ABSTRACT, INTRODUCTION AND CONCLUSIONS OF A PAPER（1.3）
ABSTRACTWe have investigated various theories on solving Dirac equation, and found that there are many new paradoxes of mathematical and physical logics such as $1=0$ concealed in a multitude of literatures, in which the problems were covered up by some specious algorithm and explanations. Such as the forced injunction to select Dirac formula of the energy levels and stealthily delete the other necessary formula in the formal solutions. The energy eigenvalues for fine structure of atom given therein were actually speciously fabricated. As examples, five typical literatures that conceal the representative problems are commented on in the present paper. Some literatures seem to be purposive to make the inauthentic mathematical calculations so that the unreal deductions were given for mixing the false with the genuine therein. Some literatures seem to be unintentional to make the incorrect calculations but the ABC of differential equations such as the existence of solution and uniqueness of solution and so on were missed. Most of the paradoxes are unable to be solved, fewness could be solved in form but the formal corrected results are not in agreement with the order of nature.
INTRODUCTION
Many incorrectly mathematical methods and logic relations have been used for solving the Dirac equation, the corresponding theories actually destroy the uniqueness of solution coupled differential equations and conceal the basic paradox , which having been covered up by the specious explanations given in various literatures. By reappearance of those mathematical processes in detail, it was found that some basic problems of mathematics are missed. We sum up a kind of simple problems on solving Dirac equation with the pure Coulomb potential to formulate again those typical problems concealed in relativistic quantum mechanics. Because of the techniques of composition, the corresponding problems of mathematical calculations and physical logics seem to be unconquerable, but it is at least necessary for us to envisage the hidden historical events of coining some physical laws with the paradoxes. In fact, some pivotal problems concealed in the critical literatures should be discovered before they were published, but the corresponding theories have not been verified for tens of years. The emblematical logic difficulties such as and for the wave function and so on are call the paradox for solving Driac equations.
The one of the most important parts of quantum mechanics is just deriving and solving the wave equation with the correct boundary condition or initial value condition. One cannot violate any basic mathematical rule to piece together some physical formula for being agreement with the corresponding experiment data. For nonrelativistic and relativistic quantum mechanics, once a wave equation is introduced and the initial value conditions or the natural boundary conditions are determined, all remnant physics should be essentially mathematics . When looking from a mathematical point of view, any differential equation has the formal solution and the real solution respectively. From general solution, only the special solution that satisfies the uniqueness theorem of solution and the boundary condition or the initial value condition does denote the real solution of the differential equations. However, in many literatures on Dirac's relativistic quantum mechanics, these basic mathematical rules seem to be missed. It mainly behaves in two aspects. One is that the original Dirac equation of first-order, by being introduced the real transformation of functions, were transformed into two second-order differential equations of second-order, which have two conflicting formal eigensolution sets, violating uniqueness theorem of solution. Another is that some strange marks were introduced for constructing the so-called Schr鰀inger-like or Klein-Gordon-like equation, which conceals the breach to the uniqueness theorem of solution and the boundary condition, to coin the distinguished solution of the Driac equation. However, all of the corresponding paradoxes were covered up by some specious explanations. The main shelter is often the distinguished Dirac solution for the hydrogen and hydrogen-like atom. In fact, one can often use some completely incorrect methods to spell backward the Dirac formula of energy levels or its likeness. Tens of methods for giving imitated distinguished deduction of theoretical physics can be found, but they are not the real mathematics and physics.
Here we comment on five examples of literatures chosen from thousands of the similar literatures. We focus on the obvious mathematical paradox concealed in solving Dirac equations with the Coulomb potential. Because Darwin and Gordon firstly gave the correct methods of finding the solution in 1928, one easily judges the right and wrong for the logic of the critical literatures. How the mathematical paradoxes in those literatures take place, how the corresponding mathematical paradoxes have been covered up, and those typical theories of coining Dirac’s relativistic quantum mechanics were disclosed here.
