derivative of 120591(119904) should benegative [30] which helps to generate the essential conditionfor any choice of proper bound state solutions In additionthe energy eigenvalues are obtained from (8) and (9)

3 Position-Dependent Effective MassSchroumldinger Equation

In general working on position-dependent effective massHamiltonians is inspired by the von Roos Hamiltonian [35]proposal with ℏ = 2119898

0= 1

[minus

1

2

(119898120572(119909) 120597119909

119898120573(119909) 120597119909

119898120574(119909)

+ 119898120574(119909) 120597119909119898120573(119909) 120597119909119898120572(119909)) + 119881 (119909) ] 120593 (119909)

= 119864120593 (119909)

(11)

where ℏ = 21198980= 1 and 119898(119909) is the dimensionless form of

the function119898(119909) = 1198981sdot119898(119909)The ambiguity parameters are

constrained by the relation 120572 + 120573 + 120574 = minus1 and we have thefollowing time-independent Schrodinger equation from (11)

119867120593 (119909) equiv [minus120597119909(

1

119898 (119909)

) 120597119909+ 119881eff (119909) minus 119864]120593 (119909) = 0 (12)

where the effective potential is

119881eff (119909) = 119881 (119909) +

1

2

(120573 + 1)

11989810158401015840(119909)

1198982(119909)

minus [120572 (120572 + 120573 + 1) + (120573 + 1)]

11989810158402(119909)

1198983(119909)

(13)

where primes denote derivatives Thus Schrodinger equationtakes the form

(minus

1

119898 (119909)

1198892

1198891199092+

1198981015840(119909)

1198982(119909)

119889

119889119909

+ 119881eff (119909) minus 119864)120593 (119909) = 0 (14)


minus10 minus5

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 5 1000

02

04

06

08

10

Mas

s fun

ctio

nm(x)



have

minus

1198892

1198891199092minus (2] minus 1)

1198981015840(119909)

119898 (119909)

119889

119889119909

minus (] (] minus 2) + 120572 (120572 + 120573 + 1) + 120573 + 1)

11989810158402

1198982

+(

1

2

(120573 + 1) minus ])11989810158401015840(119909)

119898 (119909)

+ 119898 (119909) (119881 (119909) minus 119864)120595 (119909)

= 0

(15)


119881 (119909) = 1198811

119886 + 119890120582119909

119887 + 119890120582119909

minus 1198812

119889 + 119890120582119909

119887 + 119890120582119909

(16)



119898(119909) =

1198981

1 + 119887119890minus120582119909

(17)


1is a




119887119890minus120582119909

) (0 le 119904 le 1)

119860 = ] (] minus 2) + 120572 (120572 + 120573 + 1) + 120573 + 1

119861 = (

1

2

(120573 + 1) minus ])

120577 =

1

2

minus ]


1198981(119886 minus 119887)119881

1

1198871205822

+

1198981(119889 minus 119887)119881

2

1198871205822


1198981(1198861198811minus 1198891198812minus 119887119864)

1198871205822

119877 minus 1205772= minus 119860 + 119861

119875 minus 119876 + 119877 =

1198981(1198811minus 1198812minus 119864)

1205822

= 1205762

(18)

also

1198981015840(119909)

119898 (119909)

= 120582 (1 minus 119904)

11989810158401015840(119909)

119898 (119909)

= 1205822(1 minus 119904) (1 minus 2119904)

(19)


1198892120595

1198891199042+

2] minus (2] + 1) 119904

119904 (1 minus 119904)

119889120595

119889119904

+

1

1199042(1 minus 119904)

2[(1205772minus 119875) 119904

2+ (119876 minus 2120577

2) 119904 + (120577

2minus 119877)]120595 = 0

(20)


120590 (119904) = 119904 (1 minus 119904)

(119904) = (1205772minus 119875) 119904

2+ (119876 minus 2120577

2) 119904 + (120577

2minus 119877)

120591 (119904) = 2] minus (2] + 1) 119904

(21)


120587 (119904) = (

1

2


plusmn



(22)




(23)




1015840(119904) lt 0





120582 = 120582119899= 2119899 (radic119877 + 120576) + 119899 (119899 + 1)

