Analytic Representation of The Dirac Equation. Tepper L. Gill 1,2,3,5, W. W. Zachary 1,2 & Marcus Alfred 2,4

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1 Analytic Representation of The Dirac Equation Tepper L. Gill 35 W. W. Zachary Marcus Alfred 4 Department of Electrical Computer Engineering Computational Physics Laboratory 3 Department of Mathematics 4 Department of Physics Howard University Washington D. C On leave at School of Mathematics University of Minnesota Abstract In this paper we construct an analytical separation diagonalization of the full minimal coupling Dirac equation into particle and antiparticle components. The diagonalization is analytic in that it is achieved without transforming the wave functions as is done by the Foldy-Wouthuysen method and reveals the nonlocal time behavior of the particle-antiparticle relationship. We then show explicitly that the Pauli equation is not completely valid for the study of the Dirac hydrogen atom problem in s-states hyperfine splitting. We conclude that there are some open mathematical problems with any attempt to explicitly show that the Dirac equation is insufficient to explain the full hydrogen spectrum. If the perturbation method can be justified our analysis suggests that the use of cutoffs in QED is already justified by the eigenvalue analysis that supports it. Using a new method we are able to effect separation of variables for full coupling solve the radial equation and provide graphs of the probability density function for the p and s states and compare them with those of the Dirac Coulomb case. I. Introduction It is generally agreed that quantum electrodynamics QED is an almost perfect theory that is in excellent agreement with experiments. The fact that it is very successful is without doubt. However there are still some technical and foundational issues which require clarification.

2 Historically when Lamb and Retherford confirmed suspicions that the s state hydrogen was shifted above the p state the Pauli approximation to the Dirac equation was used to essentially decide that the Dirac equation was not sufficient. II Purpose In light of the tremendous success historically of eigenvalue analysis in physics and engineering it is not inappropriate for us to reinvestigate the foundations of spin / particles with an eye towards clearly identifying the physical and mathematical limitations to our understanding of the hydrogen spectrum as an eigenvalue problem. In particular want to investigate the extent that we may believe conclusions about the Dirac equation based on analysis of the Pauli approximation. The first successful attempt to resolve the question of how best to handle the square-root equation: i" t = c p m c 4 " = I I was made by Dirac in 96. Dirac noticed that the Pauli matrices could be used to write c p m c 4 as c "p mc [ ]. The matrix is defined by = 3 where i = " i " " i = " = i " i 3 =. Thus Dirac showed that an alternative representation of equation could be taken as: i" t = c p mc ". In this case must be viewed as a vector-valued function or spinor. To be more precise "L R 3 C 4 = L R 3 C 4 is a four-component column vector t. In this approach = t represents the particle positive t represents the antiparticle negative energy = " " energy component and = component of the theory for details see Thaller 3. A fair understanding of the Dirac equation can only be claimed in recent times and as pointed out by D. Finkelstein Dirac introduced a Lorentz-invariant Clifford algebra into the complex algebra of observables of the electron. See in particular Biedenharn 4 or devries 5 and Hestenes 6. Despite successes both practical and theoretical there still remain a number of conceptual interpretational and technical misunderstandings about this equation. It is generally believed that it is not possible to separate the particle and antiparticle components directly without approximations when interactions are present. The various approximations found in the literature might have led to this belief. In addition

3 the historically important algebraic approaches of Foldy-Wouthuysen 7 Pauli 8 and Feynman and Gell-Mann 9 have no doubt further supported such ideas. We show in Section III that it is possible to directly separate the particle and antiparticle components of the Dirac equation without approximations even when scalar and vector potentials of quite general character are present. In Section IV we show that the square root operator cannot be considered physically equivalent to the Dirac operator. In addition we offer another interpretation of the zitterbewegung and the fact that expected value of a velocity measurement of a Dirac particle at any instant of time is ±c. In Section V we reconsider the hydrogen atom problem from an exact point of view and then discuss the extent that we may believe in the validity of the use of perturbation analysis to compute the hyperfine splitting separation. Finally in Section VI we show using a combination of methods developed by Harish-Chandra and Villalba that it is possible to effect a separation of variables for the hydrogen atom problem with the magnetic dipole vector potential. This allows us to provide some justification for perturbation analysis to compute the hyperfine splitting in this case. II. Complete Separation It turns out that a direct analytic separation is actually quite simple and provides additional insight into the particle and antiparticle components. In order to see this let Axt and Vx be given vector and scalar potentials and after adding Vx and making the transformation p " = p e ca write in two-component form as: i " t = V mc " c 3a Equation 3b can be written in the form: i " t = V mc " c. 3b [ ] and D = c " i " t ib = D 4 with B = V mc [ ]. From an analytical point of view we see that equation 4 is an inhomogeneous partial differential equation. This equation can be solved via the Green s function method. Thus we then must solve It is easy to see that the solution to equation 5 is " t ib ut = t. 5 3

