Established pseudo solution of second-order Dirac-Coulomb equation with position-dependent mass
aa r X i v : . [ phy s i c s . g e n - ph ] J u l Established pseudo solution of second-order Dirac-Coulomb equation withposition-dependent mass
Ruida Chen
Shenzhen Institute of Mathematics and Physics, Shenzhen, 518028, China
We show that one of exact solutions of the second-order Dirac-Coulomb equation werepseudo. In the corresponding original literature, it was considered that the mass of theelectron with a Coulomb potential was position-dependent, but the obtained eigenvalues setwas not the inevitable mathematical deduction of the given second-order Dirac equation,and the second-order Dirac equations were not the inevitable mathematical deduction of thegiven couplet first-order Dirac equation with the position-dependent mass of the electron.In the present paper, we obtain the correct solution of the introduced first-order differentialequations. This new solution would be tenable only when the wave equation is correct, butthere is not any experiment date to validate the so-called position-dependent of the electronin the Coulomb field.
PACS numbers: 03.65.Pm,03.65.GeKeywords: position-dependent mass, Dirac equation, pseudo solution.
I. INTRODUCTION
An exact solution of the Dirac equation with the so-called position-dependent mass of theelectron in the Coulomb field was introduced[1]. According to the article, in atomic units ( ~ = m = 1), the spherically symmetric singular mass distribution was taken as follows m ( r ) = 1 + µλ (cid:14) r (1)where λ is the Compton wavelength ~ / m c = c − , and µ is a real scale parameter with inverselength dimension. The author presented some reasons why he introduced the position-dependentmass of the electron. We find the reasons were independent of any physical logic. For example,it was of that the rest mass of the particle ( m = 1) was obtained either as the asymptotic limit( r → ∞ ), or the nonrelativistic limit ( λ →
0) of m ( r ), consequently, a possible interpretation forthis singular mass term might be found in relativistic quantum field theory. It was even told ofthat it should also be noted that this position-dependent mass term has a relativistic origin aswell since it was proportional to the Compton wavelength which vanishes as c → ∞ (equivalently, λ → m and relativistic result. Of course, at present, we have to transitorilyavoid such questions and only check the corresponding mathematic deduction procedure. We showthat the original solution of the second-order Dirac equation with the so-called position-dependentmass was incorrect, and the given second-order Dirac equation is not the necessary mathematicaldeduction of the given first-order Dirac equation with position-dependent mass of the electron inthe Coulomb field. We introduce the correct exact solution of the original first-order differentialequation only for further showing that the original solution includes many mathematical mistakes,and don’t think the supposition of the position-dependent mass of the electron in the Coulombfield is correct. II. ESTABLISHED SOLUTION OF DIRAC EQUATION WITHPOSITION-DEPENDENT MASS
In order to solve the Dirac equation with the so-called position-dependent mass term in theCoulomb field, the spinor wavefunction was written as follows ψ = i [ g ( r )/ r ] χ jιm [ f ( r )/ r ] ~σ · χ jιm (2)where f and g are real radial functions, ˆ r is the radial unit vector, and the angular wavefunctionwith the spherical harmonic function Y m − ι for the two-component spinor was written as χ jιm = 1 √ ι + 1 p ι ± m + 1/2 Y m − ι ∓ p ι ∓ m + 1/2 Y m +1/2 ι (3)then the following 2 × λ Z + µr − ε λ (cid:0) kr − ddr (cid:1) λ (cid:0) kr + ddr (cid:1) − λ Z − µr − ε g ( r ) f ( r ) = 0 (4)where ε is the