(25)


radic119877 + 120576 = minus(119899 +

1

2

) + radic119875 minus ] +1

4

radic119877 minus 120576 =

119876 minus 119875


(26)


1205762=

1

4


4

]

2

(27)


119864119899

= minus

1205822

41198981

times[

[

2119899 + 1

minus 2radic

1198981(1198811minus 1198812)

1205822

minus

1198981(1198811119886 minus 1198812119889)

1198871205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

+ (1198811minus 1198812) 0 le 119899 lt infin

(28)



120588 (119904) = 1199042radic119877

(1 minus 119904)2120576 (29)


120601 (119904) = 119904radic119877+120577

(1 minus 119904)120576 (30)

minus3500

minus4000

minus4500

minus5000

minus5500

n

Hyl

lera

as en

ergy

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 1 2 3 4


119889 = 201198981= 1 119881

1= 50 and 119881

2= 100


119875(120577120585)

119899(119909) =

(minus1)119899(1 minus 119909)

minus120577(1 + 119909)

minus120585

2119899119899

times

119889119899

119889119909119899[(1 minus 119909)

119899+120577(1 + 119909)

119899+120585]

119875(21205772120585)

119899(1 minus 2119904) =

(minus2)119899(119904)minus2120577

(1 minus 119904)minus2120585

2119899119899

times

119889119899

119889119909119899[119904119899+2120577

(1 minus 119904)119899+2120585

]

(31)

where 119875(119886119887)



120595119899(119904) = 119873

119899119904radic119877+120577

(1 minus 119904)120576119875(2radic1198772120576)

119899(1 minus 2119904) (32)



int

infin

minusinfin

1003816100381610038161003816120595119899 (

119909)1003816100381610038161003816

2119889119909 = 1 = int

1

0

1003816100381610038161003816120595119899 (

119904)1003816100381610038161003816

2119889119904 (33)




= 1198811 119886 = 1 + 119889 and 119887 = minus119902


119881 (119909) = minus1198811

119890minus120582119909

1 minus 119902119890minus120582119909

(34)


n = 1

n = 4 n = 3n = 2

minus03

minus02

minus01

s

00 02 04 06 08 10

00

01

02

03

04

120595(s)


120576 = 2

minus50

minus40

minus30

minus20

minus10

x

Hul

th en

pote

ntia

l

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4 6 8

0



119864119899

= minus

1205822

41198981

times[

[


11989811198811

1199021205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

(35)


1= 1

minus50

minus40

minus30

minus20

minus10

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4

0

minus4 minus2

Woo

ds-S

axon

pot

entia

l




2= 0 the potential


119881 (119909) = minus1198810

1

1 + 119887119890minus120582119909

(36)


119864119899

= minus

1205822

41198981

times[

[


0

1198871205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

minus 1198810

(37)


6 Conclusion





Acknowledgment


References
































2N-fold






















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




minus10 minus5

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 5 1000

02

04

06

08

10

Mas

s fun

ctio

nm(x)



have

minus

1198892

1198891199092minus (2] minus 1)

1198981015840(119909)

119898 (119909)

119889

119889119909

minus (] (] minus 2) + 120572 (120572 + 120573 + 1) + 120573 + 1)

11989810158402

1198982

+(

1

2

(120573 + 1) minus ])11989810158401015840(119909)

119898 (119909)

+ 119898 (119909) (119881 (119909) minus 119864)120595 (119909)

= 0

(15)


119881 (119909) = 1198811

119886 + 119890120582119909

119887 + 119890120582119909

minus 1198812

119889 + 119890120582119909

119887 + 119890120582119909

(16)



119898(119909) =

1198981

1 + 119887119890minus120582119909

(17)


1is a




119887119890minus120582119909

) (0 le 119904 le 1)

119860 = ] (] minus 2) + 120572 (120572 + 120573 + 1) + 120573 + 1

119861 = (

1

2

(120573 + 1) minus ])

120577 =

1

2

minus ]


1198981(119886 minus 119887)119881

1

1198871205822

+

1198981(119889 minus 119887)119881

2

1198871205822


1198981(1198861198811minus 1198891198812minus 119887119864)