4 ut = texp{"ibt} t = t > 6 t < so that f g is the convolution of f g t = cut "[ i ] t = ct exp{ibt }[ i ] d 7a t t = c exp{"ibt " }[ i ] d 7b " where means convolution. Using equation 7 in 3a we have i " t = V mc " c i t [ ] exp{ibt } " d. 8 In a similar manner we obtain the complete equation for : i " t = V mc " c i t [ ] exp{ib t } " d 9 where B = [V mc ] and vt = t exp{"ib t} which allows us to solve for : t t = cvt"[ i]t = c exp{ib t }[ i ]d. as L R 3 C 4 = L R 3 C L R 3 C. One contains the particle positive energy wave component while the Thus we have decomposed L R 3 C 4 copy of L R 3 C other copy contains the antiparticle negative energy wave component. Which of these copies corresponds to the components = t and which to the components = t depends to some extent on the properties of the scalar potential V. However we will not consider this problem in greater detail in the present paper. An unsettled issue is the definition of the appropriate inner product for the two subspaces which will account for the quantum constraint that the total probability integral is " = " " and normalized. We can satisfy this requirement if we set " A = A A" " A = A A " where A = cut "[ i ] t A = cvt "[ i ]t. Define the particle and antiparticle inner products by " " A }dx " p = { a R 3 " " A }dx " ap = { a R 3 4

5 so that: t " = " c exp{ibt } [ i ]" d b " = " t c exp{ib t }[ i ]" d. b The second term in equation b equation b comes directly from the definition of in terms of the definition of in terms of see equations 7 and. Thus it is clear that " dx = R 3 dx = so we have a complete separation of the particle and R 3 antiparticle wave functions. In the standard representation the charge conjugation operator is C = U C with U C = i". A simple computation establishes the following theorem. Theorem. Equations 8 and 9 are mapped into each other under the charge conjugation transformation. Equations 8 and 9 offer an interesting alternative to the many attempts to decompose the Dirac equation into particle-antiparticle and/or parity-sensitive pairs. They also offer a different approach to the study of large Z hydrogen-like atoms. IV. Interpretations Writing the Dirac equation and the direct separation in two-component matrix form we have: and i " t = V mc c c V - mc " 3 V mc i " t = c i [ ] u - [ ] V. mc [ c i ] v - [ ] ". 4 We call 4 the analytic diagonalization of the Dirac equation because the wave function has not changed. 5

6 The standard approach to the diagonalization of the Dirac equation without an external potential V is via the Foldy-Wouthuysen representation. Assuming that A does not depend on t the following generalization can be found in devries 5 : i t " " = c e c - m c 4 c e c - m c 4 " ". 5 [ ] t = U FW [" ] t and H s = U FW H D U FW In this case see Thaller 3. Equation 5 is studied elsewhere where it is shown that when A is zero H s = c p m c 4 has the following analytic representation see also Gill : H s f x = µ c" - K µ x y K µ x y x y µ x y R 3.- x y x y /- f ydy 6 - where denotes the vector norm in R 3. Here the K n are modified Bessel functions of the third kind and µ = mc. Equation 6 is of independent interest since it is the first example of a physically relevant operator which has a representation as the confinement of a composite of three singularities two negative and one hard core positive within a Compton wavelength such that at the point of singularity they cancel each other providing a finite result. The second paper in this series is devoted to the square root operator where this result discussed in detail. From equation 4 we conclude that the coupling of the particle and antiparticle wave functions in the first-order form of the Dirac equation hides the second order nonlocal time nature of the equation. From 6 we see explicitly that 5 is nonlocal in space. Thus the implicit time nonlocality of the Dirac equation is mapped into the explicit spatial nonlocality of the square-root equation by the Foldy-Wouthuysen transformation. These observations imply that the Dirac Hamiltonian H D and the squareroot Hamiltonian H s = U FW H D U FW are mathematically but not what we would normally mean by physically equivalent. Furthermore it appears that the only way we can partially justify using the square-root equation to interpret the Dirac equation is that they both square to give the Klein-Gordon equation. We conclude that they can only be viewed as physically equivalent outside a Compton wavelength where they both appear as point particles. 6