The first critical literature is the book entitled “Anatomy on Choice Problems in Quantum Mechanics”published by Science press, the first edition in 1988. For recovering the distinguished Dirac formula of the energy eigenvalues from the so-called second-order differential equation, in this book, a real transformation of functions was introduced. However the obtained two second-order equations have different formal eigenvalues sets, violating the uniqueness of solution. However, one was chosen and another was stealthily deleted. This paradox cannot be solved.
The second critical literature entitled ``Simplified solutions of the Dirac-Coulomb equation'' published by Physical Review in 1985 constructed a new different system of first-order differential equations with the strange hyperbolic functions, and it was called the radial Dirac-Coulomb equation. In fact, it is actually not the Dirac equation but only a similar system of equations to the Driac equations. In form, the new radial Dirac-like equations of first-order were also transformed into the Schr鰀inger-like equation, in which four second-order differential equations were written in one form with another unnecessary sign. Making the mathematical calculation again, from the original Dirac-like equations one will obtain two correct Schr鰀inger-like equations with the same energy eigenvalues parameters, without the unnecessary sign in form. They are essentially different from the given form in the critical paper, and solving the coupled second-order equations violates the uniqueness theorem of solution. One only obtaining the Dirac formula of energy eigenvalues, by solving one of the second-order differential equations for the components of radial wave functions and hiding the other incompatible deduction of the energy eigenvalues from solving the other second-order differential equation, results in the same paradox of mathematical logic. The paradox is unable to solved yet.
The third critical literature is the paper entitled “A simpler solution of the Dirac equation in a Coulomb potential” published by American Journal of Physics in 1997. There is a completely incorrect calculation of differential coefficient for the introduced new function with the matrix coefficients therein. All of deductions that come from this incorrect formula are unable to be recovered. It hence forms another paradox. The comment for its pivotal mathematical problem ever incurred some censures with the pleonastic unreal mathematical calculations. In Fact, using the radial wave function given in the critical paper to compare with other form that come from any real mathematical calculations, one will find how the unreal and incorrect mathematical calculation lead to specious deductions. None of the paradox in this critical paper can be solved.
The fourth critical literature is the paper entitled “Solution of the Dirac equation with position-dependent mass in the Coulomb field” published by Physics Letters A in 2004 . Of course, the theory recently obtains to be recurred and developed, seeing the paper entitled “Relativistic scattering with a spatially dependent effective mass in the Dirac equation” published by Physical Reviews A in 2007 . In Ref. 18, it was alleged that an exact solution of the Dirac equation for a particle whose potential energy and mass were inversely proportional to the distance from the force centre was found. It is considered that the bound states exist provided the length scale a which appears in the expression for the mass is smaller than the classical electron radius and the bound states also exist for negative values of a even in the absence of the Coulomb interaction. In form, the quasarelativistic expansion of the energy was carried out, and a modified expression for the fine structure of the energy levels was given. However, by strictly verifying the mathematical procedure in the critical paper, one will find that the given exact solution of the corresponding second-order Dirac-Coulomb equations yielded from the first-order Dirac equations with the so-called position-dependent mass in the Coulomb field is completely incorrect. One will obtain the different deduction by correctly solving the differential equations. The given eigenvalues set and Klein-Gordon-like equations therein are all not necessary mathematical deduction of the original first-order Dirac equation. Two corrected Klein-Gordon-like equations for the new introduced components of wave function will be given. However, their complete eigensolutions still violate the uniqueness of solution. On the other hand, one directly solves the first-order Dirac-Coulomb equation with the so-called position-dependent mass of the electron will derive the new exact solution that is different from the given result in the critical paper as well as the Driac function with the new formula of energy eigenvalues. However, there are not any experimental data satisfying the new formula, implying the so-called exact solution directly from the first-order equations is only a formal solution. It forms new example of mathematical and physical paradox, being from the chimerical Dirac-Coulomb equation with the position-dependent mass of the electron. How to solve this paradox is still unbeknown.