1198871205822

119877 minus 1205772= minus 119860 + 119861

119875 minus 119876 + 119877 =

1198981(1198811minus 1198812minus 119864)

1205822

= 1205762

(18)

also

1198981015840(119909)

119898 (119909)

= 120582 (1 minus 119904)

11989810158401015840(119909)

119898 (119909)

= 1205822(1 minus 119904) (1 minus 2119904)

(19)


1198892120595

1198891199042+

2] minus (2] + 1) 119904

119904 (1 minus 119904)

119889120595

119889119904

+

1

1199042(1 minus 119904)

2[(1205772minus 119875) 119904

2+ (119876 minus 2120577

2) 119904 + (120577

2minus 119877)]120595 = 0

(20)


120590 (119904) = 119904 (1 minus 119904)

(119904) = (1205772minus 119875) 119904

2+ (119876 minus 2120577

2) 119904 + (120577

2minus 119877)

120591 (119904) = 2] minus (2] + 1) 119904

(21)


120587 (119904) = (

1

2


plusmn



(22)




(23)




1015840(119904) lt 0





120582 = 120582119899= 2119899 (radic119877 + 120576) + 119899 (119899 + 1)

(25)


radic119877 + 120576 = minus(119899 +

1

2

) + radic119875 minus ] +1

4

radic119877 minus 120576 =

119876 minus 119875


(26)


1205762=

1

4


4

]

2

(27)


119864119899

= minus

1205822

41198981

times[

[

2119899 + 1

minus 2radic

1198981(1198811minus 1198812)

1205822

minus

1198981(1198811119886 minus 1198812119889)

1198871205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

+ (1198811minus 1198812) 0 le 119899 lt infin

(28)



120588 (119904) = 1199042radic119877

(1 minus 119904)2120576 (29)


120601 (119904) = 119904radic119877+120577

(1 minus 119904)120576 (30)

minus3500

minus4000

minus4500

minus5000

minus5500

n

Hyl

lera

as en

ergy

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 1 2 3 4


119889 = 201198981= 1 119881

1= 50 and 119881

2= 100


119875(120577120585)

119899(119909) =

(minus1)119899(1 minus 119909)

minus120577(1 + 119909)

minus120585

2119899119899

times

119889119899

119889119909119899[(1 minus 119909)

119899+120577(1 + 119909)

119899+120585]

119875(21205772120585)

119899(1 minus 2119904) =

(minus2)119899(119904)minus2120577

(1 minus 119904)minus2120585

2119899119899

times

119889119899

119889119909119899[119904119899+2120577

(1 minus 119904)119899+2120585

]

(31)

where 119875(119886119887)



120595119899(119904) = 119873

119899119904radic119877+120577

(1 minus 119904)120576119875(2radic1198772120576)

119899(1 minus 2119904) (32)



int

infin

minusinfin

1003816100381610038161003816120595119899 (

119909)1003816100381610038161003816

2119889119909 = 1 = int

1

0

1003816100381610038161003816120595119899 (

119904)1003816100381610038161003816

2119889119904 (33)




= 1198811 119886 = 1 + 119889 and 119887 = minus119902


119881 (119909) = minus1198811

119890minus120582119909

1 minus 119902119890minus120582119909

(34)


n = 1

n = 4 n = 3n = 2

minus03

minus02

minus01

s

00 02 04 06 08 10

00

01

02

03

04

120595(s)


120576 = 2

minus50

minus40

minus30

minus20

minus10

x

Hul

th en

pote

ntia

l

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4 6 8

0



119864119899

= minus

1205822

41198981

times[

[


11989811198811

1199021205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

(35)


1= 1

minus50

minus40

minus30

minus20

minus10

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4

0

minus4 minus2

Woo

ds-S

axon

pot

entia

l




2= 0 the potential


119881 (119909) = minus1198810

1

1 + 119887119890minus120582119909

(36)


119864119899

= minus

1205822

41198981

times[

[


0

1198871205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

minus 1198810

(37)


6 Conclusion





Acknowledgment


References
































2N-fold






















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






1015840(119904) lt 0





120582 = 120582119899= 2119899 (radic119877 + 120576) + 119899 (119899 + 1)