7 V. The Hydrogen Atom In this section we reconsider the standard analysis of the Dirac equation for the hydrogen atom problem from an exact point of view. We assume that A = µ I r r 3 V = c" r and = e c. Rewrite 3a and 3b in eigenvalue form: E V mc " = c 7a E V mc " = c. 7b Eliminating in terms of and vice versa we obtain the following equations: E V mc " = c pv E V mc " c E V mc " 8a E V mc " = c pv E V mc " c E V mc ". 8b We can also get 8 from equations 8 and 9 via straightforward integration. We call 8a and 8b the Slater equations since they were first used by one of his students as early as 94 and appeared in his book 3 first published in 96 see Appendix 9. It s surprising that Slater s work is not well known. For obvious reasons we concentrate on 8a. First note that if we drop the middle term and replace E V mc by mc we get the Pauli approximation to the Dirac equation: E V mc " = e B" mc m ". 9 As noted earlier the Pauli equation was used to extract the hyperfine splitting portion of the hydrogen spectrum to support the predictions of QED. It follows that the conditions that justify the Pauli approximation and the dropping of the middle term of 8a are both of importance for the foundations. There are a number of other equations and/or approximations that have been given the name and/or used in lieu of the Pauli equation see for example Greiner 4 Mizushima 5 or Bethe and Salpeter 6. We do not consider these equations since although they are related to the Dirac equation they do not give additional information and it is not obvious that they have any more mathematical or physical justification when applied to the s-states of hydrogen. Recall that there is a finite probability of finding the electron at the origin in s- states but the required condition for the validity of 9 is E V mc << mc. Thus this condition is not satisfied for s-state calculations. It follows that use of the Pauli equation to compute the hyperfine splitting of s-states is not convincing. On the other hand the condition is easily seen to be satisfied for all other states. A more 7

8 reasonable approximation is to use mc E << mc to replace E V mc by where r = e E mc e mc. The above condition is always mc r r satisfied 3ev compared to.5mev. This approach also has the additional advantage of removing the nonlinear eigenvalue problem posed by 8a without substantially affecting the final result. In this case we have E V mc " = pv 4m c r r " m r r ". This equation is new although Slater had all the tools to derive it. Using standard computations we get see Slater 3 L = r p is the angular momentum and S is the spin S = and µ I is the nuclear magnetic moment " " = e c " pv " = e S" L r - r Putting these expressions in we have: 8 3 S" µ r 3S" rµ I " r I S" µ I / -. r 5 r 3 a d dr e S" µ I c S" rµ I " r r r 4. /. b E V mc r " = S L d m r r r r dr e S µ I S r µ r I - c r r 4. / " e 8 mc r r 3 S µ r 3Srµ r I I S µ. I - r 5 r 3. / " m r r " When µ I = becomes E V mc " = r S L d m r r r r dr " p m r r ". 3 Equation 3 has using r = e E mc the same eigenvalues as the unperturbed Dirac equation so that our interest centers on the following terms op means operator e mc r r 8" 3 S µ r 3Srµ r I I S µ I - r 5 r 3 er S µ I mc r r r r " S rµ I r r 4 op. / 4a. 4b op 8

9 would The delta term in equation 4a except for the additional factor r r normally be used to compute the hyperfine splitting of s-states in the Pauli approximation. It is easy to see that with this additional factor the same calculation would give a value of zero for the splitting. In all other states this factor is small >> r r and may be dropped. Slater 3 used equation 4b to compute the s-state hyperfine splitting and obtained the correct result. Since this term is part of the focus of our investigation we repeat some of Slater s calculations. In the s-state the total angular momentum J is equal to S. Hence following standard procedures we replace S µ I r S r " S r r 4 by S µ I S [ "S rµ I r r 4 ] op [ ] op. If < A > denotes the average of the operator A it is easy to see that < S r >= < 4 r > and < S >= 3 4. The term of interest becomes er 3mc r r S µ r 4 I op. 5a The important issue is the computation of the s-state expected value of where = e 3mc r r r r 5b 4 S"µ I op. Slater 3 assumed the nonrelativistic radial wave ave function for s-states. For the s state Rr = 3/ " rexp" r and = r B where r B =.5978 " m is the Bohr radius. Using the normalization " r Rr dr = this led him to the computation of " r Rr r r r r dr. 6a 4 Setting = "r and = "r we have 3 " exp"d" ". 6b " " " By a change of variables u = and integration by parts it is easy to see that is a cutoff and that the dominant contribution to the expression 6b is " 3 as ". Note that the integral is divergent but the factor in front makes the product finite. We get the same result for all s-states while it is not hard to show that equation 4b is almost zero for all other states. 9

