Volume 2 (2022)

12. The End of Teratogenic Simplified Dirac Hydrogen Equations

The two-component radial wave function of the Dirac equation of hydrogen is decomposed by linear combination function, which leads to the difference in the range of energy eigenvalues of the new first-order differential equations and the corresponding two simplified second-order differential equations constrained by the same energy parameter, belonging to the teratogenic simplified Dirac equation. The teratogenic simplification theory of the Dirac equation of hydrogen atom introduces the term “decoupling”, which is far from scientific logic, thus deleting a recursive relationship or corresponding second-order differential equation whose eigenvalue set does not meet the expectation, and only retaining the other one whose eigenvalue set value meets the expectation. Such an obvious logic problem has not been discovered or intentionally covered up, reflecting the real background of modern physics. Here, we first use the machine proof method to prove that the coefficient of the series solution of the teratogenic simplified first-order Dirac equation system should satisfy the linear recurrence relationship system without solution, which also proves that the teratogenic simplified first-order Dirac equation system has no eigensolution. Then the mathematical proof of the absence of solution of the teratogenicity simplified first-order Dirac equation system is given from different aspects. The simple truth is that the inconsistency of the eigenvalues of the two teratogenic simplified second-order differential equations destroys the existence and uniqueness theorem of the solutions of the differential equations. The essence of decoupling is to intentionally delete one of the two parallel second-order differential equations. The methods and conclusions do not conform to the unitary principle. It is concluded that the decoupled eigensolution of the teratogenic simplified first-order Dirac equation is pseudo-solution. The teratogenic simplified Dirac equation for the hydrogen-like atoms is therefore ended, and this conclusion is irreversible.

Citation: Dongfang, X. D. The End of Teratogenic Simplified Dirac Hydrogen Equations. Mathematics & Nature 2, 202202 (2022).

13. Dongfang Solution of Induced Second Order Dirac Equations

Solving the radial Dirac equation of the hydrogen atom, it usually follows the treatment method of Schrödinger equation of the hydrogen atom and expresses the two-component wave function as two new variables divided by the radial independent variables, thus transforming the equation into the induced first-order Dirac equation system. According to the induced first-order Dirac equations, two induced second order Dirac equations constrained by the same energy parameter can be obtained. Here, I study the eigensolutions of the induced second-order Dirac equation system of the hydrogen atom, and draw several unusual conclusions. The exact solution of the first-order induced Dirac equation system of hydrogen atom satisfies the induced second-order Dirac equation, but the complexity of correlation checking increases with the increase of the radial quantum number, and even the checking process of energy states with small radial quantum numbers is very complicated; The induced second-order Dirac equation of hydrogen atom has an eigensolution, and its energy eigenvalue is the same as that of the induced first-order Dirac equation; Different from the constraint of the coefficients of the two wave function components of the first-order Dirac equation system, the exact solutions of the two induced second-order equations are independent of each other, which means that the coefficients of the two component functions have their own normalized coefficients. The independence of the component function of the second order equation poses a new challenge to the physical meaning of the multi-component wave function of Dirac theory.

Citation: Dongfang, X. D. Dongfang Solution of Induced Second Order Dirac Equations. Mathematics & Nature 2, 202203 (2022).

14. The End of Isomeric Second Order Dirac Hydrogen Equations

Biedenharn and Wong wrote the Dirac equation in the form that the combination of the differential term and the function term is equal to zero, then changed the positive and negative sign of the mass term and removed the wave function to extract a mixing operator, and then used this mixing operator to act on the first-order Dirac equation. The resulting second-order equation is called the isomeric second-order Dirac equation. Because the equations in mathematical sense can be constructed arbitrarily, the isomeric second-order Dirac equation can exist as a pure differential equation. However, as the wave equation of quantum mechanics, the isomeric second-order Dirac equation advocated by famous journals seriously lacks scientific basis and destroys mathematical principles, and the processing of isomeric second-order Dirac equation is completely false calculation. Here, the real heterogeneous second-order Dirac equation is first derived, and it is proved that it is a system of equations composed of four non-solvable second-order partial differential equations of four wave function components. Then it is proved that the highly respected second-order Dirac equation of single-component wave function isomerism is forged, and the Dirac energy level formula of hydrogen atom pieced together is only a prop to cover up the above lies. Then it is proved that the most ideal isomeric second-order radial Dirac equation of hydrogen atom is also an unsolvable differential equation system composed of at least two partial differential equations, which further illustrates the fraud of the highly respected isomeric second-order Dirac equation of single-component wave function. Finally, it is proved that the mixed operator method for constructing the isomeric second-order Dirac equation destroys the unitary principle and leads to many confused and uncertain conclusions. The results of these rigorous calculations declare the end of the heterogeneous second-order Dirac equation and the mixed operator method itself used for the construction of heterogeneous wave equations.

Citation: Dongfang, X. D. The End of Isomeric Second Order Dirac Hydrogen Equations. Mathematics & Nature 2, 202204 (2022).