relativistic energy which is real. By introducing some signs and using the globalunitary transformation, it was alleged that, the Schrodinger-like wave equation was finally obtained.The main steps are as follows. The Schrodinger-like requirement dictates that the parameter η satisfies the constraint Cµ + Sκ / λ = ± Z (5)where S = sin ( λη ), C = cos ( λη ) and − π /2 ≤ λη ≤ π /2, κ = ± , ± , · · · . The solution of theconstraint (5) gives two angles whose cosines are C = (cid:0) µ + κ (cid:14) λ (cid:1) − " ± µZ + | κ | λ r(cid:16) κλ (cid:17) + µ − Z > C − ε + (1 ± λ Zr λ (cid:0) − sλ + γr − ddr (cid:1) λ (cid:0) − sλ + γr + ddr (cid:1) − C − ε + (1 ∓ λ Zr φ + ( r ) φ − ( r ) = 0 (7)where γ = | κ | κ p κ + λ ( µ − Z ) and φ + φ − = cos λη sin λη − sin λη cos λη gf (8)equation (7) gives one spinor component in terms of the other as follows φ ± = λC ± ε (cid:18) ± Sλ ∓ γr + ddr (cid:19) φ ∓ (9)whereas, the resulting Schr¨odinger–like wave equation becomes (cid:20) − d dr + γ ( γ ± r + 2 Zε + µr − ε − λ (cid:21) φ ∓ ( r ) = 0 (10)the author comparing this equation with that of the well-known nonrelativistic Coulomb problemwith constant mass and alleged that he had new discover for the relativistic spectrum ε ιn = " (cid:18) λZn + ι + 1 (cid:19) − " − λ µZ ( n + ι + 1) ± s λ Z − µ ( n + ι + 1) (11)namely ε ιn = − λ µZ ± q ( n + ι + 1) + λ ( Z − µ ) ( n + ι + 1) ( n + ι + 1) + ( λZ ) (12)where ι stands for either one of the four possible alternative values in the following expressionassociated independently with φ ± and ± κ ≥
1. For φ + ι → γ or ι → − γ − , Z → Z ε + µ , E → (cid:0) ε − (cid:1)(cid:14) λ (13)and for φ − ι → γ − ι → − γ, Z → Z ε + µ , E → (cid:0) ε − (cid:1)(cid:14) λ (14)It is similar to the above procedure, writing the paper about the relativistic quantum mechanics,many authors did not introduce their detail operation steps on how to obtain those necessarytransition equation and their new mathematical result. They only alleged that those formulas intheir paper are necessary deduction. We don’t understand what the expression (9) means? III. EINGEVALUES-SET (12) DISOBEY UNIQUENESS OF SOLUTION OF WAVEEQUATION
For the same quantum system, it should have only one of eigenvalues set for any theory. Theformula (12) of the energy levels includes two eigenvalues set corresponding to different definition ι .It is one of the mathematical contradictions of the articles. Consequently we cannot believe theformula (12) is the real energy eigenvalues set in the Coulomb field. It seems that the author isnot up on the method of finding the eigensolutions of the second-order differential equations withvariable coefficients. Why don’t we directly solve the second-order differential equation (10) now?Now, one can write the equation (10) in the separate form h − d dr + γ ( γ +1) r + 2 Zε + µr − ε − λ i φ ( r ) = 0 h − d dr + γ ( γ − r + 2 Zε + µr − ε − λ i ψ ( r ) = 0 (15)using the asymptotic solution φ ( r ) ∼ exp (cid:16) − r √ − ε . λ (cid:17) , ψ ( r ) ∼ exp (cid:16) − r √ − ε . λ (cid:17) satisfyingthe boundary condition at r →
0, we seek the formal solution φ ( r ) = e − √ − ε uλ r u, ψ ( r ) = e − √ − ε vλ r v (16)it easily obtained that dφ ( r ) dr = e − √ − ε uλ r (cid:18) dudr − √ − ε u λ u (cid:19) d φ ( r ) dr = e − √ − ε uλ r (cid:18) d udr − √ − ε u λ dudr + − ε u λ u (cid:19) dψ ( r ) dr = e − √ − ε vλ r (cid:18) dvdr − √ − ε v λ v (cid:19) d ψ ( r ) dr = e − √ − ε vλ r (cid:18) d vdr − √ − ε v λ dvdr + − ε v λ v (cid:19) (17)substituting (16) and (17) into (15), it educes that d udr − √ − ε u λ dudr − γ ( γ +1) r u − Zε u + µr u = 0 d vdr − √ − ε v λ dvdr − γ ( γ − r v − Zε v + µr v = 0 (18)finding the power series solution of the above equations, it assumed that u = ∞ X n =0 b n r s u + n , v = ∞ X n =0 d n r s v + n (19)so that dudr = ∞ P n =0 ( s u + n ) b n r s u + n − dvdr = ∞ P n =0 ( s v + n ) d n r s v + n − d udr = ∞ P n =0 ( s u + n ) ( s u + n − b n r s u + n − d vdr = ∞ P n =0 ( s v + n ) ( s v + n − d n r s v + n − (20)substituting (19) and (20) into the equations (17), we have ∞ P n =0 [( s u + n ) ( s u + n − − γ ( γ + 1)] b n − (cid:20) √ − ε u λ ( s u + n −