10 It would appear that the correct approach for s-state hyperfine splitting gives the same results as those obtained from the Pauli equation. Furthermore equation 4b introduces a natural cutoff which removes the conceptual difficulty of a point magnetic dipole interaction as implied by use of the delta term in the Pauli equation. In addition it is not hard to show that Slater s approach goes through giving the same result if we use the correct Dirac solution for the first-order calculation. However to provide input for the precise results of QED we must first correct the normalization condition to [ " p 4m c r r ]dx =. 7 Clearly we expect that the additional term will give a very small correction. However it is not clear how small. For example if it changes the hyperfine splitting values in the eight or ninth decimal place in GHz it may well be an important correction. For example the measured value of the s state hyperfine splitting in hydrogen is Ghz see Mizushima 5. We now approach the more difficult issue facing attempts to completely understand the Dirac problem for full coupling namely the A term: e A mc r r = e µ I sin mc r rr. 8 4 In most treatments of the Dirac hydrogen atom problem this term with r = is either ignored or assumed to be small. Power counting shows that it cannot be ignored without investigation. It is easy to show that this term will be small in all except s-states. The first observation is that this term appears to be more singular than the Coulomb potential so that perturbation analysis may not be appropriate. However this is not completely clear since the sin term vanishes on the spin axis and could strongly modify the singular nature of this term. If we take an engineering approach and assume that we can treat the A perturbation then for the s- state the expected value is term as a r µ I Rr r r rr dr " sin sin " d = µ expd I ". 9 In atomic units = µ = " r = = " g N µ I = 836 g N µ I op so we can write 9 as = I op = 3 4 and ave

11 3 µ I 4 " 4 exp"d. 3 Using a table of integrals see Gradshteyn and Ryzhik 7 pages 95 and 97 and the cutoff prescription of Bethe 6 page we have exp"d = "Ei" and " "exp"d" = e " Ei" k where Ei" = C ln " " k kk C is Euler s constant. Using these results in 3 we get 3 µ I k = [ ] and -/ lim " expd 4 4 /. expd / 3/ = 3 µ I lim " Ei" 4 e 4 Ei. 3 It is clear that Ei" will diverge like ln" as ". If we fix at and note that e " then 3 µ I Ei " " 4 e" " 4 " Ei " " 3 µ I 4 " 5 4 " - C ln "./ " 3 µ exp"d I g N k = k k " k k / 3 { [ C ln " ]}. 3 If we note that " then this last term is of order > 7. Thus if there is mathematical support for the calculation procedures the A term does not make a significant contribution. In the next section we discuss an exact separation of variables for full coupling Coulomb plus magnetic dipole vector potentials which will allow us to partially answer this question. VI. Separation of Variables In this section we show that an exact separation of variables is possible via a new approach based on earlier work of Harish-Chandra 8 and Villalba 9. To begin we reconsider equation 3 in eigenvalue form with A = µ I r r 3 V = c" r and = Ze c. Following Dirac set I 4 = diag i = " i 3 = " where

12 i = " i " = I i I = ii ii. 33 It follows that i " j = " j i i j = 3 and equation 3 can be written in the form: H D = E = [ c" p e A mc c " 3 VI 4 ]. 34 Following Harish-Chandra 8 we represent the components of in spherical coordinates: r = cos" sin" sin 3 cos = r cos" sin" cos r 3 sin " = r cos" sin" sin 35 so that " p = r p r p r p rsin. 36 With the proton s magnetic moment lying along the z-axis the vector potential in spherical coordinates takes the form A r = A = and A = µ I r sin" so that: " A = A = µ I r sin. 37 It is well known that the orbital angular momentum L z = i" " about the z-axis is not a conserved quantity taking = for now. However it is easy to show that the total angular momentum about the z-axis L z I 4 3 commutes with H D : [ H D L z I 4 3 ] = 38 so that it is conserved. Therefore we can choose solutions of 34 in the form: r" = exp[im 3 ] r" 39 where r" does not depend on and m is a half odd integer with both positive and negative values allowed. Now using the exact relations:

