Research ArticleExact Solutions of the Razavy Cosine Type Potential
Shishan Dong 1 Qian Dong2 Guo-Hua Sun3 S Femmam4 and Shi-Hai Dong 2
1 Information and Engineering College Dalian University Dalian 116622 China2Laboratorio de Informacion Cuantica CIDETEC Instituto Politecnico Nacional UPALM CDMX 07700 Mexico3Catedratica CONACYT CIC Instituto Politecnico Nacional CDMX 07738 Mexico4UHA University and Polytechnic Engineers School Sceaux France
Correspondence should be addressed to Shi-Hai Dong dongsh2yahoocom
Received 5 June 2018 Revised 3 August 2018 Accepted 5 August 2018 Published 2 October 2018
Academic Editor Chun-Sheng Jia
Copyright copy 2018 Shishan Dong 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 solve the quantum system with the symmetric Razavy cosine type potential and find that its exact solutions are given by theconfluent Heun function The eigenvalues are calculated numerically The properties of the wave functions which depend on thepotential parameter 119886 are illustrated for a given potential parameter 120585 It is shown that the wave functions are shrunk to the originwhen the potential parameter 119886 increases We note that the energy levels 120598
119894(119894 isin [1 3]) decrease with the increasing potential
parameter 119886 but the energy levels 120598119894(119894 isin [4 7]) first increase and then decrease with the increasing 119886
1 Introduction
As we know the exact solutions of quantum systems havebeen playing an important role since the foundation ofquantum mechanics The hydrogen atom and harmonicoscillator have been taken as typical and seminal examplesto explain the classic quantum phenomena in almost allquantum mechanics textbooks [1 2] Generally speakingsome popular methods are used to solve these quantumsoluble systems First we call the functional analysis methodwith which one solves the second-order differential equationand obtains their solutions [3] expressed by somewell-knownspecial functions Second it is called the algebraic methodand can be realized by analyzing theHamiltonian of quantumsystem This method is relevant for the SUSYQM [4] andessentially connected to the factorization method [5] Thirdwe call the exact quantization rule method [6] and furtherdeveloped as the proper quantization rule method [7] Thelatter approach shows more beauty and symmetry than theformer one It should be recognized that almost all solublepotentials mentioned above belong to single well potentialsexcept for the double well potentials [8ndash10]
More than thirty years ago Razavy proposed a cosine typepotential [11 12]
119881(119898 119909)= ℏ2
2120583 181205852 [1 minus cos (2119898119909)] minus (119886 + 1) 120585 cos (119898119909) (1)
with 119881(119898 minus119909) = 119881(119898 119909) and 119881(minus119898 119909) = 119881(minus119898119909) Herethe parameters 119886 119898 are positive integers and 120585 is a positivereal number (The potential taken here is slightly differentfrom original expression [11 12] in which a proportionalcoefficient was included In addition the factor (119886 + 1)120585is extracted from the originally proposed Razavy potential[11 12] to incorporate the energy level 119864 Such a treatmentdoes not affect the property of the quantum system) InFigure 1 we plot it as a function of the variable 119909with various119886 in which we take 120585 = 3 and 119898 = 1 for simplicity Wefind that the minimum value of the potential 119881min(119898 119909) =minus(119886 + 1)120585 which is independent of the parameter 119898 Razavypresented the so-called exact solutions by using the seriesmethod [11 12] After studying it carefully it is found thatthe solutions cannot be given exactly due to the complicatedthree-term recurrence relation The method used by himis nothing but the Bethe ansatz method as summarizedin [13] In this case the solutions cannot be expressed as
HindawiAdvances in High Energy PhysicsVolume 2018 Article ID 5824271 5 pageshttpsdoiorg10115520185824271
2 Advances in High Energy Physics
a=0a=5a=10
minus30
minus20
minus10
0
10
2V(x)ℏ2
10 2 3minus2 minus1minus3
x
Figure 1 A plot of potential as function of the variables 119909 and 119886
