Hydrodynamical equations for the probability density \ensuremath{\rho} and the local velocity v of a non-Markovian stochastic process have been recently obtained. The corresponding complex equation in terms of the wave function \ensuremath{\psi} coincides with the Schr\"odinger equation if the first-order terms only are retained. The second-order terms vanish and the third-order terms could be considered as QED-type corrections. To obtain the coefficient ${\mathit{D}}_{4}$ of the third-order term the probability density \ensuremath{\rho} for a free particle is obtained by stochastic electrodynamics. This solution is substituted in the new complex equation, thus obtaining ${\mathit{D}}_{4}$ . By this coefficient the calculation of the Lyman-\ensuremath{\alpha} wavelength is in agreement with the best direct measurements for the transition 2P\ensuremath{\rightarrow}1S. The 1S displacement is 1% of the QED value for the Lamb shift. It is suggested how the new negative contribution could be the difference between the (1+0.01) QED value for the 3S state and the corresponding experimental value, which is 20% less than that given by QED.Hydrodynamical equations for the probability density \ensuremath{\rho} and the local velocity v of a non-Markovian stochastic process have been recently obtained. The corresponding complex equation in terms of the wave function \ensuremath{\psi} coincides with the Schr\"odinger equation if the first-order terms only are retained. The second-order terms vanish and the third-order terms could be considered as QED-type corrections. To obtain the coefficient ${\mathit{D}}_{4}$ of the third-order term the probability density \ensuremath{\rho} for a free particle is obtained by stochastic electrodynamics. This solution is substituted in the new complex equation, thus obtaining ${\mathit{D}}_{4}$. By this coefficient the calculation of the Lyman-\ensuremath{\alpha} wavelength is in agreement with the best direct measurements for the transition 2P\ensuremath{\rightarrow}1S. The 1S displacement is 1% of the QED value for the Lamb shift. It is suggested how the new negative contribution could be the difference between the (1+0.01) QED value for the 3S state and the corresponding experimental value, which is 20% less than that given by QED.

Physical review. B, Condensed matterFeb 1, 199112 citations

Towards the Schrödinger Equation

Jochen Pade

Undergraduate lecture notes in physicsJan 1, 20180 citations

Towards the Schrödinger Equation

Jochen Pade

Undergraduate lecture notes in physicsNov 7, 20130 citations

Derivation of the Schrödinger equation from QED

Spyros Efthimiades

Abstract The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.Abstract The Schrödinger equation relates the emergent quantities of wavefunction and electric potential and is postulated as a principle of quantum physics or obtained heuristically. However, physical consistency requires that the Schrödinger equation is a low-energy dynamical condition we can derive from the foundations of quantum electrodynamics. Due to the small value of the electromagnetic coupling constant, we show that the electric potential accurately represents the contributions of intermediate low-energy photon exchanges. Then, from the total nonrelativistic energy relation, we see that the dominant term of the electron wavefunction is a superposition of plane waves that satisfies the Schrödinger equation. Our derivation shows that the Schrödinger equation is not an energy conservation relation because its middle term does not represent the electron kinetic energy as assumed. We analyze the physical content of the Schrödinger equation and verify our assessments by calculating and evaluating the physical quantities in the ground state of the hydrogen atom. Furthermore, we explain why nonrelativistic quantum dynamics differs from classical dynamics. Undergraduate students can follow the derivation because it involves fundamental physical concepts and mathematical expressions, and we explain every step.

arXiv:2212.13103European Journal of PhysicsFeb 22, 20242 citations

The energy level structure of the modified Schrodinger equation can be consistent with Lamb shift

Yukuo Zhao, Dong, Yu-xin

In the literature of calculating atomic and molecular structures, most Schrodinger equations are described by Coulomb potential. However, there are also a few literatures that discuss some magnetic correction methods, such as Pauli and Shortley's early work. But in fact, the calculation accuracy of these Schrodinger equations is not consistent with Lamb shift. Therefore, in the traditional ab initio calculation of quantum mechanics, it is common and necessary to use Dirac theory or quantum electrodynamics (QED) to correct the energy level of Schrodinger equation. However, the calculation of Feynman diagram is a daunting problem, including the application of self-consistent field in relativity and density functional theory. So recently, we have noticed the simplicity of the modified Newtonian mechanics, and we think that quantum mechanics will have similar properties. Here, we state this and improve the correction function in our previous action potential. In addition, through the demonstration of hydrogen-like and helium-like systems here, it can be proved that this conclusion is a potential application, that is, the energy level structure of our modified Schrodinger equation is consistent with Lamb shift.In the literature of calculating atomic and molecular structures, most Schrodinger equations are described by Coulomb potential. However, there are also a few literatures that discuss some magnetic correction methods, such as Pauli and Shortley's early work. But in fact, the calculation accuracy of these Schrodinger equations is not consistent with Lamb shift. Therefore, in the traditional ab initio calculation of quantum mechanics, it is common and necessary to use Dirac theory or quantum electrodynamics (QED) to correct the energy level of Schrodinger equation. However, the calculation of Feynman diagram is a daunting problem, including the application of self-consistent field in relativity and density functional theory. So recently, we have noticed the simplicity of the modified Newtonian mechanics, and we think that quantum mechanics will have similar properties. Here, we state this and improve the correction function in our previous action potential. In addition, through the demonstration of hydrogen-like and helium-like systems here, it can be proved that this conclusion is a potential application, that is, the energy level structure of our modified Schrodinger equation is consistent with Lamb shift.

arXiv:2201.10174Jan 25, 20220 citations

Beyond the Schrödinger Equation

Luis de la Peña, Ana Marı́a Cetto, Andrea Valdés Hernández

Springer eBooksJul 15, 20141 citation

{\mathit{D}}_{4}

of the third-order term the probability density \ensuremath{\rho} for a free particle is obtained by stochastic electrodynamics. This solution is substituted in the new complex equation, thus obtaining

{\mathit{D}}_{4}

. By this coefficient the calculation of the Lyman-\ensuremath{\alpha} wavelength is in agreement with the best direct measurements for the transition 2P\ensuremath{\rightarrow}1S. The 1S displacement is 1% of the QED value for the Lamb shift. It is suggested how the new negative contribution could be the difference between the (1+0.01) QED value for the 3S state and the corresponding experimental value, which is 20% less than that given by QED.