The fifth critical literature, entitled “Nuclear size corrections to the energy levels of single-electron and muon atoms” published by Journal of Physics B: Atomic, Molecular and Optical Physics in 2005 , is still commented on here. In form, the paper gave the nuclear size corrections to the energy levels of single-electron and–muon atoms. However, the given solutions of the Dirac equations exterior to the nucleus in the critical paper still violate the uniqueness of solution but the two formal formulas of energy eigenvalues was forcedly written in one form. Comparing two formulas results in the typical paradox, , in mathematical method. On the other hand, the formal exact solutions dissatisfy the boundary condition because of being divergent at the origin of coordinate system for state, but the complete functions were separated in some factors so that the paradox from the boundary condition has been covered up. In particular, combining the potentials exterior to the nucleus and inside the nucleus to give a new potential is not a real potential to the Dirac equation. In the original paper, it seems to be unknown that the Dirac equations with harmonic potential inside the nucleus have no eigensolutions, however the papers gave some formal numerical solutions mixed the false with the genuine by the datasheets for the Dirac equations with nonexistence of eigensolutions in the introduced new field. Since many new formulas of the energy eigenvalues for the hydrogen atom have the same order of magnitude to the Dirac formula of energy eigenvalues, almost all pertinent special problems concealed in those papers are difficult to be discovered. CONCLUSIONS
Up to now, we have found that there are four kinds of mathematical prodexes concealed in solving the Dirac equation with the Coulomb potential. (a) Modify the original Dirac-Hamiltonian and use some contrived differential coefficient arithmetic without logic argumentation to construct second-order Dirac coulomb equations (b) Construct similar first-order Dirac equation that is essentially not the original Dirac equation with the Coulomb potential firstly to transform the unreal first-order wave equation into the second-order differential equations. (c) Introduce some transformations of functions to transform the original first-order differential equation into second-order differential equations. (d) Introduce incorrect potential to newly solve the Dirac equation and give some formal correction to the energy eigenvalues. In fact, introducing any transformation of functions for the original first-order Dirac equations must yield two uncoupled second-order differential equations for two new components of wave functions. Of course, those unreal second-order Dirac equations should be spurned. Almost all formal solutions of second-order differential equations conceal two logic paradoxes: violating the uniqueness theorem of solution and destroying the boundary condition. It clearly denotes that those theories on second-order Dirac equation are completely incorrect.
One can find that if someone fakes a formula that is just as the distinguished Dirac formula and similar to the some other famous formula of energy eigenvalues, the incorrect mathematics concealed in the theory are usually regarded as the progress of relativistic quantum mechanics. It is the important causation why the corresponding mathematical and physical paradoxes are very difficult to be discovered, and the incorrect methods are more and more widely used by later theories on solving the Dirac equation. We insist on that, all of theories on quantum mechanics must keep to the basic mathematical rules and physical logics, but not only for making up some specious deductions that are in form agreement with the known experimental data. It should be pointed out that many papers could not distinguish the formal solution and the real solution of the wave equation, and often consider the formal solution being the real solution. The formal solutions are those mathematical general solutions that maybe violate the uniqueness of solution and the condition for the determining solution. If the formal solutions of a wave equation with some introduced potential violate the uniqueness theorem of solution and the boundary condition, it should demonstrate the nonexistence of solution of the wave equation with the given potential. Stealthily deleting the other eigenvalues set for two uncoupled second-order Dirac-Coulomb equation and covering up the divergence of the formal wave function for S state are not real physical science. They are all the very serious logic mistakes. By this token, only by comparing the corresponding mathematical deduction with some experimental data can one judge if a theory is correct is usually disabled. For the wave equations, the boundary condition, the uniqueness theorem of solution and the existence of solution are the incontestable test standards in logic. All deductions given in the critical literatures are completely incorrect. Up to here, it has been found that many details for solving the Dirac equations are very important and should be soberly treated, such as the boundary conditions, the uniqueness theorem of solution, the existence of eigensolutions and the harmonic analysis and their homogeneous spaces and so on. For the Dirac equation in the Coulomb field, there still exist many problems which need to be completely solved.
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