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Coulomb expectation values in $D=3$ and $D=3-2\epsilon$ dimensions

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Pith's one-line read The paper establishes that the dimensionally regularized Coulomb problem in D=3−2ε dimensions yields finite regularized values for divergent expectation values such as ⟨V̄³⟩ and ⟨(V̄′)²⟩, and that the tabulated finite values include ε→0…

desk verdict A valuable reference for D-dimensional Coulomb expectation values, but Eq. (III.17) has a real coefficient error that undercuts the printed consistency checks. read the letter →

arxiv 1908.02324 v2 pith:G6KKCXCY submitted 2019-08-06 quant-ph hep-ph

Gregory S. Adkins , Md Faisal Alam , Conor Larison , Ruosi Sun This is my paper · ORCID
classification quant-phhep-ph MSC 81Q0581V4533C4581T15 PACS 03.65.-w31.15.-p12.20.-m
keywords CoulombexpectationvaluesdimensionalregularizationD=3−2εdimensionsassociatedLaguerrepolynomialssubtractedintegralsmomentumspacebracketsNRQEDboundstatesdiharmonicnumbers
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes that the Coulomb bound-state problem in D=3−2ε dimensions works as a dimensional-regularization scheme for expectation values that diverge in three dimensions. Divergent S-state quantities such as ⟨V̄³⟩ and ⟨(V̄′)²⟩ are shown to have well-defined Laurent expansions in ε, with the 1/ε pole encoding the short-distance divergence and the finite part serving as the regularized value. The paper also identifies a subtlety: for some finite operators such as ⟨$r^{{−2+4ε}}$∂_r²⟩, the ε→0 limit of the D-dimensional expectation value differs from the value computed directly in three dimensions. A systematic table of finite and regularized expectation values and momentum-space brackets is provided, along with general integration formulas for associated Laguerre polynomials and their subtracted versions. These results give practitioners a ready-made toolkit for dimensional regularization in QED corrections to Coulombic bound-state energies.

What carries the argument

The key object is the D-dimensional radial wave function, written as φ̄_{nℓ} Ω_{D−1}^{1/2} $e^{{−ρ/2}}$ ρ^ℓ L_{nℓ}(ρ) with ρ=2γ̄_{nℓ}r. Here L_{nℓ}(ρ) is not a standard Laguerre polynomial but a generalized power series L_{nℓ}(ρ)=Σ_{j,k} a_{jk} n̄_{nℓ}^k $ρ^{{j+2εk}}$, with coefficients fixed by a two-index recursion relation; in the ε→0 limit it reduces to the usual associated Laguerre polynomial. For divergent expectation values, the paper isolates the short-distance part of L_{nℓ} (the first few terms) as \widehat L_{nℓ}, treats its integrals analytically in ε, and evaluates the remainder at ε=0 using subtracted associated Laguerre polynomials—polynomials with low powers of the variable removed. General formulas for integrals of one or two (subtracted) associated Laguerre polynomials times powers of x and ln x, together with D-dimensional Fourier transforms of momentum-space brackets, carry the tabulation.

What would settle it

For a chosen state, say n=2, ℓ=0, evaluate ⟨(V̄′)²⟩ by numerically solving the D=3−2ε radial Schrödinger equation with a small-distance cutoff, extract the 1/ε and constant terms, and compare them with the Laurent expansion in (A.13e); a mismatch would show that the short-distance isolation of L_{n0} misses contributions. A second check: compute ⟨$r^{{−2+4ε}}$∂_r²⟩ for ℓ=0 using a different regulator such as a momentum cutoff and test whether the δℓ=0 term in (III.8) appears.

Watch

Extended reading notes

Core claim

The central claim is that every divergent Coulomb expectation value relevant to bound-state QED corrections can be assigned a finite regularized value by working in D=3−2ε dimensions. The paper shows, for example, that ⟨(V̄′)²⟩_{n0} = π m_r (Zα)³ φ̄_n² μ̄^{2ε} {−2/ε −8 ln(μ n/(2 m_r Zα)) + 8H_n + 4/(3n²) −4/n −16/3 + O(ε)}, and that ℓ>0 cases remain finite. Similarly, ⟨V̄³⟩ has a 1/ε pole for S states and a finite closed form for ℓ>0. The paper further demonstrates that taking D→3 does not always commute with taking the expectation value: entries like ⟨$r^{{−2+4ε}}$∂_r²⟩ and ⟨$r^{{4ε}}$p⁴⟩ differ from their three-dimensional counterparts in explicit δℓ=0 terms. It also gives exact D-dimensional relations, such as 2m_r⟨(V̄′)²⟩ = ⟨p²V̄p²⟩ − ⟨p⁴V̄⟩, and a large table of coordinate-space expectation values and momentum-space brackets in both three and D dimensions.

