In this paper, we coupled the Quantum Mechanics conventional Schrödinger’s equation, for the particles, with the Maxwell’s wave equation, in order to study the potential’s role on the conversion of the electromagnetic field energy to mass and vice versa. We show that the dissipation (“conductivity”) factor and the particle implicit proper frequency are both related to the potential energy. We have also derived a new expression for the Schrödinger’s Equation considering the potential energy into this equation not as an ad hoc term, but also as an operator (Hermitian), which has the scalar potential energy as a natural eigenvalue of this operator.In this paper, we coupled the Quantum Mechanics conventional Schrödinger’s equation, for the particles, with the Maxwell’s wave equation, in order to study the potential’s role on the conversion of the electromagnetic field energy to mass and vice versa. We show that the dissipation (“conductivity”) factor and the particle implicit proper frequency are both related to the potential energy. We have also derived a new expression for the Schrödinger’s Equation considering the potential energy into this equation not as an ad hoc term, but also as an operator (Hermitian), which has the scalar potential energy as a natural eigenvalue of this operator.

Journal of Electromagnetic Analysis and ApplicationJan 1, 20150 citations

The Schrödinger Equation in Potential Representation

WORLD SCIENTIFIC eBooksAug 1, 20200 citations

Notes on quantum mechanics: the Schrodinger Equation

Nielsen, M. P., Laugesen, Jacob, Sanjayan, J. G.

Jan 1, 20010 citations

Theoretical Background

Anne‐Marie Sapse

Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation, $\left( { - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + v} \right)\Psi = {\rm E}\Psi,\hbar = \frac{h}{{2\pi }},$ or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression $H = - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + V;$ h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and $\frac{{ - {\hbar ^2}}}{{2m}}{\nabla ^2}$ the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation, $$ \left( { - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + v} \right)\Psi = {\rm E}\Psi,\hbar = \frac{h}{{2\pi }}, $$ or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression $$ H = - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + V; $$ h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and $$ \frac{{ - {\hbar ^2}}}{{2m}}{\nabla ^2} $$ the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.

Birkhäuser Boston eBooksJan 1, 20002 citations

INTRODUCTORY QUANTUM MECHANICS II: THE SCHRÖDINGER EQUATION

WORLD SCIENTIFIC eBooksJan 1, 20100 citations

Persistence of solvability in quantum systems deformed by Dunkl operators

Axel Schulze‐Halberg

We study persistence of solvability in nonrelativistic quantum systems with positiondependent mass upon introduction of a deformation by Dunkl operators. Conditions are derived for the governing Schrödinger equation of the conventional system to admit the same solutions as in the deformed case, up to a reparametrisation of coupling constants. These conditions require the positiondependent mass or the potential of the system to have a specific form. If this is the case for a particular system, then the Schrödinger equations for its conventional version and for the Dunkl-deformed partner share solutions in the same functional form.We study persistence of solvability in nonrelativistic quantum systems with positiondependent mass upon introduction of a deformation by Dunkl operators. Conditions are derived for the governing Schrödinger equation of the conventional system to admit the same solutions as in the deformed case, up to a reparametrisation of coupling constants. These conditions require the positiondependent mass or the potential of the system to have a specific form. If this is the case for a particular system, then the Schrödinger equations for its conventional version and for the Dunkl-deformed partner share solutions in the same functional form.

Acta PolytechnicaMay 7, 20250 citations

\left( { - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + v} \right)\Psi = {\rm E}\Psi,\hbar = \frac{h}{{2\pi }},

or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression

H = - \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2} + V;

h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and

\frac{{ - {\hbar ^2}}}{{2m}}{\nabla ^2}

the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.