one of the special functions due to the complicated three-term recurrence relations One must take some constraintson the coefficients in the recurrence relations as shownin [11 12] to obtain quasi-exact solutions Recently it isfound that the solutions of the hyperbolic type potentials[14ndash21] are given explicitly by the confluent Heun function[22] Just recently we have carried out the Razavy cosinehyperbolic type 119881(119909) = (ℏ212057322120583)[(18)1205852 cosh(4120573119909) minus (119898 +1)120585 cosh(2120573119909) minus (18)1205852] which was studied by Razavy in[11 12] and found that its solutions can be written as theconfluent Heun function [23] The purpose of this work isto study the solutions of the Razavy cosine type potential(1) [11 12] and to see whether its solutions can be written asthe confluent Heun function or not The answer is yes butthe energy spectra must be calculated numerically since theenergy level term is involved inside the parameter 120578 of theconfluent Heun function 119867119888(120572 120573 120574 120575 120578 119911) Even though theHeun functions have been studied well since 1889 its maintopics are focused on the mathematical area The reason whyRazavy did not find its solutions related to this function is thatonly recent connectionswith the physical problemshave beendiscovered in particular for those hyperbolic type potentials[14ndash21]
This paper is organized as follows In Section 2 we showhow to obtain the solutions of the Schrodinger equationwith the Razavy cosine type potential This is realized bytransforming the Schrodinger equation into a confluentHeun differential equation through taking some variabletransformations In Section 3 some fundamental propertiesof the solutions are studied and illustrated graphically Theenergy levels for different parameter values 119886 are calculatednumerically We summarize our results and conclusions inSection 4
2 Exact Solutions
Let us consider the one-dimensional Schrodinger equation
119911 + 1 + 120574119911 minus 1) 119889119867 (119911)
119889119911+ (120583
119911 + ]119911 minus 1)119867 (119911) = 0
(8)
we find the solution to (7) is given by the acceptable confluentHeun function 119867119888(120572 120573 120574 120575 120578 119911) with the following parame-ters
120572 = 2120585119898
120573 = minus12
120574 = minus12
120583plusmn = 120585 (minus2119886 + 119898 minus 2) plusmn 212059821198982
] = 120583minus
(9)
Advances in High Energy Physics 3
Table 1 Spectra of the Schrodinger equation with potential (1)
120578 = 12120572 (120573 + 1) minus 120583+ minus 1
2 (120573 + 120574 + 120573120574)
= 8 (119886 + 1) 120585 + 31198982 minus 812059881198982
(10)
which implies the parameter 120578 involved in the confluentHeunfunction is related to energy levels The wave function givenby this Heun function seems to be analytical but the key issueis how to first get the energy levels Otherwise the solutionbecomes unsolvable Generally the confluent Heun functioncan be expressed as a series expansion
The explicit expression of this determinant can refer to [16ndash19] for some detail
For present case there is a problem for the first conditionThat is 120583
+ + 120583minus + 120572 = 0 when 119873 = 1 From this we have119898 = 2(1 + 119886)(1 + 4120585) This is contrary to the assumption 119898is positive integer Therefore how to obtain the eigenvaluesbecomes a challenging task Due to 119911 isin [0 1] we would liketo solve this problem via series expansion method as shownin [15] Unfortunately the calculation results are not ideal Wehave to solve it in another way as shown in [14]
3 Fundamental Properties
Now let us study some basic properties of the solutions asshown in Figures 2 and 3 We find that the wave functionsare shrunk to the origin when the potential parameter 119886increases This makes the amplitude of the wave function beincreasedWe list the energy levels 120598
119894 (119894 isin [1 7]) inTable 1 andillustrate them in Figure 3 We notice that the energy levels 120598119894
4 Advances in High Energy Physics
a=0
a=5
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
(x)
1 2 3
a=1
a=10
10
05
00
minus05
minus10minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
x
=3
Figure 2 The characteristics of wave functions as a function of the position 119909 We take 120585 = 3
minus30
minus20
minus25
minus10
minus15
minus5
5
0
15
10
20
spectra
2 3 4 5 6 71
i
a=1a=2a=3a=4
a=0a=7a=8a=9a=10
a=6
a=5
Figure 3The variation of the energy spectra 120598119894and 120585 = 3
(119894 isin [1 3]) decrease with the increasing potential parameter119886 but 120598119894 (119894 isin [4 7]) first increase and then decrease with theincreasing potential parameter 119886
4 Conclusions
In this work we have studied the quantum system withthe Razavy cosine type potential and found that its exactsolutions are given by confluent Heun function 120595(119911) =exp[(2119911 minus 1)1205852]119867
