Exact and approximate solutions to Schrödinger’s equation with decatic potentials

• Authors: David Brandon , Nasser Saad
• Languages of publication: EN

Abstracts

EN

The one-dimensional Schrödinger’s equation is analysed with regard to the existence of exact solutions for decatic polynomial potentials. Under certain conditions on the potential’s parameters, we show that the decatic polynomial potential V (x) = ax
10 + bx
8 + cx
6 + dx
4 + ex
2, a > 0 is exactly solvable. By examining the polynomial solutions of certain linear differential equations with polynomial coefficients, the necessary and sufficient conditions for corresponding energy-dependent polynomial solutions are given in detail. It is also shown that these polynomials satisfy a four-term recurrence relation, whose real roots are the exact energy eigenvalues. Further, it is shown that these polynomials generate the eigenfunction solutions of the corresponding Schrödinger equation. Further analysis for arbitrary values of the potential parameters using the asymptotic iteration method is also presented.

Keywords

EN

differential equations; anharmonic oscillators; polynomial potentials; decatic potentials; energy spectrum; exact solutions

• Publisher: De Gruyter Open
• Journal: Open Physics
• Year: 2013
• Volume: 11
• Issue: 3
• Pages: 279-290

Dates

published

1 - 3 - 2013

online

28 - 3 - 2013

Contributors

author

David Brandon

• Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PEI, Canada, C1A 4P3

author

Nasser Saad

• Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PEI, Canada, C1A 4P3

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Identifiers

DOI

10.2478/s11534-013-0179-3