13 r = exp{" i 3}exp{" i } 3 exp{ i }exp{ i 3} = rexp{" i 3 }exp{" i } exp{ i }exp{ i 3 } = rsin exp{" i 3 }exp{" i } exp{ i }exp{ i 3 } 4 and the easily proved relations: and exp{ i "} " = " i exp{ i "} 4a exp{ i "}exp{ i 3 } = i cos" sin" 3 exp{ i "}exp{ i 3} 4b equation 34 takes the form: ic" 3 r r r cot I mcsc e 4 3 c µ sin - 3 I / 4. 5 mc " 3 V EI 4 ]6 = 4 where r" exp{ i " imi 4 } r". In order to complete our separation we now follow the method due to Villalba 9. Setting = " where is an operator to be determined so that and K = i "ic r 3 r. r r mc 3 V " EI 4 - / = "3 43a - K = i"ic cot I " mcsc " e 4 3 c m I sin /. 3 = 4. 43b Then K K =. Using our definition of we have K = c r " "r r ir mc 3 3 ir V E 44a 3

14 and - K = c " cot" i m csc" e c m I sin " /.. 44b We obtain a complete separation provided that [K K ] =. It is easy to see that this requirement imposes the following conditions recall that i = " i. [ 3 " "] =. [ 3 " "] = 3. [ 3 " "] = 45a. [ 3 " "] =. [" ] = 3. [" ] =. 45b If " exists then from 45b- we obtain [" ] = and from 45b-3 [" ] =. From 45a- [ 3 " "] = [ 3 ]" [ 3 "]" = i " [ 3 "]". 46 Then since i " = i i = 3 [ 3 "] = i " = i " { } = = i i 3 " = 3 " 3 " i.e. 3 and anticommute. Similar procedures yield the same conditions from 45a- and 45b-. Now write " = A B C D where A B C and D are matrices. Expanding these matrices as series of the form A = a I a i " i with similar expressions for B C and D and using [" ] = [" ] = {" 3 } = shows that 3 i= = b " 3 b = const. 47 " 3 Setting b equal to this completes the separation of variables for system 4. 4

15 We now consider the eigenvalue equations: - K = c" r r. r 3 ir mc " 3 ir V E = 3 48 /. " K = c " cot i m csc - e c m I sin 3 = 4 49 / for the radial and angular equations respectively. Solution of the Radial Equations Putting = [ ] t with and -spinors we obtain from equation 48 reinstating the relevant factors of : " r "r r ir mc E c r = 3 c 5a " r r " r ir mc E c r = 3 c. 5b Now put = "ar = "br where " are two-spinors and ar br are functions. Substitution into equations 5a and 5b gives: " r "r r ar ir mc E c r 3br = - c ar " r r " r br ir mc E c r 3ar = - c br. If we choose = [ ] t and =[" " ] t we can eliminate the spinors from the above equations to get 3 " = 3 " = : " r "r r ar ir mc E c r br = ar 5a c " r r " r br ir mc E c r ar = c br. 5b 5

16 We can now apply standard methods putting ar = ur r br = vr r in 5a and 5b yields the equations: du dr " u c r i dv dr v c r i mc E c mc E c r v = 5a " r u =. 5b These equations are very close to those for the Dirac-Coulomb problem except for the factor of i. As in that problem make the change of independent variable y = r with = m c 4 " E c and change dependent variables via: we follow Greiner 4 p. 79 u = e y/ mc E " " v = e y / mc E " " from which the following equations for and are obtained: d dy = " i cy mc " cy i c " i c y E 53a d dy = " cy i c " i c y E i cy mc. 53b We now look for series solutions of these equations in the form: = " q y q = q y q q= where is to be determined. Substitution into equations 53 gives the following relations: q= q q " i mc - c y q = q" = i -. q y q" / c i E c -. q y q " - q y q " q= q= q= 54a 6

17 i mc q q " c y q = q" - = q- y q" - i. q- y q" - - / c c i E c. q y q " -. q= q= q= 54b For q = equations 54a and 54b give c " i E i mc c " c - = i" mc c c i" E c =. In order that " the determinant of the coefficients must vanish. This which is analogous to the Dirac-Coulomb result. There are two gives = " c possible values for. Taking the positive one for obvious reasons we get " = c i E c i mc c Returning to 54 we obtain the following recursion relations: = i mc c c i E c. 55 i mc q " c = " i q c " - q" c " i E c q 56a q" i" mc q c = q i q c - q c i" E c q. 56b Now by some tedious and uninteresting algebraic manipulations similar to those in Greiner 4 we get: a equals a conjugate q " q = q K iq K q K = i c i c mc ie 57 q = q "" n qq Iterating equation 58 gives: q K q " K q" q n = E ". 58 c- 7