119888(120572 120573 120574 120575 120578 119911) by transforming the orig-inal differential equation into a confluent type Heun differ-ential equation The fact that the energy levels are involvedinside the parameter 120578 makes us calculate the eigenvaluesnumerically The properties of the wave functions dependingon the potential parameter 119886 have been illustrated graphicallyfor a given potential parameter 120585 We have also noticed thatthe energy levels 120598
119894 (119894 isin [1 3]) decrease with the increasingpotential parameter 119886 but 120598119894 (119894 isin [4 7]) first increase and thendecrease with the increasing 119886Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Advances in High Energy Physics 5
Acknowledgments
This work is supported by project 20180677-SIP-IPNCOFAA-IPN Mexico and partially by the CONACYTproject under grant No 288856-CB-2016
References
[1] L D Landau and E M Lifshitz Quantum mechanics non-relativistic theory Pergamon New York NY USA 1977
[2] L I Schiff Quantum Mechanics McGraw-Hill Book Co NewYork NY USA 3rd edition 1955
[3] D terHaarProblems inQuantumMechanics Pion Ltd London3rd edition 1975
[4] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[5] S H Dong Factorization Method in Quantum Mechanics vol150 Springer Kluwer Academic Publisher 2007
[6] Z Q Ma and B W Xu ldquoQuantum correction in exact quanti-zation rulesrdquo EPL (Europhysics Letters) vol 69 p 685 2005
[7] W C Qiang and S H Dong ldquoProper quantization rulerdquo EPLvol 89 article 10003 2010
[8] H Konwent P Machnikowski and A Radosz ldquoA certaindouble-well potential related to SU(2) symmetryrdquo Journal ofPhysics A Mathematical and General vol 28 no 13 pp 3757ndash3762 1995
[9] Q-T Xie ldquoNew quasi-exactly solvable double-well potentialsrdquoJournal of Physics A Mathematical and General vol 45 no 17Article ID 175302 2012
[10] B Chen Y Wu and Q Xie ldquoHeun functions and quasi-exactly solvable double-well potentialsrdquo Journal of Physics AMathematical andTheoretical vol 46 no 3 2013
[11] M Razavy ldquoA potential model for torsional vibrations ofmoleculesrdquo Physics Letters A vol 82 no 1 pp 7ndash9 1981
[12] M Razavy ldquoAn exactly soluble Schrodinger equation with abistable potentialrdquo American Journal of Physics vol 48 no 4p 285 1980
[14] S Dong Q Fang B J Falaye G Sun C Yanez-Marquez andS Dong ldquoExact solutions to solitonic profile mass Schrodingerproblem with a modified PoschlndashTeller potentialrdquo ModernPhysics Letters A vol 31 no 04 p 1650017 2016
[15] S Dong G H Sun B J Falaye and S H Dong ldquoSemi-exactsolutions to position-dependent mass Schrodinger problemwith a class of hyperbolic potential V0tanh(ax)rdquoThe EuropeanPhysical Journal Plus vol 131 no 5 p 176 2016
[16] G H Sun S H Dong K D Launey T Dytrych J P Draayerand J Quan ldquoShannon information entropy for a hyperbolicdouble-well potentialrdquo International Journal ofQuantumChem-istry vol 115 no 14 article 891 2015
[17] C ADowning ldquoOn a solution of the Schrodinger equationwitha hyperbolic double-well potentialrdquo Journal of MathematicalPhysics vol 54 no 7 072101 8 pages 2013
[18] P P Fiziev ldquoNovel relations and new properties of confluentHeunrsquos functions and their derivatives of arbitrary orderrdquoJournal of Physics A Mathematical and General vol 43 no 3article 035203 2010
[19] R R Hartmann and M E Portnoi ldquoQuasi-exact solution tothe Dirac equation for the hyperbolic-secant potentialrdquoPhysical
Review A Atomic Molecular and Optical Physics vol 89 no 12014
[20] D Agboola ldquoOn the solvability of the generalized hyperbolicdouble-well modelsrdquo Journal of Mathematical Physics vol 55no 5 Article ID 052102 8 pages 2014
[21] F-K Wen Z-Y Yang C Liu W-L Yang and Y-Z ZhangldquoExact Polynomial Solutions of Schrodinger equation withvarious hyperbolic potentialsrdquo Communications in TheoreticalPhysics vol 61 no 2 pp 153ndash159 2014
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
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Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
2 Advances in High Energy Physics
a=0a=5a=10
minus30
minus20
minus10
0
10
2V(x)ℏ2
10 2 3minus2 minus1minus3
x
Figure 1 A plot of potential as function of the variables 119909 and 119886