Load-bearing premise

The load-bearing premise is that a divergent S-state expectation value can be computed by isolating the first few terms of the generalized Laguerre series as the only short-distance-sensitive part, then evaluating the remaining infinite series at ε=0.

Editorial extensions

If this is right

  • Any QED correction to Coulomb bound states that uses dimensional regularization can quote the tabulated expectation values directly, including the divergent S-state ⟨V̄³⟩ and ⟨(V̄′)²⟩.
  • Operators whose radial derivatives generate negative powers of ρ acquire δℓ=0 correction terms in the ε→0 limit, so dimensional regularization and the D→3 limit must be taken in the right order.
  • The exact D-dimensional identities from recursion relations, the Feynman–Hellmann theorem, and momentum-space brackets let a user check or rederive any tabulated entry without a separate integration.
  • A sample NRQED calculation shows the CX1 contribution to bound-state energies reduces to −4m_r(...)⟨(V̄′)²⟩, so the tables plug directly into order-mα⁶ energy calculations.
  • The subtracted-Laguerre integral formulas provide a general technique for evaluating short-distance divergent expectation values for any potential whose wave function has a generalized Laguerre expansion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to apply the same subtracted-Laguerre technique to other D=3−2ε central potentials; the harmonic oscillator would show whether the δℓ=0 non-commutativity of limits is generic or Coulomb-specific.
  • A consequence the paper leaves implicit is that any NRQED mα⁶ calculation using these tables inherits the ε→0-limit subtleties; a different regulator such as point-splitting should reproduce the same finite parts, which would be a direct check.
  • One could extend the tables to relativistic corrections by inserting the D-dimensional wave function into the Dirac–Coulomb problem and taking the nonrelativistic expansion; the same short-distance split would regulate the newly divergent operators.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 3 minor

Summary. This paper develops the D=3−2ε dimensional Coulomb problem as a tool for regularizing divergent expectation values in atomic bound-state calculations. It constructs the radial wave function as a double power series in ρ and ε, derives perturbative energy and wave-function corrections, and tabulates a large set of coordinate-space and momentum-space expectation values in Appendix A. The central technical claims are that the dimensionally regularized divergent S-state expectation values such as ⟨V̄³⟩ and ⟨(V̄′)²⟩ are correctly obtained, that several finite D-dimensional expectation values such as ⟨r^{-2+4ε}∂_r²⟩ have ε→0 limits different from their direct three-dimensional limits, and that the tabulated entries are consistent with recursion relations, the Feynman-Hellmann theorem, and numerical integration. The paper also presents general integral formulas for ordinary and subtracted associated Laguerre polynomials, including logarithmic and double-logarithmic cases, and applies the results to an NRQED energy-level correction.

Significance. If the tabulated results are correct, this is a very useful reference for dimensional-regularization calculations of Coulombic bound-state properties, particularly for mα⁶ NRQED corrections. The paper is systematic and unusually cross-checked: it verifies many entries against recursion relations, Feynman-Hellmann identities, and direct numerical integration in three dimensions, and it clearly warns that S-state divergent entries are quoted only through O(ε⁰). The new formulas for subtracted Laguerre integrals and for diharmonic sums are independently valuable. The main weakness identified below is a concrete coefficient error in an exact D-dimensional identity that is cited in support of the consistency claims; this is localized and correctable, but it must be fixed and re-verified before the paper can serve as a trusted reference.