18 q = q "" n q " " n " n q K qq q " K. 59 We now must express in terms of. To do this set q = in equations 56a and 56b and get: i " mc c = i c i- c " E c 6a i" mc c = i c - c i" E c. 6b Use one of these equations to eliminate so that the other can be used to determine in terms of and. Since is proportional to from equation 55 we obtain = K K i c " " i mc c - q " " q = n " n q K K " i "c qq " i mc "c-. 6 Recall that the confluent hypergeometric function is defined by see Gradshteyn and Ryzhik 7 We can write: a F a;b; x = " n a n = b n n n = aa a n = x n q q " a n. 6 a = q " = q and using the decomposition q K = q n " [ n " K n "] n " in the numerator of equation 6 we get: We now have that q n "q n " n "q K = [ n " K n "] n " n " n " q = n " q [ n " K n "] n " q. 8

19 K i c y = y " " i mc c - 3 / n. q [ n. K n.] n. q q= 4 y q q " q 63 K i c - = y " 7 / " i mc c F n.; "; y n. K : F n.; "; y ;. 9 n. < This leads to In a similar fashion we can solve for the coefficients to get: q Kq " " q = i n " n K " i "c qq "c " i E "c y K i c = y " c i E c i- 5 6 F. n ; "; y 78. / n K 4 n. 3 F. n ; "; y 9 : ;8. 65 We can now obtain the eigenfunctions using ". " = = " ar br = - / ar / br where ar = y mc " E / e " y/ " / e " y / K " i c = y " mc " E " i mc c - 6 K ik. 7 i F " n /; ; y " F " n /; ; y " n / 3 F " / n ; ; y 9 : 8 ; and 67 9

20 br = y mc E / e " y/ / e " y / K " i c = y " mc E " i mc c - 6 K "ik. 7" i F " n /; ; y " F " n /; ; y " n / 3 F " / n ; ; y 9 :. 8 ; Thus we see that the solutions of the radial equations are linear combinations of the same types of confluent hypergeometric functions that occur as eigenfunctions of the Dirac- Coulomb problem. However our parameters n and are different from corresponding ones that appear in the Dirac-Coulomb case. Furthermore our coefficients are complex numbers. To obtain the eigenvalues E we follow the standard approach and require that n is an integer so that each hypergeometric series reduce to polynomial and consequently the eigenfunctions are normalizable compare 39 the definition of following 4 the definitions of ar and br following and 68. This gives 68 n = ±mc E " n c " -. / / 69 and following the argument of Greiner 4 p. 8 in the Dirac-Coulomb case we discard the minus sign. From the above we see that the energy eigenvalues are of the same form as in the Dirac-Coulomb case. The only difference is that the quantity j in the Dirac-Coulomb problem is replaced by c momentum quantum number. where j is the total angular In order to identify the similarities and differences between the functions ar br in respectively with the corresponding Dirac-Coulomb eigenfunctions see e.g. Greiner 4 chapter 9 eq. 36 we compare the corresponding probability densities of the radial equations. Recall that 67 and 68 are complexvalued whereas the corresponding Dirac-Coulomb eigenfunctions are real-valued so we should expect some differences. Before computing the absolute square of ar and br let

21 E = E n = mc " n c " A = mc E " E c " mc c c " E c c B = mc E " E c " mc c c " E c c - -. A = " mc E n c 3 B =. / mc " mc E n c 3 E / mc E A 3 = c " E c " B 3 = c " E c " mc E mc E. c " / mc E mc E. c " / mc E 3 " mc E mc E c mc E 3 " mc E mc E c. 7 Now using the definition of K from 57 we can write the absolute square of 67 and 68 as: ar = A { F I y F I yf II y A F II y A A n " 3 }y e y 7 { B n " 3 } y e y 7 br = B F II y F I yf II y B F I y B where F I y = F n "; ; y and F II y = F n "; ; y. In order to facilitate quantitative comparison with the Dirac-Coulomb radial wave functions as in Greiner we need to convert equations 7 and 7 to normalized form by requiring that " ar br r dr =. This condition determines the quantity. After some simplifying calculations if we set = n " n " = n " n " A " = 4 3 mc 3 [ n " ] 4 D B" = 73 A "