one of the special functions due to the complicated three-term recurrence relations One must take some constraintson the coefficients in the recurrence relations as shownin [11 12] to obtain quasi-exact solutions Recently it isfound that the solutions of the hyperbolic type potentials[14ndash21] are given explicitly by the confluent Heun function[22] Just recently we have carried out the Razavy cosinehyperbolic type 119881(119909) = (ℏ212057322120583)[(18)1205852 cosh(4120573119909) minus (119898 +1)120585 cosh(2120573119909) minus (18)1205852] which was studied by Razavy in[11 12] and found that its solutions can be written as theconfluent Heun function [23] The purpose of this work isto study the solutions of the Razavy cosine type potential(1) [11 12] and to see whether its solutions can be written asthe confluent Heun function or not The answer is yes butthe energy spectra must be calculated numerically since theenergy level term is involved inside the parameter 120578 of theconfluent Heun function 119867119888(120572 120573 120574 120575 120578 119911) Even though theHeun functions have been studied well since 1889 its maintopics are focused on the mathematical area The reason whyRazavy did not find its solutions related to this function is thatonly recent connectionswith the physical problemshave beendiscovered in particular for those hyperbolic type potentials[14ndash21]
This paper is organized as follows In Section 2 we showhow to obtain the solutions of the Schrodinger equationwith the Razavy cosine type potential This is realized bytransforming the Schrodinger equation into a confluentHeun differential equation through taking some variabletransformations In Section 3 some fundamental propertiesof the solutions are studied and illustrated graphically Theenergy levels for different parameter values 119886 are calculatednumerically We summarize our results and conclusions inSection 4
2 Exact Solutions
Let us consider the one-dimensional Schrodinger equation
119911 + 1 + 120574119911 minus 1) 119889119867 (119911)
119889119911+ (120583
119911 + ]119911 minus 1)119867 (119911) = 0
(8)
we find the solution to (7) is given by the acceptable confluentHeun function 119867119888(120572 120573 120574 120575 120578 119911) with the following parame-ters
120572 = 2120585119898
120573 = minus12
120574 = minus12
120583plusmn = 120585 (minus2119886 + 119898 minus 2) plusmn 212059821198982
] = 120583minus
(9)
Advances in High Energy Physics 3
Table 1 Spectra of the Schrodinger equation with potential (1)
120578 = 12120572 (120573 + 1) minus 120583+ minus 1
2 (120573 + 120574 + 120573120574)
= 8 (119886 + 1) 120585 + 31198982 minus 812059881198982
(10)
which implies the parameter 120578 involved in the confluentHeunfunction is related to energy levels The wave function givenby this Heun function seems to be analytical but the key issueis how to first get the energy levels Otherwise the solutionbecomes unsolvable Generally the confluent Heun functioncan be expressed as a series expansion
The explicit expression of this determinant can refer to [16ndash19] for some detail
For present case there is a problem for the first conditionThat is 120583
+ + 120583minus + 120572 = 0 when 119873 = 1 From this we have119898 = 2(1 + 119886)(1 + 4120585) This is contrary to the assumption 119898is positive integer Therefore how to obtain the eigenvaluesbecomes a challenging task Due to 119911 isin [0 1] we would liketo solve this problem via series expansion method as shownin [15] Unfortunately the calculation results are not ideal Wehave to solve it in another way as shown in [14]
3 Fundamental Properties
Now let us study some basic properties of the solutions asshown in Figures 2 and 3 We find that the wave functionsare shrunk to the origin when the potential parameter 119886increases This makes the amplitude of the wave function beincreasedWe list the energy levels 120598
119894 (119894 isin [1 7]) inTable 1 andillustrate them in Figure 3 We notice that the energy levels 120598119894
4 Advances in High Energy Physics
a=0
a=5
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
(x)
1 2 3
a=1
a=10
10
05
00
minus05
minus10minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
x
=3
Figure 2 The characteristics of wave functions as a function of the position 119909 We take 120585 = 3
minus30
minus20
minus25
minus10
minus15
minus5
5
0
15
10
20
spectra
2 3 4 5 6 71
i
a=1a=2a=3a=4
a=0a=7a=8a=9a=10
a=6
a=5
Figure 3The variation of the energy spectra 120598119894and 120585 = 3
(119894 isin [1 3]) decrease with the increasing potential parameter119886 but 120598119894 (119894 isin [4 7]) first increase and then decrease with theincreasing potential parameter 119886
4 Conclusions
In this work we have studied the quantum system withthe Razavy cosine type potential and found that its exactsolutions are given by confluent Heun function 120595(119911) =exp[(2119911 minus 1)1205852]119867
119888(120572 120573 120574 120575 120578 119911) by transforming the orig-inal differential equation into a confluent type Heun differ-ential equation The fact that the energy levels are involvedinside the parameter 120578 makes us calculate the eigenvaluesnumerically The properties of the wave functions dependingon the potential parameter 119886 have been illustrated graphicallyfor a given potential parameter 120585 We have also noticed thatthe energy levels 120598