major comments (1)
  1. [§III, Eq. (III.17)] The coefficient of ⟨(V̄′)²⟩ in Eq. (III.17) is misprinted. Starting from Eq. (III.13) with s=−2+4ε and using ⟨r^s⟩=⟨V̄²⟩/β², ⟨r^{s−1+2ε}⟩=−⟨V̄³⟩/β³, and ⟨r^{s−2}⟩=⟨(V̄′)²⟩/[(1−2ε)²β²], multiplication by −β²/2 gives the coefficient [3(1−2ε)²−4ℓ(ℓ+1−2ε)]/(1−2ε), not [3(1−2ε)−4ℓ(ℓ+1−2ε)]/(1−2ε). The printed form is therefore not exact in D dimensions as claimed immediately below the equation. For ℓ=0 the correct coefficient is 3−6ε rather than 3; since ⟨(V̄′)²⟩_{n0} has a −2/ε pole (A.13e), this changes the O(ε⁰) finite part of the identity by +12πφ̄²_n m_r(Zα)³μ̄^{2ε}. Because the identity is called exact and is cited again in the list of exact relations before Eq. (A.15), the error invalidates the stated recursion-relation consistency check at precisely the order at which the S-state results are quoted. The authors should correct the coefficient and confirm that all table entries satisfy the corrected relation, or amend the consistency claim.
minor comments (3)
  1. [Appendix A, after Eq. (A.13)] The warning that S-state expectation values involving V̄ are shown only through O(ε⁰) is essential and should also be repeated at the first use of these entries, e.g., in Eq. (IV.5), so that readers do not mistake the displayed finite part for a complete all-orders D-dimensional result.
  2. [§II, Eq. (II.30)] The symbols H_n, H_n^{(2)}, and diH±(n,m) are defined only in Appendix D; a forward reference at the first occurrence would improve readability, since Eq. (II.30) is otherwise hard to parse.
  3. [Fig. 2 caption] The caption uses diH⁺(7,5), diH⁺(10,−3), and diH⁻(7,12) without defining the symbols; either define diH± in the caption or explicitly refer the reader to Eq. (D.29).

Circularity Check

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No circularity found: the regularized expectation values follow from the D-dimensional Schrödinger equation and Laguerre integral identities, with prior work used only as tools and checks.

full rationale

The paper's central claims are derived rather than assumed. In Sec. II the D=3-2epsilon Coulomb wave function is obtained by solving the radial Schrödinger equation with the power-series ansatz of Eq. (II.25), and the recursion relation (II.26) is derived from the differential equation itself. The divergent expectation values in Sec. III are computed by isolating the short-distance parts of L_n0, evaluating the gamma-function integrals for Re(epsilon)>1/2, and analytically continuing to epsilon=0, as described around Eqs. (III.3)-(III.6). No fitted parameter, target expectation value, or previously tabulated result is inserted into the derivation to force the quoted values. The finite expectation values in Appendix A are supported by the general Laguerre integration formulas of Sec. V, and the paper states that they were checked by numerical integration and by recursion relations; these checks are independent consistency tests, not circular inputs. The paper does cite prior work by the same group, notably the generalized power-series solution in Ref. [2] and NRQED applications in Refs. [11] and [40], but these citations are used as background, tools, or checks rather than as the sole justification for the central regularized values. The assumption that only the first few terms of L_n0 are short-distance sensitive, stated near Eqs. (III.4)-(III.5), is an analytical approximation about convergence of the series near the origin; it is not a definition of the expectation values in terms of themselves. The skeptic's concern about Eq. (III.17) is a potential algebraic/typo error in an exact identity at order epsilon^0, which would affect consistency checks, but an algebraic mistake is not a circularity loop: the tabulated expectation values are not constructed so that Eq. (III.17) holds trivially. Overall, the derivation chain is self-contained against the stated assumptions, and the self-citations are not load-bearing in a circular sense.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The calculation rests on standard Schrodinger-Coulomb machinery, dimensional regularization conventions, and analytic continuation of gamma-function integrals. The main paper-specific additions are the double power-series ansatz and the short-distance subtraction assumption; neither is an independently verified theorem, but both are internally consistent and cross-checked.