22 A = " n c A = 3 c B = 3 c / B = " n c n " " c " c n " " c " c / / / n " [ n "] n " [ n " ]. 74 The normalized version of 7 becomes: ar n = D F I y " F I yf II y A { A F II y A A n 3 }y " e " y 75 br n = B D { F I y " F I yf II y B F II y B B n 3 }y " e "y. 76 The constant D in equations 75 and 76 comes from the normalization requirement and is defined by: h = n " [ n " ]h = n " [ n " ] -/ D = J h n "c n" n " h. / -/ n " 4. n " c / 56 n " n " n " c h / n " h 7 n " 89 h 4 n "c n " c h / n " n " n " c h / / 3/ J / / J 3 J = [ F II y] y " e " y dy n " n " n " n " n m = - -" n m n m n m J = n = m = F I yf II yy " e " y dy n " n n " n = - -" n m n m n = m = n m n m 3/

23 J 3 = [ F I y] y " e " y dy n n n n = --" n m n m n= m = n m. n m Equations 75 and 76 are to be compared to the normalized Dirac-Coulomb probability densities Greiner 4 p.8 eq. 36: = " n f r n gr n = 3 mc = 3 mc 3 " " 5 " 3 " " 5 " n n " F y n F y I II [ ] y e y 77 n n " F y n F y I II [ ] y e y. 78 We have converted Greiner s notation to our own in order to make comparison easy. In the above two equations one should replace c by in the definition of see the discussion prior to 55 where " j j = l j j = l. In figures 6 we have restricted our comparison of densities to the S / and P / states. These are the most important cases for two reasons because they are the ones involved in the Lamb shift and these are the states where we expect equations 75 and 76 to differ the most from 77 and 78. In the figures we have plotted the quantities: Ar = Fr = ar n " 3 mc 3 y max " e y max f r n " 3 mc 3 y max " e y max Br = Gr = br n " 3 mc 3 y max " e y max gr n " 3 mc 3 y max " e y max verses y = [ m c 4 E c]r where y max 5 for all cases. This allows one to clearly see the difference between our results and those for the Dirac-Coulomb problem especially in the region about the origin. For the P / and S / states = and = " respectively; the quantum number n = in both cases so that the confluent hypergeometric functions in 75 and 76 simplify to F I y = y " and F II y =. Note because we replaced c by our eigenvalues are the same as in the Dirac Coulomb case and we don t need to solve the angular equations for our graphs. 3

24 For the p / -state = we have plotted Ar verses Fr Br verses Gr and Ar Br verses Fr Gr. The same approach is also followed for the s / -state = ". We see that as expected the graph of Ar differs considerably from Fr and Br differs from Gr for small values of r but approach each other for large values of r. The important point is that in both the p / -state and the s / -state the probability density Ar Br diverges at an earlier stage compared to Fr Gr. This reflects the additional singular nature of the equation when the magnetic moment is included and further supports our contention that treating the magnetic moment as a perturbation may not be valid. Fig = p / -state Ar thin verses Fr thick 4

25 Fig = p / -state Br thin verses Gr thick It appears in Fig that Gr goes to zero but this is not the case. It actually diverges near r = but the scale is so small that it cannot be shown on the graph. Fig 3 = p / -state Ar Br thin verses Fr Gr thick 5

26 Fig 4 = " s / -state Ar thin verses Fr thick Fig 5 = " s / -state Br thin verses Gr thick 6

27 Fig 6 = " s / -state Ar Br thin verses Fr Gr thick The Angular Equations We now investigate equation 49. Using the matrix 47 for the angular equations become upon reinserting the appropriate factors of and using the spinor relations: m = m z = eµ I c " d d cot mcsc zsin = "c 79a " d d cot mcsc zsin = "c. 79b Making the change of variable x = cos transforms 79 to " d x / dx m x x z / x = "c " d x / dx m x x z / x = "c. 8a 8b 7

28 These are generalizations of equations that lead to the associated Legendre equations. In order to see the differences use 8a to solve for and put this in 8b to obtain an equation for the equation is identical so we drop the subscripts and = c : x "" x " m mx 4 x -. z m x x / z x } =. 8a The corresponding equation for is: x "" x " m mx 4 x -. z m x x / z x } =. 8b The term in each of the above braces is close to the general form of Legendre s differential equation but is clearly different see Gradshteyn and Ryzhik 7. The deviation from Legendre s equation reflects the lack of spherical symmetry caused in part by the magnetic moment potential as reflected by the terms multiplied by z. It should be noted that the equations are not of Legendre type even when z =. This reflects the fact that the Dirac equation does not conserve angular momentum. Thus the solutions of 8 will be interesting generalizations of the Legendre functions even when z =. We are currently unable to construct an exact analytic solution of equation 8. Interestingly it is the square root term that prevents the use of standard analytic solution methods. A detailed study of this equation is in progress. Problems Physical We can now identify the problems that cloud our complete understanding of the Dirac equation and its relation to the hydrogen spectrum. The basic physical problem is to construct a complete solution for the angular eigenvalue problem for the Dirac equation with the Coulomb and magnetic dipole interaction equation 8. Mathematical A basic mathematical problem is to prove or disprove that perturbation theory can or cannot be applied to equation 5a and the A term equation 8 using 7. 8

