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The Schrödinger Equations and Their Consequences

Salvatore Capozzıello, Wladimir-Georges Boskoff

UNITEXT for physicsJan 1, 20210 citations

Schrödinger Equation as a Self-Consistent Field

Victor Valentinovich Vedenyapin, Татьяна Семеновна Казакова, V. Ya. Kiselevskaya-Babinina +1 more

It is well known that the Schrödinger equation can be reduced to the Hamilton–Jacobi equation in Bohmian mechanics. Corresponding new equations of the Vlasov and Lamb types are derived, and their stationary solutions are investigated.

Doklady MathematicsMay 1, 201812 citations

Schrödinger Equation II

Sergio Cecotti

UNITEXT for physicsJan 1, 20250 citations

Characteristic solutions of the chain of Vlasov equations

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

Abstract A new method has been presented of constructing a class of exact solutions of an infinite self-linking chain of the Vlasov equations for distribution functions of kinematic quantities of all orders. Using the characteristic transformation of variables proposed in this paper, any equation from the Vlasov chain can be reduced to the mathematical form of the first Vlasov equation. Since the solution of the first Vlasov equation can be found by the solution of the Schrödinger equation, the authors have proposed an algorithm for constructing characteristic solutions for an arbitrary equation from the Vlasov chain. The proposed method of construction of exact solutions has been successfully implemented on an example of time-dependent quantum system with thermodynamic parameter in the form of inverse temperature. These found exact solutions are also applicable to quantum dot systems.Abstract A new method has been presented of constructing a class of exact solutions of an infinite self-linking chain of the Vlasov equations for distribution functions of kinematic quantities of all orders. Using the characteristic transformation of variables proposed in this paper, any equation from the Vlasov chain can be reduced to the mathematical form of the first Vlasov equation. Since the solution of the first Vlasov equation can be found by the solution of the Schrödinger equation, the authors have proposed an algorithm for constructing characteristic solutions for an arbitrary equation from the Vlasov chain. The proposed method of construction of exact solutions has been successfully implemented on an example of time-dependent quantum system with thermodynamic parameter in the form of inverse temperature. These found exact solutions are also applicable to quantum dot systems.

arXiv:2506.22377Physica ScriptaJun 25, 20260 citations

Equations with physical applications arising from the Vlasov chain

Paul Bracken

The first equation in a hierarchy of equations due to Vlasov is studied. The equation can be described as a Schrödinger equation for the probabilistic description of a system. It has applications in other areas as well. A physical meaning can be assigned to the phase of the wavefunction. The approach allows construction of solutions of the Schrödinger equation from known solutions and conversely much like a Bäcklund transformation. Finally, the process of introducing a potential into the Schrödinger equation is generalized to the case of Bohmian mechanics.The first equation in a hierarchy of equations due to Vlasov is studied. The equation can be described as a Schrödinger equation for the probabilistic description of a system. It has applications in other areas as well. A physical meaning can be assigned to the phase of the wavefunction. The approach allows construction of solutions of the Schrödinger equation from known solutions and conversely much like a Bäcklund transformation. Finally, the process of introducing a potential into the Schrödinger equation is generalized to the case of Bohmian mechanics.

International Journal of Modern Physics BAug 20, 20200 citations

A new derivation of the schrödinger equation

H. N. Ioannidou

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoAug 1, 19827 citations

Schrödinger equation as a consequence of new Vlasov type equations

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