free parameters (1)
  • MS renormalization scale mu (mu^2 = muMS^2 e^{gamma_E}/(4 pi))
    Introduced in Eqs. (II.1)-(II.2) to keep alpha dimensionless; it appears in the finite parts of divergent expectation values such as (A.13c) and (A.13e), but it is not fitted to data.
assumptions (6)
  • domain assumption The D-dimensional Schrodinger-Coulomb equation with potential -beta/r^{1-2epsilon} is the correct dimensionally regularized extension of the three-dimensional Coulomb problem.
    Used throughout Section II, especially Eqs. (II.5) and (II.9), as the starting point for the calculation.
  • standard math The angular momentum operator has eigenvalue ell(ell+D-2) on D-dimensional spherical harmonics.
    Used in Eq. (II.8) and around Eq. (II.9) to separate radial and angular variables.
  • standard math The D-dimensional Fourier transform formulas of Appendix B, including (B.1) and (B.2), are valid.
    Used in Section III and Appendix E to relate momentum-space brackets to coordinate-space expectation values.
  • ad hoc to paper The double power series ansatz Ln0(rho) = sum a_jk rho^{j+2epsilon k} has independent powers and solves the radial equation.
    Introduced in Eq. (II.25) with the comment that a bit of trial and error reveals this form; the recursion relation (II.26) depends on treating each power as independent.
  • ad hoc to paper For divergent S-state expectation values, only the first few terms of the Ln0 series are short-distance sensitive and need separate treatment.
    Assumed in Eqs. (III.4)-(III.5) and used for all divergent S-state expectation values in the paper.
  • standard math Analytic continuation of gamma functions defines the regularized values of divergent radial integrals.
    Used throughout Section V and Appendix D when integrals have poles at integral values of s.

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Pith. "Pith review of Coulomb expectation values in $D=3$ and $D=3-2\epsilon$ dimensions." pith.science (2026). https://pith.science/paper/G6KKCXCY

@misc{pith2026190802324,
  author       = {Pith},
  title        = {Pith review of: Coulomb expectation values in $D=3$ and $D=3-2\epsilon$ dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/G6KKCXCY}},
  note         = {Machine review of arXiv:1908.02324}
}
abstract

We explore the quantum Coulomb problem for two-body bound states, in $D=3$ and $D=3-2\epsilon$ dimensions, in detail, and give an extensive list of expectation values that arise in the evaluation of QED corrections to bound state energies. We describe the techniques used to obtain these expectation values and give general formulas for the evaluation of integrals involving associated Laguerre polynomials. In addition, we give formulas for the evaluation of integrals involving subtracted associated Laguerre polynomials--those with low powers of the variable subtracted off--that arise when evaluating divergent expectation values. We present perturbative results (in the parameter $\epsilon$) that show how bound state energies and wave functions in $D=3-2\epsilon$ dimensions differ from their $D=3$ dimensional counterparts and use these formulas to find regularized expressions for divergent expectation values such as $\big \langle \bar V^3 \big \rangle$ and $\big \langle (\bar V')^2 \big \rangle$ where $\bar V$ is the $D$-dimensional Coulomb potential. We evaluate a number of finite $D$-dimensional expectation values such as $\big \langle r^{-2+4\epsilon} \partial_r^2 \big \rangle$ and $\big \langle r^{4\epsilon} p^4 \big \rangle$ that have $\epsilon \rightarrow 0$ limits that differ from their three-dimensional counterparts $\big \langle r^{-2} \partial_r^2 \big \rangle$ and $\big \langle p^4 \big \rangle$. We explore the use of recursion relations, the Feynman-Hellmann theorem, and momentum space brackets combined with $D$-dimensional Fourier transformation for the evaluation of $D$-dimensional expectation values. The results of this paper are useful when using dimensional regularization in the calculation of properties of Coulomb bound systems.

Figures

Figures reproduced from arXiv: 1908.02324 by the authors.

Figure 1
Figure 1. FIG. 1: Diagrams showing the [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Regions in the plane of integral-valued ( [PITH_FULL_IMAGE:figures/full_fig_p036_2.png] view at source ↗

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Works this paper leans on

56 extracted references · 56 canonical work pages

  1. [1]

    Czarnecki, K

    A. Czarnecki, K. Melnikov, and A. Yelkhovsky, Physical R eview A 59, 4316 (1999)

    work page 1999

  2. [2]

    G. S. Adkins, Physics Letters B 382, 1545 (2018)

    work page 2018

  3. [3]

    L. S. Brown, Quantum Field Theory (Cambridge University Press, Cambridge, 1992)

    work page 1992

  4. [4]

    Pineda and J

    A. Pineda and J. Soto, Nuclear Physics B (Proc. Suppl.) 64, 428 (1998)

    work page 1998

  5. [5]