29 Since equation 8 is not one of the standard forms we may be forced to use approximation methods to solve it. It appears to require some new ideas for its analytic solution. This would be preferred since our separation of variables method does not allow us to directly explore the questions posed. Conclusion In this paper we have shown that the full minimal coupling Dirac equation can be analytically separated diagonalized into particle and antiparticle components without transforming the wave functions as is done by the Foldy-Wouthuysen method. This diagonalization reveals the nonlocal time behavior of the particle-antiparticle relationship. This discovery naturally raises the questions about the zitterbewegung. Although we have not considered this problem in detail preliminary analysis leads us to believe that a more physically reasonable interpretation of the zitterbewegung and the result that a velocity measurement of a Dirac particle at any instant in time is ±c are reflections of the fact that the Dirac equation makes a spatially extended particle appear as a point in the present by forcing it to oscillate between the past and future at speed c. From this we infer that although the form of the Dirac equation serves to make time and space appear on an equal footing mathematically it is clear that they are not on an equal footing from a physical point of view. On the other hand the Foldy-Wouthuysen transformation which connects the Dirac and square root operator is unitary. Reflection on these results suggests that a more refined notion than the mathematical one of unitary equivalence may be required for physical systems. We have also shown that one of the difficult issues facing attempts to completely understand the Dirac problem for full coupling is the singular nature of the A term. This term is small in all but s- states where it diverges when treated as a perturbation. If we introduce a cutoff the contribution is of order 7 so one might be inclined to dismiss the term as is traditionally done. However this term appears to be more singular than the Coulomb potential so that perturbation analysis and indeed the whole eigenvalue approach may be called into doubt. On the other hand this is not completely clear since the sin term vanishes on the spin axis and could strongly modify the singular nature of this term. This problem must be solved in order to determine the exact extent that the Dirac equation contributes to spectrum of hydrogen. If the problems posed in this paper can be solved in the positive it would appear that the correct approach for s-state hyperfine splitting gives the same results as those obtained from using the Pauli equation. Furthermore equation 4b introduces a natural cutoff which removes the conceptual difficulty of a point magnetic dipole interaction as implied by use of the delta term in the Pauli equation. This also suggests that the use of cutoffs in QED are already justified by the eigenvalue analysis that supports it. Using a different method we are able to effect separation of variables for full coupling and solve the radial equation. Since the behavior of the radial equation at the 9

30 origin is the same as in the Dirac-Coulomb case we can say that the A term does increase the singular nature of the radial equation. We have not been able to solve the angular equation. Although we strongly believe that contribution is small we still cannot say how much the Dirac equation contributes to the hydrogen spectrum. ACKNOWLEDGMENT The authors would like to acknowledge important comments and encouragement from Professor David Finkelstein of Georgia Tech. REFERENCES. W. E. Lamb Jr. and R. C. Retherford Phys. Rev P. A. M. Dirac Proc. Roy. Soc London A A B. Thaller The Dirac Equation Springer-Verlag New York L. C. Biedenharn Phys. Rev E. devries Fortsch. d. Physik D. Hestenes Space Time Algebra Gordon and Breach New York L. L. Foldy and S. A. Wouthuysen Phys. Rev W. Pauli Handbuch der Physik d ed R. P. Feynman and M. Gell-Mann Phys. Rev T. L. Gill FermiLab-Pub-8/6-THY.. E. H. Lieb and M. Loss Analysis Grad. Studies in Math. 4 Amer. Math. Soc. 99 see pages A. O. Williams Jr. Phys. Rev J. C. Slater Quantum Theory of Atomic Structure Vol. II McGraw-Hill New York W. Greiner Relativistic Quantum Mechanics Springer-Verlag New York M. Mizushima Quantum Mechanics of Atomic Spectra and Atomic Structure Benjamin Inc. New York H. Bethe and E. E. Salpeter Quantum Mechanics of One-and Two-Electron Atoms Springer-Verlag New York I. S. Gradshteyn and I. M. Ryzhik Tables of Integrals Series and Products Academic Press New York Harish-Chandra Phys. Rev

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