15. The End of True Second Order Dirac Hydrogen Equation

The original Coulomb field radial Dirac equation is essentially a first order differential system of two-component wave functions. The second order differential equation of the original wave function component directly converted from the first order differential equation set is called the true second order Dirac equation. Relativistic quantum mechanics usually ignores the physical meaning of wave function and only focuses on the energy eigenvalue. So, the main expectation of solving the true second order Dirac equation of hydrogen-like atoms is that the Dirac energy level formula is the eigenvalue of the equation. Here I derive two true second order Dirac equations that are mutually independent in form but actually constrained by common energy parameters, and then use the traditional boundary conditions of hydrogen-like atoms to solve the true second order Dirac equation. The conclusion drawn from this is not exactly the same as the traditional understanding. 1) The formal solution of the true second order Dirac equation satisfying the traditional boundary conditions takes the Dirac hydrogen level formula as the energy eigensolution, which seems to meet the expectation; 2) However, when the radial quantum number is 0, regardless of the value of the angular quantum number, the complete expression of the wave function as the exact solution of the equation diverges at the coordinate origin, which does not meet the traditional boundary conditions, which means that the universe is collapsed and does not conform to the fact of the universe structure. From this it is concluded that the Dirac energy level formula is only the formal eigenvalue of the true second order Dirac equation that does not conform to the physical meaning. This announced the end of the expectation of using traditional boundary conditions to solve the true second order Dirac equation to naturally obtain the Dirac energy level formula. This result will promote the re-study of the exact solution of the original Dirac equation.

Citation: Dongfang, X. D. The End of True Second Order Dirac Hydrogen Equation. Mathematics & Nature 2, 202205 (2022).

16. Dongfang Challenge Solution of Dirac Hydrogen Equation

In quantum mechanics, the Schrödinger equation is used to describe the bound state system, and the exact solution of the equation needs to be determined by boundary conditions. However, the size of an atomic nucleus is usually not considered, and the boundary condition that the proposed wave function should meet is only a rough form. The rough boundary condition causes the S-state wave function of the exact solution of the Klein-Gordon equation and Dirac equation of the so-called relativistic quantum mechanics describing the bound state of the Coulomb field to diverge at the coordinate origin and makes the hydrogen-like atom with the nuclear charge number Z>137 appear unreal virtual energy. The divergence of the wave function means that the probability density of the electron or meson appearing near the nucleus will increase rapidly. What it predicts is the untrue conclusion that the hydrogen-like atom in S-state will collapse rapidly into a neutron-like atom. Considering that the atomic nucleus has a certain radius, the exact boundary conditions that the wave function should satisfy are given here, and then the Dirac equation of hydrogen-like atom is solved again, and a new exact solution without wave function divergence and virtual energy is obtained. Surprisingly, the exact boundary condition makes the angular quantum number naturally regress to the eigenvalue determined by the exact solution of the equation. It has no contribution to the quantized energy, which means that the angular quantum number constructed by Dirac electron theory is denied. Unlike the Dirac energy level formula, the new energy level formula corresponding to the solution of the Dirac equation for the hydrogen-like atom satisfying the exact boundary conditions has no so-called fine structure, and its accuracy is equivalent to the accuracy of the Bohr energy level. The exact boundary condition solution poses a serious challenge to the Dirac relativistic quantum mechanics, which is called the challenging solution of the Dirac equation for the hydrogen atom.

Citation: Dongfang, X. D. Dongfang Challenge Solution of Dirac Hydrogen Equation. Mathematics & Nature 2, 202206 (2022).

17. Neutron State Solution of Dongfang Modified Dirac Equation

The challenging solution of the Dirac equation of the Coulomb field satisfying exact boundary conditions is further studied. If the Dirac equation is effective, then the intrinsic angle quantum number determined by the exact solution of the equation must be introduced to modify the Dirac equation to make it self-consistent. The solution of the modified Coulomb field Dirac equation satisfying the exact boundary conditions leads to a variety of breakthrough conclusions that overturns the traditional physical thinking. 1) The modified Dirac equation of Coulomb field has a neutron state solution corresponding to the neutron structure, and the binding energy of the neutron has a certain value, while the calculation result of the intrinsic radius written in the accurate boundary condition is equivalent to the size of the atomic nucleus; 2) The energy eigenvalue formula of the modified Coulomb field Dirac equation contains only radial quantum numbers and is independent of the intrinsic angular quantum numbers, where the zero radial quantum number energy level corresponds to the neutron state, and the nonzero radial quantum number energy level corresponds to the atomic state, and the accuracy of each atomic state energy level is equivalent to the Bohr energy level, while the Dirac energy level formula as the expected solution no longer exists; 3) The intrinsic angular quantum number of the modified Coulomb field Dirac equation indirectly negates the Dirac algebra theory that constructs the Dirac angular quantum number beyond the mathematical calculation rules; 4) The neutron state wave function component of the modified Coulomb Dirac equation is the terminated Yukawa potential function, which reflects the physics dilemma that the wave function is wrongly described as a potential function to establish a Yukawa pseudo-scientific theory that can also be infinitely developed and admired by physicists around the world, exposing the false prosperity of modern physics. It is concluded that the Dirac equation of the Coulomb field defined by Dirac algebra is not self-consistent, and the exact boundary condition solution of the modified self-consistent Dirac equation of the angular quantum number regression intrinsic physical quantity negates the Dirac electronic theory of fabricating the energy level formula of the fine structure of the hydrogen atom spectrum, and the microscopic quantum theory urgently needs to find a more reasonable wave equation that describes the fine spectral structure of the hydrogen atom.

