Relativistic Invariance in Direct Derivation of a 4x4 Diagonal Matrix Schrodinger Equation from Relativistic Total Energy

Han J.F. Geurdes, Koji Nagata

Elena M. Ovsiyuk, A. Safronov, Alina V. Ivashkevich +1 more

The St¨uckelberg equation for a particle with two spin states, S = 1 and S = 0, is studied in the presence of an external uniform magnetic field. In relativistic case, the particle is described by an 11-component wave function. On the solutions of the equation, the operators of energy, the third projection of the total angular momentum, and the third projection of the linear momentum along the direction of the magnetic field are diagonalized. After separation of variables, we derive a system for 11 functions depending on one variable. We perform the nonrelativistic approximation in this system. For this we apply the known method of deriving nonrelativistic equations from relativistic ones, which is based on projective operators related to the matrix Γ0 of the relativistic equation. The nonrelativistic wave function turns out to be 4-dimensional. We derive the system for 4 functions. It is solved in terms of confluent hypergeometric functions. There arise three series of energy levels with corresponding solutions. This result agrees with that obtained for the relativistic St¨uckelberg equation.

H. N. Ioannidou

Eugene Paul Wigner

Mobarak Tag, Wail Hessen ALawad, Adil Elrayah +1 more

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).

Mobarak Tag, Adil Elrayah, Wael Hessen +1 more

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.