Reformulating the Schroedinger equation as a Shabat-Zakharov system
- Department of Mathematics, Faculty of Science, Chulalongkorn University, Phayathai Rd., Pathumwan, Bangkok 10330 (Thailand)
- School of Mathematics, Statistics, and Operations Research, Victoria University of Wellington, PO Box 600, Wellington 6140 (New Zealand)
We reformulate the second-order Schroedinger equation as a set of two coupled first-order differential equations, a so-called 'Shabat-Zakharov system' (sometimes called a 'Zakharov-Shabat' system). There is considerable flexibility in this approach, and we emphasize the utility of introducing an 'auxiliary condition' or 'gauge condition' that is used to cut down the degrees of freedom. Using this formalism, we derive the explicit (but formal) general solution to the Schroedinger equation. The general solution depends on three arbitrarily chosen functions, and a path-ordered exponential matrix. If one considers path ordering to be an 'elementary' process, then this represents complete quadrature, albeit formal, of the second-order linear ordinary differential equation.
- OSTI ID:
- 21335902
- Journal Information:
- Journal of Mathematical Physics, Vol. 51, Issue 2; Other Information: DOI: 10.1063/1.3282847; (c) 2010 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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