119894 (119894 isin [1 3]) decrease with the increasingpotential parameter 119886 but 120598119894 (119894 isin [4 7]) first increase and thendecrease with the increasing 119886Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Advances in High Energy Physics 5
Acknowledgments
This work is supported by project 20180677-SIP-IPNCOFAA-IPN Mexico and partially by the CONACYTproject under grant No 288856-CB-2016
References
[1] L D Landau and E M Lifshitz Quantum mechanics non-relativistic theory Pergamon New York NY USA 1977
[2] L I Schiff Quantum Mechanics McGraw-Hill Book Co NewYork NY USA 3rd edition 1955
[3] D terHaarProblems inQuantumMechanics Pion Ltd London3rd edition 1975
[4] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[5] S H Dong Factorization Method in Quantum Mechanics vol150 Springer Kluwer Academic Publisher 2007
[6] Z Q Ma and B W Xu ldquoQuantum correction in exact quanti-zation rulesrdquo EPL (Europhysics Letters) vol 69 p 685 2005
[7] W C Qiang and S H Dong ldquoProper quantization rulerdquo EPLvol 89 article 10003 2010
[8] H Konwent P Machnikowski and A Radosz ldquoA certaindouble-well potential related to SU(2) symmetryrdquo Journal ofPhysics A Mathematical and General vol 28 no 13 pp 3757ndash3762 1995
[9] Q-T Xie ldquoNew quasi-exactly solvable double-well potentialsrdquoJournal of Physics A Mathematical and General vol 45 no 17Article ID 175302 2012
[10] B Chen Y Wu and Q Xie ldquoHeun functions and quasi-exactly solvable double-well potentialsrdquo Journal of Physics AMathematical andTheoretical vol 46 no 3 2013
[11] M Razavy ldquoA potential model for torsional vibrations ofmoleculesrdquo Physics Letters A vol 82 no 1 pp 7ndash9 1981
[12] M Razavy ldquoAn exactly soluble Schrodinger equation with abistable potentialrdquo American Journal of Physics vol 48 no 4p 285 1980
[14] S Dong Q Fang B J Falaye G Sun C Yanez-Marquez andS Dong ldquoExact solutions to solitonic profile mass Schrodingerproblem with a modified PoschlndashTeller potentialrdquo ModernPhysics Letters A vol 31 no 04 p 1650017 2016
[15] S Dong G H Sun B J Falaye and S H Dong ldquoSemi-exactsolutions to position-dependent mass Schrodinger problemwith a class of hyperbolic potential V0tanh(ax)rdquoThe EuropeanPhysical Journal Plus vol 131 no 5 p 176 2016
[16] G H Sun S H Dong K D Launey T Dytrych J P Draayerand J Quan ldquoShannon information entropy for a hyperbolicdouble-well potentialrdquo International Journal ofQuantumChem-istry vol 115 no 14 article 891 2015
[17] C ADowning ldquoOn a solution of the Schrodinger equationwitha hyperbolic double-well potentialrdquo Journal of MathematicalPhysics vol 54 no 7 072101 8 pages 2013
[18] P P Fiziev ldquoNovel relations and new properties of confluentHeunrsquos functions and their derivatives of arbitrary orderrdquoJournal of Physics A Mathematical and General vol 43 no 3article 035203 2010
[19] R R Hartmann and M E Portnoi ldquoQuasi-exact solution tothe Dirac equation for the hyperbolic-secant potentialrdquoPhysical
Review A Atomic Molecular and Optical Physics vol 89 no 12014
[20] D Agboola ldquoOn the solvability of the generalized hyperbolicdouble-well modelsrdquo Journal of Mathematical Physics vol 55no 5 Article ID 052102 8 pages 2014
[21] F-K Wen Z-Y Yang C Liu W-L Yang and Y-Z ZhangldquoExact Polynomial Solutions of Schrodinger equation withvarious hyperbolic potentialsrdquo Communications in TheoreticalPhysics vol 61 no 2 pp 153ndash159 2014
120578 = 12120572 (120573 + 1) minus 120583+ minus 1
2 (120573 + 120574 + 120573120574)
= 8 (119886 + 1) 120585 + 31198982 minus 812059881198982
(10)
which implies the parameter 120578 involved in the confluentHeunfunction is related to energy levels The wave function givenby this Heun function seems to be analytical but the key issueis how to first get the energy levels Otherwise the solutionbecomes unsolvable Generally the confluent Heun functioncan be expressed as a series expansion
The explicit expression of this determinant can refer to [16ndash19] for some detail
For present case there is a problem for the first conditionThat is 120583
+ + 120583minus + 120572 = 0 when 119873 = 1 From this we have119898 = 2(1 + 119886)(1 + 4120585) This is contrary to the assumption 119898is positive integer Therefore how to obtain the eigenvaluesbecomes a challenging task Due to 119911 isin [0 1] we would liketo solve this problem via series expansion method as shownin [15] Unfortunately the calculation results are not ideal Wehave to solve it in another way as shown in [14]
