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Journal of Karary University for Engineering and Science

Expression Time Dependent Schrödinger Equation from De Broglie Hypothesis and Hamiltonian function

Mobarak Tag, Adil Elrayah, Wael Hessen, Thora Abd Elradi

Published Dec 31, 2022• 0 citations
Publisher

Abstract

The Schrödinger and de Broglie hypothesis are directly derived from classical force and blank hypothesis by the quantum of total energy relation () and de Broglie relation. New time dependent Schrödinger equation is obtained by using wave function and non – relativistic wave function beside plank hypothesis .Also obtained new equation from TDSE to on additional kinetic energy and potential energy in classical equation which the result is corresponding to TDSE in 3D by using de Broglie hypothesis.

DOI
10.54388/jkues.vi.150

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The Schrödinger and de Broglie hypothesis are directly derived from classical force and blank hypothesis by the quantum of total energy relation () and de Broglie relation. New time dependent Schrödinger equation is obtained by using wave function and non – relativistic wave function beside plank hypothesis .Also obtained new equation from TDSE to on additional kinetic energy and potential energy in classical equation which the result is corresponding to TDSE in 3D by using de Broglie hypothesis.
The Schrodinger and de Broglie hypothesis are directly derived from classical force and blank hypothesis. Thus, depending on the quantum equations the quantum of total energy relation (E = T + V ) and de Broglie relation (? = h=p). Accordingly, new time dependent Schrodinger equation is obtained. As a result, by using the wave function and nonrelativistic wave function beside plank hypothesis. Moreover, TDSE was obtained, thus by used additional kinetic energy and potential energy (classical equation). Finally, a result corresponding to TDSE in 3D was confirmed by using De Broglie hypothesis.
In this study, both the Klein-Gordon equation and time-dependent Schrodinger equation are directly derived from De Broglie's (\lamda=h/p). Besides that, a new formula of the Klein-Gordon equation was obtained by using two equations, i.e., the classical wave function and relativistic total energy relation [[hw] = [(h^2 k^2)/2m] + V]. Moreover, a new equation of time-dependent Schrodinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, the previous result corresponds to (TDSE) in 3D calculated using two equations, (i.e., Einstein energy and Newton's second law of mechanics).
In this study, both of Klein-Gordon equation and time dependent Schrdinger equation are directly derived from De Broglie hypothesis (? = h=p). Besides that, a new formula of Klein- ordon equation was obtained by used two equations, i.e., the classical wave function and relativistic total energy relation hw = (h2k2)=2m+V ?. Moreover, a new equation of time dependent Schrdinger (TDSE) in three-dimension 3D was derived and evidently obtained. Thus, previous result is corresponding to (TDSE) in 3D calculated by using two equations, (i.e., Einstein energy and Newton second law of mechanics).
The Schrödinger equation was really postulated rather than derived. But we can see why it was postulated. It has been proven to be valid, in that it provides information about small, slow moving particles subject to a wide variety of potentials. The derivation of the time-dependent Schrödinger equation (TDSE) follows rather directly from assuming a free particle wave: Ψ = e i(kx-ωt) . If this wave function is plugged into the classical wave equation, and quantum principles are used, the two sides of the equation are not equal (this means this function does not "solve" the equation). The quantum principles required are an understanding of the photoelectric effect (by which E = ħω) and the de Broglie hypothesis (λ = h/p). Using these relationships, we show that one derivative of time is equal to a double derivative with respect to position, with a few proportionality factors mixed in. Thus, the TDSE for a free particle is just a statement that kinetic energy equals total energy . For a non-free particle (subject to a potential), potential must merely be added to the left-hand side of the equation, to yield kinetic energy plus potential energy equals total energy .
This article describes the concept of the time-dependent Schrödinger equation (TDSE) as a review to add a detailed understanding of the steps to formulate TDSE which can be seen from the classical mechanic's concept as a mechanical wave function. In this article, a comprehensive approach to the concepts of momentum and energy in particles will be described using operators working on wave functions. The results show that the use of momentum and energy operators can show that TDSE is a Hamiltonian system.
The proposed work presents the derivation of the wave function and the Schrödinger equation based on the dual tangential equation of time with an imaginary rate. The derivation uses de Broglie formulas. The wave and complex-conjugate functions are considered, which participate in the probabilistic description of the position of a particle in a potential field. Another interpretation of the product of these functions is presented. Based on it, the classical structure of the electron is derived. The derivation determines the quantum value of the imaginary incident vector. When deriving wave functions, simple solutions immediately arise for energies corresponding to radiation emitted by portions and a microscopic harmonic oscillator. In quantum mechanics, the first solution was used by Planck as a hypothesis when deriving his famous formula for describing equilibrium thermal radiation. The second energy solution was obtained from the Schrödinger equation by cumbersome calculations.