    Pineda and J

    A. Pineda and J. Soto, Physical Review D 59, 016005 (1998)

    work page 1998

  6. [6]

    Pineda and J

    A. Pineda and J. Soto, Physics Letters B 420, 391 (1998)

    work page 1998

  7. [7]

    Czarnecki, K

    A. Czarnecki, K. Melnikov, and A. Yelkhovsky, Physical R eview Letters 82, 311 (1999)

    work page 1999

  8. [8]

    Czarnecki, U

    A. Czarnecki, U. D. Jentschura, and K. Pachucki, Physica l Review Letters 95, 180404 (2005)

    work page 2005

Show all 56 references
  1. [9]

    U. D. Jentschura, A. Czarnecki, and K. Pachucki, Physica l Review A 72, 062102 (2005)

    2005

  2. [10]

    B. J. Wundt and U. D. Jentschura, Physics Letters B 659, 571 (2008)

    2008

  3. [11]

    G. S. Adkins, J. Phys.: Conf. Series 1138, 012005 (2018)

    2018

  4. [12]

    A. H. Hoang, in At the Frontier of Particle Physics/Handbook of QCD , edited by M. Shifman (World Scientific, Singapore, 2002), vol. 4, chap. 37, pp. 2215–2331, [arXiv:hep-ph/0204 299v1]

    2002

  5. [13]

    Brambilla, A

    N. Brambilla, A. Pineda, J. Soto, and A. Vairo, Reviews o f Modern Physics 77, 1423 (2005)

    2005

  6. [14]

    Pineda, Progress in Particle and Nuclear Physics 67, 735 (2012)

    A. Pineda, Progress in Particle and Nuclear Physics 67, 735 (2012)

    2012

  7. [15]

    S. G. Samko, Soviet Mathematics 22, 60 (1978)

    1978

  8. [16]

    Andrew and J

    K. Andrew and J. Supplee, American Journal of Physics 58, 1177 (1990)

    1990

  9. [17]

    D. J. Morales, International Journal of Quantum Chemis try 57, 7 (1996)

    1996

  10. [18]

    Dong, Wave Equations in Higher Dimensions (Springer, Dordrecht, 2011)

    S.-H. Dong, Wave Equations in Higher Dimensions (Springer, Dordrecht, 2011)

    2011

  11. [19]

    Bureˇ s, Doctoral thesis, Masaryk University, Brno, Czech Republic (2015)

    M. Bureˇ s, Doctoral thesis, Masaryk University, Brno, Czech Republic (2015)

    2015

  12. [20]

    Avery, Hyperspherical Harmonics: Applications in Quantum Theory (Klewer Academic Publishers, Dordrecht, 1989)

    J. Avery, Hyperspherical Harmonics: Applications in Quantum Theory (Klewer Academic Publishers, Dordrecht, 1989)

    1989

  13. [21]

    J. S. Avery, Journal of Computational and Applied Mathe matics 233, 1366 (2010)

    2010

  14. [22]

    S. P. Alliluev, Soviet Physics JETP 6, 156 (1958), [J. Exptl. Theoret. Phys. (U.S.S.R.) 33, 200-2 03 (July, 1957)]

    1958

  15. [23]

    M. M. Nieto, American Journal of Physics 47, 1067 (1979)

    1979

  16. [24]

    Schr¨ odinger, Annalen der Physik 80, 437 (1926)

    E. Schr¨ odinger, Annalen der Physik 80, 437 (1926)

    1926

  17. [25]

    Waller, Zeitschrift f¨ ur Physik 38, 635 (1926)

    I. Waller, Zeitschrift f¨ ur Physik 38, 635 (1926)

    1926

  18. [26]

    J. H. Van Vleck, Proceedings of the Royal Society of Lond on, Series A 143, 679 (1934)

    1934

  19. [27]

    Pasternack, Proceedings of the National Academy of S cience (USA) 23, 91 (1937)

    S. Pasternack, Proceedings of the National Academy of S cience (USA) 23, 91 (1937)

    1937

  20. [28]

    H. A. Bethe and E. E. Salpeter, Quantum mechanics of one- and two-electron atoms (Springer-Verlag, Berlin, 1957)

    1957

  21. [29]

    J. H. Epstein and S. T. Epstein, American Journal of Phys ics 30, 266 (1962)

    1962

  22. [30]