3 Fundamental Properties
Now let us study some basic properties of the solutions asshown in Figures 2 and 3 We find that the wave functionsare shrunk to the origin when the potential parameter 119886increases This makes the amplitude of the wave function beincreasedWe list the energy levels 120598
119894 (119894 isin [1 7]) inTable 1 andillustrate them in Figure 3 We notice that the energy levels 120598119894
4 Advances in High Energy Physics
a=0
a=5
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
(x)
1 2 3
a=1
a=10
10
05
00
minus05
minus10minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
x
=3
Figure 2 The characteristics of wave functions as a function of the position 119909 We take 120585 = 3
minus30
minus20
minus25
minus10
minus15
minus5
5
0
15
10
20
spectra
2 3 4 5 6 71
i
a=1a=2a=3a=4
a=0a=7a=8a=9a=10
a=6
a=5
Figure 3The variation of the energy spectra 120598119894and 120585 = 3
(119894 isin [1 3]) decrease with the increasing potential parameter119886 but 120598119894 (119894 isin [4 7]) first increase and then decrease with theincreasing potential parameter 119886
4 Conclusions
In this work we have studied the quantum system withthe Razavy cosine type potential and found that its exactsolutions are given by confluent Heun function 120595(119911) =exp[(2119911 minus 1)1205852]119867
119888(120572 120573 120574 120575 120578 119911) by transforming the orig-inal differential equation into a confluent type Heun differ-ential equation The fact that the energy levels are involvedinside the parameter 120578 makes us calculate the eigenvaluesnumerically The properties of the wave functions dependingon the potential parameter 119886 have been illustrated graphicallyfor a given potential parameter 120585 We have also noticed thatthe energy levels 120598
119894 (119894 isin [1 3]) decrease with the increasingpotential parameter 119886 but 120598119894 (119894 isin [4 7]) first increase and thendecrease with the increasing 119886Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Advances in High Energy Physics 5
Acknowledgments
This work is supported by project 20180677-SIP-IPNCOFAA-IPN Mexico and partially by the CONACYTproject under grant No 288856-CB-2016
References
[1] L D Landau and E M Lifshitz Quantum mechanics non-relativistic theory Pergamon New York NY USA 1977
[2] L I Schiff Quantum Mechanics McGraw-Hill Book Co NewYork NY USA 3rd edition 1955
[3] D terHaarProblems inQuantumMechanics Pion Ltd London3rd edition 1975
[4] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[5] S H Dong Factorization Method in Quantum Mechanics vol150 Springer Kluwer Academic Publisher 2007
[6] Z Q Ma and B W Xu ldquoQuantum correction in exact quanti-zation rulesrdquo EPL (Europhysics Letters) vol 69 p 685 2005
[7] W C Qiang and S H Dong ldquoProper quantization rulerdquo EPLvol 89 article 10003 2010
[8] H Konwent P Machnikowski and A Radosz ldquoA certaindouble-well potential related to SU(2) symmetryrdquo Journal ofPhysics A Mathematical and General vol 28 no 13 pp 3757ndash3762 1995
[9] Q-T Xie ldquoNew quasi-exactly solvable double-well potentialsrdquoJournal of Physics A Mathematical and General vol 45 no 17Article ID 175302 2012
[10] B Chen Y Wu and Q Xie ldquoHeun functions and quasi-exactly solvable double-well potentialsrdquo Journal of Physics AMathematical andTheoretical vol 46 no 3 2013
[11] M Razavy ldquoA potential model for torsional vibrations ofmoleculesrdquo Physics Letters A vol 82 no 1 pp 7ndash9 1981
[12] M Razavy ldquoAn exactly soluble Schrodinger equation with abistable potentialrdquo American Journal of Physics vol 48 no 4p 285 1980
[14] S Dong Q Fang B J Falaye G Sun C Yanez-Marquez andS Dong ldquoExact solutions to solitonic profile mass Schrodingerproblem with a modified PoschlndashTeller potentialrdquo ModernPhysics Letters A vol 31 no 04 p 1650017 2016
[15] S Dong G H Sun B J Falaye and S H Dong ldquoSemi-exactsolutions to position-dependent mass Schrodinger problemwith a class of hyperbolic potential V0tanh(ax)rdquoThe EuropeanPhysical Journal Plus vol 131 no 5 p 176 2016
[16] G H Sun S H Dong K D Launey T Dytrych J P Draayerand J Quan ldquoShannon information entropy for a hyperbolicdouble-well potentialrdquo International Journal ofQuantumChem-istry vol 115 no 14 article 891 2015
[17] C ADowning ldquoOn a solution of the Schrodinger equationwitha hyperbolic double-well potentialrdquo Journal of MathematicalPhysics vol 54 no 7 072101 8 pages 2013
[18] P P Fiziev ldquoNovel relations and new properties of confluentHeunrsquos functions and their derivatives of arbitrary orderrdquoJournal of Physics A Mathematical and General vol 43 no 3article 035203 2010
[19] R R Hartmann and M E Portnoi ldquoQuasi-exact solution tothe Dirac equation for the hyperbolic-secant potentialrdquoPhysical
Review A Atomic Molecular and Optical Physics vol 89 no 12014
[20] D Agboola ldquoOn the solvability of the generalized hyperbolicdouble-well modelsrdquo Journal of Mathematical Physics vol 55no 5 Article ID 052102 8 pages 2014
[21] F-K Wen Z-Y Yang C Liu W-L Yang and Y-Z ZhangldquoExact Polynomial Solutions of Schrodinger equation withvarious hyperbolic potentialsrdquo Communications in TheoreticalPhysics vol 61 no 2 pp 153ndash159 2014
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
4 Advances in High Energy Physics
a=0
a=5
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
(x)
1 2 3
a=1
a=10
10
05
00
minus05
minus10minus3 minus2 minus1 0 1 2 3
10
05
00
minus05
minus10
minus3 minus2 minus1 0 1 2 3
x
=3
Figure 2 The characteristics of wave functions as a function of the position 119909 We take 120585 = 3
minus30
minus20
minus25
minus10
minus15
minus5
5
0
15
10
20
spectra
2 3 4 5 6 71
i
a=1a=2a=3a=4
a=0a=7a=8a=9a=10
a=6
a=5
Figure 3The variation of the energy spectra 120598119894and 120585 = 3
(119894 isin [1 3]) decrease with the increasing potential parameter119886 but 120598119894 (119894 isin [4 7]) first increase and then decrease with theincreasing potential parameter 119886
4 Conclusions
In this work we have studied the quantum system withthe Razavy cosine type potential and found that its exactsolutions are given by confluent Heun function 120595(119911) =exp[(2119911 minus 1)1205852]119867
119888(120572 120573 120574 120575 120578 119911) by transforming the orig-inal differential equation into a confluent type Heun differ-ential equation The fact that the energy levels are involvedinside the parameter 120578 makes us calculate the eigenvaluesnumerically The properties of the wave functions dependingon the potential parameter 119886 have been illustrated graphicallyfor a given potential parameter 120585 We have also noticed thatthe energy levels 120598
119894 (119894 isin [1 3]) decrease with the increasingpotential parameter 119886 but 120598119894 (119894 isin [4 7]) first increase and thendecrease with the increasing 119886Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Advances in High Energy Physics 5
Acknowledgments
This work is supported by project 20180677-SIP-IPNCOFAA-IPN Mexico and partially by the CONACYTproject under grant No 288856-CB-2016
References
[1] L D Landau and E M Lifshitz Quantum mechanics non-relativistic theory Pergamon New York NY USA 1977
[2] L I Schiff Quantum Mechanics McGraw-Hill Book Co NewYork NY USA 3rd edition 1955
[3] D terHaarProblems inQuantumMechanics Pion Ltd London3rd edition 1975
[4] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[5] S H Dong Factorization Method in Quantum Mechanics vol150 Springer Kluwer Academic Publisher 2007
[6] Z Q Ma and B W Xu ldquoQuantum correction in exact quanti-zation rulesrdquo EPL (Europhysics Letters) vol 69 p 685 2005
[7] W C Qiang and S H Dong ldquoProper quantization rulerdquo EPLvol 89 article 10003 2010
[8] H Konwent P Machnikowski and A Radosz ldquoA certaindouble-well potential related to SU(2) symmetryrdquo Journal ofPhysics A Mathematical and General vol 28 no 13 pp 3757ndash3762 1995
[9] Q-T Xie ldquoNew quasi-exactly solvable double-well potentialsrdquoJournal of Physics A Mathematical and General vol 45 no 17Article ID 175302 2012
[10] B Chen Y Wu and Q Xie ldquoHeun functions and quasi-exactly solvable double-well potentialsrdquo Journal of Physics AMathematical andTheoretical vol 46 no 3 2013
[11] M Razavy ldquoA potential model for torsional vibrations ofmoleculesrdquo Physics Letters A vol 82 no 1 pp 7ndash9 1981
[12] M Razavy ldquoAn exactly soluble Schrodinger equation with abistable potentialrdquo American Journal of Physics vol 48 no 4p 285 1980
[14] S Dong Q Fang B J Falaye G Sun C Yanez-Marquez andS Dong ldquoExact solutions to solitonic profile mass Schrodingerproblem with a modified PoschlndashTeller potentialrdquo ModernPhysics Letters A vol 31 no 04 p 1650017 2016
[15] S Dong G H Sun B J Falaye and S H Dong ldquoSemi-exactsolutions to position-dependent mass Schrodinger problemwith a class of hyperbolic potential V0tanh(ax)rdquoThe EuropeanPhysical Journal Plus vol 131 no 5 p 176 2016
[16] G H Sun S H Dong K D Launey T Dytrych J P Draayerand J Quan ldquoShannon information entropy for a hyperbolicdouble-well potentialrdquo International Journal ofQuantumChem-istry vol 115 no 14 article 891 2015