Citation: Dongfang, X. D. Neutron State Solution of Dongfang Modified Dirac Equation. Mathematics & Nature 2, 202207 (2022).

18. Ground State Solution of Dongfang Modified Dirac Equation

In order to deal with the mathematical contradiction that the Dirac wave function does not meet the definite solution condition, an effective and reasonable solution is to replace the traditional rough boundary condition with the precise boundary condition in which the nuclear radius is written. The exact solution of the hydrogen-like atom self-consistent Dirac equation satisfying the exact boundary conditions has subversive physical significance. It also shows a new mathematical point of view, that is, the boundary parameters become one of the eigensolutions of the equation, and the solutions of the equation may be completely different due to the slight difference of the boundary conditions. Dongfang modified Dirac hydrogen equation replaces the angular quantum number defined by the illogical Dirac electron theory with the intrinsic angular quantum number determined by the exact solution of the equation. Here I further study of the ground state solution of the modified Dirac equation which satisfies the exact boundary conditions. The results show that the ground state of the modified Dirac equation for hydrogen-like atoms is a triple degenerate state with three intrinsic angular momentum and three intrinsic wave functions; the intrinsic angular momentum of the ground state is neither the angular momentum constructed by the anti-logic of Dirac electron theory nor the angular momentum of Schrödinger equation. The two components of one of the intrinsic wave functions of the ground state are linearly related. The existence of the exact solution of the intrinsic ground state and the essential difference between the intrinsic ground state energy level and the Dirac ground state energy level further illustrate that the angular momentum constructed by the so-called Dirac algebra is not a corollary of scientific logic, and the Dirac equation cannot explain the fine structure of hydrogen-like atoms. It is only one of the most puzzling representatives of modern physics as the basic equation of quantum field theory. However, because Dirac equation contains rich mathematical problems and unique processing technology, it will help the development of mathematical theory to incorporate it into mathematics textbooks as a new mathematical model.

Citation: Dongfang, X. D. Ground State Solution of Dongfang Modified Dirac Equation. Mathematics & Nature 2, 202208 (2022).

19. The End of Dirac Hydrogen Equation in One Dimension

The angular quantum number and radial momentum operator of Dirac’s electron theory belongs to the formal mathematical definition, rather than the result of standard mathematical calculation. Whether such formal mathematical definition that changes the fundamental nature of physical logic is reasonable or not can be judged according to whether the definitions of the radial momentum operator and angular momentum eigenvalue are consistent with the calculation results of standard mathematics. Selecting a specific physical model can find the exact answer to the problem, which forces us to seriously deal with the similarities and differences between the formal mathematical definition and the standard mathematical calculation results. Here I discuss the one-dimensional Dirac equation model of hydrogen-like atom and give the standard mathematical calculation conclusion: the one-dimensional Dirac equation has four non-equivalent first-order differential equation systems; the four first-order differential equation systems can also be converted into a system of differential equations composed of two second-order differential equations with the same energy parameter. The mathematical process of accurately solving the general solution of the second-order differential equations is beyond the existing mathematical basis. However, it can be proved that the exact solution of the ground state and a specified excited state of the first-order differential equations does not exist, It means that the first-order Dirac differential equations and the corresponding second-order differential equations of the hydrogen-like atom in the one-dimensional case have no energy quantized eigensolutions. The expression of the radial momentum operator here has a clear conclusion that is different from the definition of Dirac electron theory. The one-dimensional Dirac hydrogen equation is actually a special case of zero angular momentum. The specific zero angular momentum is intentionally avoided by Dirac electron theory because its existence exposes the serious non-consistency of the Dirac equation. This proves that a large number of mathematical calculations on the exact solution of the one-dimensional Dirac equation of the hydrogen atom, which is respected by various scientific documents, belong to the spurious calculation of the expected results. The one-dimensional Dirac equation is thus ended.

Citation: Dongfang, X. D. The End of Dirac Hydrogen Equation in One Dimension. Mathematics & Nature 2, 202209 (2022).

20. Multiple Morbid Mathematics of Dirac Electron Theory

Citation: Dongfang, X. D. Multiple Morbid Mathematics of Dirac Electron Theory. Mathematics & Nature 2, 202210 (2022).