1) + ( Zε u + µ ) (cid:21) b n − r s u + n − = 0 ∞ P n =0 [( s v + n ) ( s v + n − − γ ( γ − d n − (cid:20) √ − ε v λ ( s v + n −
1) + ( Zε v + µ ) (cid:21) d n − r s v + n − = 0 (21)finally we obtain the two recursive relation for the power series (19)[( s u + n ) ( s u + n − − γ ( γ + 1)] b n − (cid:20) √ − ε u λ ( s u + n −
1) + ( Zε u + µ ) (cid:21) b n − = 0[( s v + n ) ( s v + n − − γ ( γ − d n − (cid:20) √ − ε v λ ( s v + n −
1) + ( Zε v + µ ) (cid:21) d n − = 0 (22)using the initial value condition b − = b − = · · · = 0, d − = d − = · · · = 0and b = 0, d = 0, let n = 0 and substitute it into the above recursive relation, it educes that s u ( s u − − γ ( γ + 1) = 0 s v ( s v − − γ ( γ −
1) = 0 (23)this gives s u = 1 + γ, s u = − γ ; s v = γ, s v = 1 − γ (24)in order to that the whole solutions (2) of the original equation satisfy the boundary condition, wehave to choose s u = 1 + γ, s v = γ (25)hence u = ∞ X n =0 b n r γ + n , v = ∞ X n =0 d n r γ + n (26)and φ ( r ) = e − √ − ε uλ r ∞ X n =0 b n r γ + n , ψ ( r ) = e − √ − ε vλ r ∞ X n =0 d n r γ + n (27)the power series must be cut off so that the whole wave function is limit at r → ∞ . It is assumedthat b n r = 0 , d n r = 0 and b n r +1 = b n r +2 = · · · = 0, d n r +1 = d n r +2 = · · · = 0. According to therecursive relation (22), let n = n r + 1, we obtain − (cid:20) √ − ε u λ ( s u + n r ) + ( Zε u + µ ) (cid:21) b n r = 0 − (cid:20) √ − ε v λ ( s v + n r ) + ( Zε v + µ ) (cid:21) d n r = 0 (28)it requires that √ − ε u λ (1 + γ + n r ) + ( Zε u + µ ) = 0 √ − ε v λ ( γ + n r ) + ( Zε v + µ ) = 0 (29)if we think little of, we would obtain the formal solution ε u = − λ Zµ + q (1+ γ + n r ) + λ Z [ λ µ +(1+ γ + n r ) ] λ Z +(1+ γ + n r ) ε u = − λ Zµ − q (1+ γ + n r ) + λ Z [ λ µ +(1+ γ + n r ) ] λ Z +(1+ γ + n r ) ε v = − λ Zµ + q ( γ + n r ) + λ Z [ λ µ +( γ + n r ) ] λ Z +( γ + n r ) ε v = − λ Zµ − q ( γ + n r ) + λ Z [ λ µ +( γ + n r ) ] λ Z +( γ + n r ) (30)in form, these results as the inevitable deductions of the second-order differential equations (14)include omnifarious logic problems.a) The solutions (30) are different from the formula (12). It shows that the original formula (12)is incorrect for the second-order differential equations (4).b) For the same quantum system described by the equations (14), the four eigenvalues setsof the energy levels disobey the uniqueness of the solution of the differential equations. Whicheigenvalues set is correct?c) Is the relativistic energy the positive number or the negative number? If we delete thenegative energy solution, we also have two eigenvales set corresponding the positive energy. Theyalso disobey the uniqueness of the eignesolutions set for the differential equation.d) It is the most serious that, the solutions (30) implying the (12) are the formal solution.Because of the definition γ = | κ | κ p κ + λ ( µ − Z ) given in the original article, only when κ < κ valuesconstructed by Dirac are actually ± , ± , · · · , also given in the original articles.All of these problems are the mathematical and physical contradictions. Consequently, we don’tthink the original solution (12) is not deceitful solution. IV. SCHR ¨ODINGER-LIKE EQUATION (10) IS BOGUS