[17] C ADowning ldquoOn a solution of the Schrodinger equationwitha hyperbolic double-well potentialrdquo Journal of MathematicalPhysics vol 54 no 7 072101 8 pages 2013
[18] P P Fiziev ldquoNovel relations and new properties of confluentHeunrsquos functions and their derivatives of arbitrary orderrdquoJournal of Physics A Mathematical and General vol 43 no 3article 035203 2010
[19] R R Hartmann and M E Portnoi ldquoQuasi-exact solution tothe Dirac equation for the hyperbolic-secant potentialrdquoPhysical
Review A Atomic Molecular and Optical Physics vol 89 no 12014
[20] D Agboola ldquoOn the solvability of the generalized hyperbolicdouble-well modelsrdquo Journal of Mathematical Physics vol 55no 5 Article ID 052102 8 pages 2014
[21] F-K Wen Z-Y Yang C Liu W-L Yang and Y-Z ZhangldquoExact Polynomial Solutions of Schrodinger equation withvarious hyperbolic potentialsrdquo Communications in TheoreticalPhysics vol 61 no 2 pp 153ndash159 2014
Applied Bionics and BiomechanicsHindawiwwwhindawicom Volume 2018
Advances inOptoElectronics
Hindawiwwwhindawicom
Volume 2018
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Hindawiwwwhindawicom Volume 2018
ChemistryAdvances in
Hindawiwwwhindawicom Volume 2018
Journal of
Chemistry
Hindawiwwwhindawicom Volume 2018
Advances inPhysical Chemistry
International Journal of
RotatingMachinery
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Submit your manuscripts atwwwhindawicom
Advances in High Energy Physics 5
Acknowledgments
This work is supported by project 20180677-SIP-IPNCOFAA-IPN Mexico and partially by the CONACYTproject under grant No 288856-CB-2016
References
[1] L D Landau and E M Lifshitz Quantum mechanics non-relativistic theory Pergamon New York NY USA 1977
[2] L I Schiff Quantum Mechanics McGraw-Hill Book Co NewYork NY USA 3rd edition 1955
[3] D terHaarProblems inQuantumMechanics Pion Ltd London3rd edition 1975
[4] F Cooper A Khare and U Sukhatme ldquoSupersymmetry andquantummechanicsrdquo Physics Reports vol 251 no 5-6 pp 267ndash385 1995
[5] S H Dong Factorization Method in Quantum Mechanics vol150 Springer Kluwer Academic Publisher 2007
[6] Z Q Ma and B W Xu ldquoQuantum correction in exact quanti-zation rulesrdquo EPL (Europhysics Letters) vol 69 p 685 2005
[7] W C Qiang and S H Dong ldquoProper quantization rulerdquo EPLvol 89 article 10003 2010
[8] H Konwent P Machnikowski and A Radosz ldquoA certaindouble-well potential related to SU(2) symmetryrdquo Journal ofPhysics A Mathematical and General vol 28 no 13 pp 3757ndash3762 1995
[9] Q-T Xie ldquoNew quasi-exactly solvable double-well potentialsrdquoJournal of Physics A Mathematical and General vol 45 no 17Article ID 175302 2012
[10] B Chen Y Wu and Q Xie ldquoHeun functions and quasi-exactly solvable double-well potentialsrdquo Journal of Physics AMathematical andTheoretical vol 46 no 3 2013
[11] M Razavy ldquoA potential model for torsional vibrations ofmoleculesrdquo Physics Letters A vol 82 no 1 pp 7ndash9 1981
[12] M Razavy ldquoAn exactly soluble Schrodinger equation with abistable potentialrdquo American Journal of Physics vol 48 no 4p 285 1980
[14] S Dong Q Fang B J Falaye G Sun C Yanez-Marquez andS Dong ldquoExact solutions to solitonic profile mass Schrodingerproblem with a modified PoschlndashTeller potentialrdquo ModernPhysics Letters A vol 31 no 04 p 1650017 2016
[15] S Dong G H Sun B J Falaye and S H Dong ldquoSemi-exactsolutions to position-dependent mass Schrodinger problemwith a class of hyperbolic potential V0tanh(ax)rdquoThe EuropeanPhysical Journal Plus vol 131 no 5 p 176 2016
[16] G H Sun S H Dong K D Launey T Dytrych J P Draayerand J Quan ldquoShannon information entropy for a hyperbolicdouble-well potentialrdquo International Journal ofQuantumChem-istry vol 115 no 14 article 891 2015
[17] C ADowning ldquoOn a solution of the Schrodinger equationwitha hyperbolic double-well potentialrdquo Journal of MathematicalPhysics vol 54 no 7 072101 8 pages 2013
[18] P P Fiziev ldquoNovel relations and new properties of confluentHeunrsquos functions and their derivatives of arbitrary orderrdquoJournal of Physics A Mathematical and General vol 43 no 3article 035203 2010
[19] R R Hartmann and M E Portnoi ldquoQuasi-exact solution tothe Dirac equation for the hyperbolic-secant potentialrdquoPhysical
Review A Atomic Molecular and Optical Physics vol 89 no 12014
[20] D Agboola ldquoOn the solvability of the generalized hyperbolicdouble-well modelsrdquo Journal of Mathematical Physics vol 55no 5 Article ID 052102 8 pages 2014
[21] F-K Wen Z-Y Yang C Liu W-L Yang and Y-Z ZhangldquoExact Polynomial Solutions of Schrodinger equation withvarious hyperbolic potentialsrdquo Communications in TheoreticalPhysics vol 61 no 2 pp 153ndash159 2014