(25)


radic119877 + 120576 = minus(119899 +

1

2

) + radic119875 minus ] +1

4

radic119877 minus 120576 =

119876 minus 119875


(26)


1205762=

1

4


4

]

2

(27)


119864119899

= minus

1205822

41198981

times[

[

2119899 + 1

minus 2radic

1198981(1198811minus 1198812)

1205822

minus

1198981(1198811119886 minus 1198812119889)

1198871205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

+ (1198811minus 1198812) 0 le 119899 lt infin

(28)



120588 (119904) = 1199042radic119877

(1 minus 119904)2120576 (29)


120601 (119904) = 119904radic119877+120577

(1 minus 119904)120576 (30)

minus3500

minus4000

minus4500

minus5000

minus5500

n

Hyl

lera

as en

ergy

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 1 2 3 4


119889 = 201198981= 1 119881

1= 50 and 119881

2= 100


119875(120577120585)

119899(119909) =

(minus1)119899(1 minus 119909)

minus120577(1 + 119909)

minus120585

2119899119899

times

119889119899

119889119909119899[(1 minus 119909)

119899+120577(1 + 119909)

119899+120585]

119875(21205772120585)

119899(1 minus 2119904) =

(minus2)119899(119904)minus2120577

(1 minus 119904)minus2120585

2119899119899

times

119889119899

119889119909119899[119904119899+2120577

(1 minus 119904)119899+2120585

]

(31)

where 119875(119886119887)



120595119899(119904) = 119873

119899119904radic119877+120577

(1 minus 119904)120576119875(2radic1198772120576)

119899(1 minus 2119904) (32)



int

infin

minusinfin

1003816100381610038161003816120595119899 (

119909)1003816100381610038161003816

2119889119909 = 1 = int

1

0

1003816100381610038161003816120595119899 (

119904)1003816100381610038161003816

2119889119904 (33)




= 1198811 119886 = 1 + 119889 and 119887 = minus119902


119881 (119909) = minus1198811

119890minus120582119909

1 minus 119902119890minus120582119909

(34)


n = 1

n = 4 n = 3n = 2

minus03

minus02

minus01

s

00 02 04 06 08 10

00

01

02

03

04

120595(s)


120576 = 2

minus50

minus40

minus30

minus20

minus10

x

Hul

th en

pote

ntia

l

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4 6 8

0



119864119899

= minus

1205822

41198981

times[

[


11989811198811

1199021205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

(35)


1= 1

minus50

minus40

minus30

minus20

minus10

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4

0

minus4 minus2

Woo

ds-S

axon

pot

entia

l




2= 0 the potential


119881 (119909) = minus1198810

1

1 + 119887119890minus120582119909

(36)


119864119899

= minus

1205822

41198981

times[

[


0

1198871205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

minus 1198810

(37)


6 Conclusion





Acknowledgment


References
































2N-fold






















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




n = 1

n = 4 n = 3n = 2

minus03

minus02

minus01

s

00 02 04 06 08 10

00

01

02

03

04

120595(s)


120576 = 2

minus50

minus40

minus30

minus20

minus10

x

Hul

th en

pote

ntia

l

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4 6 8

0



119864119899

= minus

1205822

41198981

times[

[


11989811198811

1199021205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

(35)


1= 1

minus50

minus40

minus30

minus20

minus10

x

120582 = 1

120582 = 2

120582 = 3

120582 = 4

0 2 4

0

minus4 minus2

Woo

ds-S

axon

pot

entia

l




2= 0 the potential


119881 (119909) = minus1198810

1

1 + 119887119890minus120582119909

(36)


119864119899

= minus

1205822

41198981

times[

[


0

1198871205822

minus 120572 (120572 + 120573 + 1)


(120573 + 1)

2

+ (

1

2

minus ])2

]

]

2

minus 1198810

(37)


6 Conclusion





Acknowledgment


References
































2N-fold






















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






Acknowledgment


References
































2N-fold






















FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics























FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics








FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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Research Article Analytical Solution of the Schrödinger ...downloads.hindawi.com/journals/ahep/2014/952597.pdf · Analytical Solution of the Schrödinger Equation with Spatially