Exact solutions of the Schrödinger equation with a complex periodic potential
The exact solutions of 1D Schrödinger equation subject to a complex periodic potential V ( x ) = - [ i a sin ( b x ) + c ] 2 ( a , b , c ∈ R ) are found as a confluent Heun function (CHF) H C ( α , β , γ , δ , η ; z ) . The energy spectra which are solved exactly by calculating the Wronskian determi...
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| Published in | Journal of mathematical chemistry Vol. 61; no. 8; pp. 1684 - 1695 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Cham
Springer International Publishing
01.09.2023
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0259-9791 1572-8897 |
| DOI | 10.1007/s10910-023-01483-7 |
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| Abstract | The exact solutions of 1D Schrödinger equation subject to a complex periodic potential
V
(
x
)
=
-
[
i
a
sin
(
b
x
)
+
c
]
2
(
a
,
b
,
c
∈
R
) are found as a confluent Heun function (CHF)
H
C
(
α
,
β
,
γ
,
δ
,
η
;
z
)
. The energy spectra which are solved exactly by calculating the Wronskian determinant are found as real except for complex values. It is found that the eigenvalues obtained by two constraints on the CHF are not reliable or complete any more since they are only one small part of those evaluated by the Wronskian determinant. The wave functions are illustrated when eigenvalues are substituted into the eigenfunctions. We also notice that the energy spectra remain invariant when one substitutes
a
→
-
a
or
b
→
-
b
or
c
→
-
c
due to the
P
T
symmetry with the property
V
(
x
)
=
V
(
-
x
)
∗
. |
|---|---|
| AbstractList | The exact solutions of 1D Schrödinger equation subject to a complex periodic potential V(x)=-[iasin(bx)+c]2 (a,b,c∈R) are found as a confluent Heun function (CHF) HC(α,β,γ,δ,η;z). The energy spectra which are solved exactly by calculating the Wronskian determinant are found as real except for complex values. It is found that the eigenvalues obtained by two constraints on the CHF are not reliable or complete any more since they are only one small part of those evaluated by the Wronskian determinant. The wave functions are illustrated when eigenvalues are substituted into the eigenfunctions. We also notice that the energy spectra remain invariant when one substitutes a→-a or b→-b or c→-c due to the PT symmetry with the property V(x)=V(-x)∗. The exact solutions of 1D Schrödinger equation subject to a complex periodic potential V ( x ) = - [ i a sin ( b x ) + c ] 2 ( a , b , c ∈ R ) are found as a confluent Heun function (CHF) H C ( α , β , γ , δ , η ; z ) . The energy spectra which are solved exactly by calculating the Wronskian determinant are found as real except for complex values. It is found that the eigenvalues obtained by two constraints on the CHF are not reliable or complete any more since they are only one small part of those evaluated by the Wronskian determinant. The wave functions are illustrated when eigenvalues are substituted into the eigenfunctions. We also notice that the energy spectra remain invariant when one substitutes a → - a or b → - b or c → - c due to the P T symmetry with the property V ( x ) = V ( - x ) ∗ . |
| Author | Sun, Guo-Hua Dong, Shi-Hai |
| Author_xml | – sequence: 1 givenname: Shi-Hai surname: Dong fullname: Dong, Shi-Hai email: dongsh2@yahoo.com organization: Research Center for Quantum Physics, Huzhou University, CIC, Instituto Politécnico Nacional, UPALM – sequence: 2 givenname: Guo-Hua surname: Sun fullname: Sun, Guo-Hua organization: CIC, Instituto Politécnico Nacional, UPALM |
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| Keywords | The Wronskian determinant 1D Schrödinger equation Complex periodic potential Confluent Heun differential equation (CHDE) |
