Constructions of the Soluble Potentials for the Nonrelativistic Quantum System by Means of the Heun Functions
Joint Authors
Sun, Guo-Hua
Mejía-García, Concepción
Dong, Shishan
Yáñez-Navarro, G.
Mercado Sanchez, M. A.
Dong, Shi-Hai
Source
Advances in High Energy Physics
Issue
Vol. 2018, Issue 2018 (31 Dec. 2018), pp.1-8, 8 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2018-06-03
Country of Publication
Egypt
No. of Pages
8
Main Subjects
Abstract EN
The Schrödinger equation ψ ′ ′ ( x ) + κ 2 ψ ( x ) = 0 where κ 2 = k 2 - V ( x ) is rewritten as a more popular form of a second order differential equation by taking a similarity transformation ψ ( z ) = ϕ ( z ) u ( z ) with z = z ( x ) .
The Schrödinger invariant I S ( x ) can be calculated directly by the Schwarzian derivative z , x and the invariant I ( z ) of the differential equation u z z + f ( z ) u z + g ( z ) u = 0 .
We find an important relation for a moving particle as ∇ 2 = - I S ( x ) and thus explain the reason why the Schrödinger invariant I S ( x ) keeps constant.
As an illustration, we take the typical Heun’s differential equation as an object to construct a class of soluble potentials and generalize the previous results by taking different transformation ρ = z ′ ( x ) as before.
We get a more general solution z ( x ) through integrating ( z ′ ) 2 = α 1 z 2 + β 1 z + γ 1 directly and it includes all possibilities for those parameters.
Some particular cases are discussed in detail.
The results are also compared with those obtained by Bose, Lemieux, Batic, Ishkhanyan, and their coworkers.
It should be recognized that a subtle and different choice of the transformation z ( x ) also related to ρ will lead to difficult connections to the results obtained from other different approaches.
American Psychological Association (APA)
Dong, Shishan& Yáñez-Navarro, G.& Mercado Sanchez, M. A.& Mejía-García, Concepción& Sun, Guo-Hua& Dong, Shi-Hai. 2018. Constructions of the Soluble Potentials for the Nonrelativistic Quantum System by Means of the Heun Functions. Advances in High Energy Physics،Vol. 2018, no. 2018, pp.1-8.
https://search.emarefa.net/detail/BIM-1118355
Modern Language Association (MLA)
Dong, Shishan…[et al.]. Constructions of the Soluble Potentials for the Nonrelativistic Quantum System by Means of the Heun Functions. Advances in High Energy Physics No. 2018 (2018), pp.1-8.
https://search.emarefa.net/detail/BIM-1118355
American Medical Association (AMA)
Dong, Shishan& Yáñez-Navarro, G.& Mercado Sanchez, M. A.& Mejía-García, Concepción& Sun, Guo-Hua& Dong, Shi-Hai. Constructions of the Soluble Potentials for the Nonrelativistic Quantum System by Means of the Heun Functions. Advances in High Energy Physics. 2018. Vol. 2018, no. 2018, pp.1-8.
https://search.emarefa.net/detail/BIM-1118355
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1118355