Abstract
From the eigenvalue equationHλ\ψn(λ)〉 =En(λ)\ψn(λ)〉 withHλ ≡H0 +λV one can derive an autonomous system of first order differential equations for the eigenvaluesEn (λ) and the matrix elementsVmn(λ) whereλ is the independent variable. To solve the dynamical system we need the initial valuesEn(λ = 0) and \ψn(λ = 0)〉. Thus one finds the “motion” of the energy levelsEn(λ). 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 onλ is derived. This means we calculate the differential equations which these quantities obey. Finally we derive the equations of motion for the extended caseHλ =H0 +λV1 +λ2V2 and give an application to a supersymmetric Hamiltonian.
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