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Phys.: Conference Series2008961 CY Chen (1483_CR42) 2021; 70 F Tellander (1483_CR53) 2017; 95 SH Dong (1483_CR8) 2008; 96 AM Ishkhanyan (1483_CR34) 2018; 133 1483_CR13 CY Chen (1483_CR41) 2020; 120 KG Makris (1483_CR58) 2008; 100 M Sakhdari (1483_CR54) 2019; 123 XY Gu (1483_CR9) 2008; 372 BH Chen (1483_CR20) 2013; 46 JK Boyd (1483_CR46) 2001; 42 1483_CR52 H Ciftci (1483_CR15) 2003; 36 F Cannata (1483_CR49) 1998; 246 Z Ahmed (1483_CR51) 2001; 286 1483_CR26 CY Chen (1483_CR44) 2021; 24 1483_CR27 CA Downing (1483_CR23) 2013; 54 1483_CR4 D ter Haar (1483_CR2) 1975 XY Gu (1483_CR12) 2008; 42 Q Dong (1483_CR38) 2020; 58 ZH Musslimani (1483_CR57) 2008; 100 AF Nikiforov (1483_CR16) 1988 PP Fiziev (1483_CR33) 2010; 43 HS Vieira (1483_CR29) 2016; 373 JR Choi (1483_CR61) 2020; 8 B Midya (1483_CR60) 2010; 374 HS Vieira (1483_CR28) 2015; 362 M Kermani (1483_CR63) 2017; 23 QT Xie (1483_CR19) 2012; 45 Q Dong (1483_CR31) 2019; 383 EN Tsoy (1483_CR50) 2014; 39 S Dong (1483_CR25) 2016; 131 CY Chen (1483_CR43) 2022; 34 C Hang (1483_CR55) 2013; 110 1483_CR39 F Cooper (1483_CR5) 1995; 251 H Panahi (1483_CR14) 2016; 68 LD Landau (1483_CR3) 1977 LI Schiff (1483_CR1) 1955 Q Dong (1483_CR36) 2019; 57 SH Dong (1483_CR17) 1998; 31 AM Ishkhanyan (1483_CR35) 2018; 388 ZH Musslimani (1483_CR59) 2008; 41 SH Dong (1483_CR6) 2007 1483_CR32 SH Dong (1483_CR18) 2011 1483_CR48 AE Sitnitsky (1483_CR22) 2017; 93 S Dong (1483_CR24) 2016; 31 CM Bender (1483_CR47) 1998; 80 ZQ Ma (1483_CR7) 2007; 371 B Bagchi (1483_CR56) 2008; 41 AE Sitnitsky (1483_CR21) 2018; 1138 1483_CR40 WC Qiang (1483_CR10) 2010; 89 Q Dong (1483_CR30) 2019; 71 (1483_CR45) 1995 Q Dong (1483_CR37) 2019; 134 M Pawlak (1483_CR62) 2018; 14 ZQ Ma (1483_CR11) 2005; 69 |
| References_xml | – reference: LandauLDLifshitzEMQuantum Mechanics (Non-Relativistic Theory)19773New YorkPergamon – reference: DongSSunGHFalayeBJDongSHThe European Physical Journal Plus20161315176 – reference: MaZQXuBWEurophys. Lett.2005696851:CAS:528:DC%2BD2MXis1Wgu7Y%3D – reference: XieQTJ. Phys. A: Math. Theor.201245 – reference: G. H. Sun, C. Y. Chen, Hind Taud, C. Yáñez-Márquez, S. H. Dong, Phys. Lett. A 384 (19), 126480 (2020) – reference: FizievPPJ. Phys. A: Math. Theor.201043 – reference: Q. Dong, F. Serrano, G. H. Sun, J. Jing, S. H. Dong, Advances in High Energy Physics, Article ID 9105825, 7 pages (2018) – reference: DowningCAJ. Math. Phys.201354 – reference: ter HaarDProblems in Quantum Mechanics19753LondonPion Ltd – reference: DongSHFactorization Method in Quantum Mechanics2007Kluwer Academic PublisherSpringer – reference: S. Dong, Q. Dong, G. H. Sun, S. Femmam, S. H. Dong, Advances in High Energy Physics Article ID 5824271, 5 pages (2018) – reference: AhmedZPhys. Lett. 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| Snippet | The exact solutions of 1D Schrödinger equation subject to a complex periodic potential
V
(
x
)
=
-
[
i
a
sin
(
b
x
)
+
c
]
2
(
a
,
b
,
c
∈
R
) are found as a... The exact solutions of 1D Schrödinger equation subject to a complex periodic potential V(x)=-[iasin(bx)+c]2 (a,b,c∈R) are found as a confluent Heun function... |
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| SubjectTerms | Chemistry Chemistry and Materials Science Eigenvalues Eigenvectors Energy spectra Exact solutions Math. Applications in Chemistry Mathematical analysis Original Paper Physical Chemistry Schrodinger equation Substitutes Theoretical and Computational Chemistry Wave functions |
| Title | Exact solutions of the Schrödinger equation with a complex periodic potential |
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