We even doubt that the second-order equation (10) is the correct deduction of the originalcoupled first-order equation (4). By all appearances, from (4) to (10), it actually introduces thetransformation (8) with (9), namely λC + ε (cid:0) Sλ − γr + ddr (cid:1) φ − λC − ε (cid:0) − Sλ + γr + ddr (cid:1) φ + = cos λη sin λη − sin λη cos λη gf (31)it gives the separate form g = λC + ε ( Sλ − γr + ddr ) φ − − λC − ε ( − Sλ + γr + ddr ) φ + cos λη +sin λη f = λC + ε ( Sλ − γr + ddr ) φ − + λC − ε ( − Sλ + γr + ddr ) φ + cos λη +sin λη (32)these transformations cannot translate the equations (4) namely (cid:16) λ Z + µr − ε (cid:17) g ( r ) + λ (cid:0) kr − ddr (cid:1) f ( r ) = 0 λ (cid:0) kr + ddr (cid:1) g ( r ) − (cid:16) − λ Z − µr − ε (cid:17) f ( r ) = 0 (33)into any Schr¨odinger-like second-order equations, and the second-order equation (10) give in theoriginal paper is not correct. V. CORRECT SOLUTION OF THE CORRESPONDING FIRST-ORDER EQUATION
Because there has been not any experiment data to approve the imagination, we don’t thinkthat the mass of electron in the Coulomb field is really dependent on position. Consequently, inprinciple, it should be meaningles to find the correct solution of the equation (4) or (33). When onlylooking from a mathematical point of view, we give the correct exact solution of the equations (4) or(33). It is well known that directly solving the original coupled first-order equation (4) namely (33)is simpler out and away than translating it into the so-called Schr¨odinger-like equation to obtainthe exact solution. One firstly note the behavior of f ( r ) and g ( r ) for g → ∞ , since neglecting theterms proportional to 1/ r the differential equations (33) read(1 − ε ) g − λ dfdr = 0 , λ dgdr + (1 + ε ) f = 0 (34)it follows immediately that d fdr + 1 − ε λ f ∼ , d gdr + 1 − ε λ g ∼ ε . However, it isconsidered all along that the relativistic energy in the Coulomb field satisfy the condition 0 < E 1. According to (13) and (14), it only gives 1 < ε < λ + 1. For the moment,it is considered that ε > 1. We obtain the asymptotic solutions of the equations (33) f ∼ e − √ ε − λ r , g ∼ e − √ ε − λ r (36)and the exact solution of the equations (33) take form f = e − √ ε − λ r u, g = e − √ ε − λ r v (37)they give dfdr = e − √ ε − λ r dudr − √ ε − λ e − √ ε − λ r u dgdr = e − √ ε − λ r dvdr − √ ε − λ e − √ ε − λ r v (38)substituting for equations (33), we have λ dudr − (cid:16) λkr + √ ε − (cid:17) u + h ( ε − − λ ( Z + µ ) r i v = 0 λ dvdr + (cid:16) λkr − √ ε − (cid:17) v + h ( ε + 1) − λ ( Z − µ ) r i u = 0 (39)finding the power series solution, put v = ∞ X n =0 b n r σ + n , u = ∞ X n =0 d n r σ + n (40)substitute it into the above equations, we obtain λ ∞ P n =0 ( σ + n ) d n r σ + n − − (cid:16) λkr + √ ε − (cid:17) ∞ P n =0 d n r σ + n + h ( ε − − λ ( Z + µ ) r i ∞ P n =0 b n r σ + n = 0 λ ∞ P n =0 ( σ + n ) b n r σ + n − + (cid:16) λkr − √ ε − (cid:17) ∞ P n =0 b n r σ + n + h ( ε + 1) − λ ( Z − µ ) r i ∞ P n =0 d n r σ + n = 0 (41)it predigests that ∞ P n =0 h λ ( σ + n − k ) d n − √ ε − d n − + ( ε − b n − − λ ( Z + µ ) b n i r σ + n − = 0 ∞ P n =0 h λ ( Z − µ ) d n − ( ε + 1) d n − − λ ( σ + n + k ) b n + √ ε − b n − i r σ + n − = 0 (42)so the coefficients of the power series satisfy the recursive relations λ ( σ + n − k ) d n − √ ε − d n − + ( ε − b n − − λ ( Z + µ ) b n = 0 λ ( Z − µ ) d n − ( ε + 1) d n − − λ ( σ + n + k ) b n + √ ε − b n − = 0 (43)Solving the above recursive relations and using the initial conditions that b = 0, b − = b − = · · · = 0 and d = 0, d − = d − = · · · = 0, put n = 0 in (43), we obtain λ ( σ − k ) d − λ ( Z + µ ) b = 0 λ ( Z − µ ) d − λ ( σ + k ) b = 0 (44)it requests that the determinant of coefficient is equivalent to zero, so that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) λ ( σ − k ) − λ ( Z + µ ) λ ( Z − µ ) − λ ( σ + k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 (45)it educes that value of the index σ = ± p k + λ ( Z − µ ), since the wave function must be limitedat r → 0, it can be only taken σ = p k + λ ( Z − µ ) (46)on the other hand, the formal whole wave function should be combined by expressions (2), (3),(37), (40) and (46), that is to say ψ = ie − √ ε − λ r ∞ P n =0 b n r √ k + λ ( Z − µ )+ n − χ jιm e − √ ε − λ r ∞ P n =0 d n r √ k + λ ( Z − µ )+ n − ~σ · χ jιm (47)the boundary condition at r → ∞ requests that the parts of the power series in the formal wholewave function must be cut off. It is assumed that b n r = 0 , d n r = 0 and b n r +1 = b n r +2 = · · · = 0, d n r +1 = d n r +2 = · · · = 0, according to the recursive relations (43), put n = n r + 1, we obtain −√ ε − d n r + ( ε − b n r = 0 − ( ε + 1) d n r + √ ε − b n r = 0 (48)it indicates that the ε should take some special values so that b n r +1 = 0 and d n r +1 = 0. Note thatthe recursive relations (43). Multiplying the first relation by √ ε − ε − 1, we have λ ( σ + n r − k ) √ ε − d n r − (cid:0) ε − (cid:1) d n r − − λ ( Z + µ ) √ ε − b n r + ( ε − √ ε − b n r − = 0 λ ( Z − µ ) ( ε − d n r − (cid:0) ε − (cid:1) d n r − − λ ( σ + n r + k ) ( ε − b n r + ( ε − √ ε − b n r − = 0 (49)this deduces that d n r = − ( σ + n r + k ) ( ε − − λ ( Z + µ ) √ ε − σ + n r − k ) √ ε − − λ ( Z − µ ) ( ε − b n r (50)0combing this relations and one of the system of the recursive relations (48), we obtain( σ + n r ) p ε − λZε + λµ (51)it gives that ε = λ µZ ± ( σ + n r ) q ( σ + n r ) + λ ( µ − Z )( σ + n r ) − λ Z (52)we find that it is different from not only the formula (11) given in the original articles but also thecorrect formula (30) of the second-order differential equations. This shows that the original exactsolution of the Dirac equation with position-dependent mass is the pseudo solution. VI. CONCLUSIONS We have used the basal knowledge of the differential equation with variable coefficient to showthat, almost every step of the mathematical operation in the mentioned paper for introducing theDirac theory with the position-dependent mass of the electron in the Coulomb field is incorrect,and the energy eigenvalues so the eigen-wave-function given in the original paper are pseudo.Only in a mathematical signification, we give the correct solution and eigenvalue set of the first-order differential equations. However we don’t regard the formula (52) of the energy-levels as thenecessary result of the development of quantum mechanics. Because the position-dependent massof the electron in the Coulomb field make us discredit its authenticity.Some other papers also alleged that they found the exact solution of the Dirac equation for aparticle with position-dependent mass[2], which might be useful in the study of the correspondingnon-relativistic problem as a reference result. It was even considered that the next terms, whichthey have neglected in this work (in particular the dipolar one) and which are responsible for thesuper-fine structure of the energy spectrum, can be taken into account by means of standard per-turbation theory. Although the corresponding paper constructed the second-order Dirac equationby using unconventionality methods which also can be seen in the some papers published 25 yearsago[3]. However, is this theory correct? Without solving the corresponding differential equation,one can find some methods to directly conclude that some other papers about the Dirac theory arenot correct. [1] A. D. Alhaidari, Solution of the Dirac equation with position-dependent mass in the Coulomb field, Phys. Lett. A , , 72(2004) [2] I. O. Vakarchuk, The Kepler problem in Dirac theory for a particle with position-dependent mass, J.Phys. A: Math. Gen. , 4727 (2005).[3] M. K. F. Wong , H. Y. Yeh, Simplified solution of the Dirac equation with a Coulomb potential, Phys.Rev.25,