Energy level crossings in quantum mechanics |
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Authors: | W. H. Steeb A. J. van Tonder C. M. Villet S. J. M. Brits |
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Affiliation: | (1) Department of Applied Mathematics, Rand Afrikaans University, P.O. Box 524, 2000 Johannesburg, South Africa |
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Abstract: | From the eigenvalue equationH![lambda](/content/n0v21275351qq658/xxlarge955.gif) n( ) =En( ) n( ) withH H0 + V one can derive an autonomous system of first order differential equations for the eigenvaluesEn ( ) and the matrix elementsVmn( ) where is the independent variable. To solve the dynamical system we need the initial valuesEn( = 0) and n( = 0) . Thus one finds the motion of the energy levelsEn( ). We discuss the question of energy level crossing. Furthermore we describe the connection with the stationary state perturbation theory. The dependence of the survival probability as well as some thermodynamic quantities on is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseH =H0 + V1 + 2V2 and give an application to a supersymmetric Hamiltonian. |
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Keywords: | Quantum mechanics dynamical system energy level crossing |
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