    H. M. Srivastava, H. A. Mavromatis, and R. S. Alassar, Ap plied Mathematics Letters 16, 1131 (2003)

    2003

  23. [31]

    S. K. Suslov and B. Trey, Journal of Mathematical Physic s 49, 012104 (2008)

    2008

  24. [32]

    Gonzalez, K

    I. Gonzalez, K. T. Kohl, I. Kondrashuk, and V. H. Moll, Sy mmetry, Integrability and Geometry: Methods and Applicati ons 13, 001 (2017)

    2017

  25. [33]

    R. A. Swainson and G. W. F. Drake, Journal of Physics A: Ma th. Gen. 24, 79 (1991)

    1991

  26. [34]

    Titard and F

    S. Titard and F. J. Yndur´ ain, Physical Review D 49, 6007 (1994)

    1994

  27. [35]

    Zatorski, Physical Review A 78, 032103 (2008)

    J. Zatorski, Physical Review A 78, 032103 (2008)

    2008

  28. [36]

    Weinberg, Lectures on Quantum Mechanics (Cambridge University Press, Cambridge, 2015), 2nd ed

    S. Weinberg, Lectures on Quantum Mechanics (Cambridge University Press, Cambridge, 2015), 2nd ed

    2015

  29. [37]

    W. E. Caswell and G. P. Lepage, Physics Letters B 167, 437 (1986)

    1986

  30. [38]

    R. J. Hill, G. Lee, G. Paz, and M. P. Solon, Physical Revie w D 87, 053017 (2013)

    2013

  31. [39]

    Wolfram Research, Inc., Mathematica 11.1 (Wolfram Research, Inc., Champaign, Illinois, 2017)

    2017

  32. [40]

    G. S. Adkins, B. Akers, M. F. Alam, L. M. Tran, and X. Zhang , Proceedings of Science FFK2019, 004 (2019)

    2019

  33. [41]

    G. S. Adkins, Bulletin of the Allahabad Mathematical So ciety 31, Part 2 , 215 (2016)

    2016

  34. [42]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972)

    1972

  35. [43]

    I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, New York, 1980)

    1980

  36. [44]

    Galindo and P

    A. Galindo and P. Pascual, Quantum Mechanics I (Springer-Verlag, Berlin, 1990)

    1990

  37. [45]

    G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic Press, San Diego, CA, 2001)

    2001

  38. [46]

    F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clar k, eds., NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010)

    2010

  39. [47]

    MATLAB (The MathWorks, Inc., Natick, MA, 2019)

    2019

  40. [48]

    R. L. Liboff, Introductory Quantum Mechanics (Addison-Wesley, Reading, Massachusetts, 1998)

    1998

  41. [49]

    Merzbacher, Quantum Mechanics (Wiley, New York, 1961), 2nd ed

    E. Merzbacher, Quantum Mechanics (Wiley, New York, 1961), 2nd ed

    1961

  42. [50]

    Podolsky and L

    B. Podolsky and L. Pauling, Physical Review 34, 109 (1929)

    1929

  43. [51]

    Fock, Zeitschrift f¨ ur Physik 98, 145 (1936)

    V. Fock, Zeitschrift f¨ ur Physik 98, 145 (1936)

    1936

  44. [52]

    Schwinger, Journal of Mathematical Physics 5, 1606 (1964)

    J. Schwinger, Journal of Mathematical Physics 5, 1606 (1964). 41

    1964

  45. [53]

    Lieber, in Relativistic, Quantum Electrodymanic, and Weak Interacti on Effects in Atoms , edited by W

    M. Lieber, in Relativistic, Quantum Electrodymanic, and Weak Interacti on Effects in Atoms , edited by W. Johnson, P. Mohr, and J. Sucher (American Institute of Physics, New Yo rk, 1989), AIP Conference Proceedings vol. 189, pp. 445-459

    1989

  46. [54]

    G. S. Adkins and J. D. Cammerata, Physical Review A 45, 1314 (1992)

    1992

  47. [55]

    Spieß, Mathematics of Computation 55, 839 (1990)

    J. Spieß, Mathematics of Computation 55, 839 (1990)

    1990

  48. [56]

    R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics: A Foundation for Computer Science (Addison- Wesley, Reading, Massachusetts, 1994), 2nd ed

    1994

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