Non-relativistic Effective Quantum Mechanics of the Coulomb Interaction
NNon-relativistic Effective Quantum Mechanics of the Coulomb Interaction
David M. Jacobs ∗ and Matthew Jankowski Physics Department, Norwich University158 Harmon Dr, Northfield, VT 05663 Physics Department, Hamilton College198 College Hill Rd., Clinton, NY 13323 (Dated: February 2, 2021)We apply the ideas of effective field theory to nonrelativistic quantum mechanics. Utilizing anartificial boundary of ignorance as a calculational tool, we develop the effective theory using bound-ary conditions to encode short-ranged effects that are deliberately not modeled; thus, the boundaryconditions play a role similar to the effective action in field theory. Unitarity is temporarily violatedin this method, but is preserved on average. As a demonstration of this approach, we consider theCoulomb interaction and find that this effective quantum mechanics can predict the bound stateenergies to very high accuracy with a small number of fitting parameters. It is also shown to beequivalent to the theory of quantum defects, but derived here using an effective framework. Themethod respects electromagnetic gauge invariance and also can describe decays due to short-rangedinteractions, such as those found in positronium. Effective quantum mechanics appears applicablefor systems that admit analytic long-range descriptions, but whose short-ranged effects are not re-liably or efficiently modeled. Potential applications of this approach include atomic and condensedmatter systems, but it may also provide a useful perspective for the study of blackholes.
I. INTRODUCTION
Effective field theory (EFT) has had many successeswithin subfields of physics that include condensed mat-ter, particle physics, astrophysics, and cosmology. Thesuccess of an effective field theory depends on a hierarchyof scales; the momenta or wavelengths of the experimen-tal probes or observations of a system must be markedlydifferent from the scale(s) of the processes not described,at least in detail, by the effective theory.In this work we apply many of the ideas of EFT toquantum mechanics. The starting point of our discus-sion begins with the description of contact interactions,or delta-functions potentials, in quantum mechanics intwo and three dimensions. It is known that such po-tentials sometimes require elaborate regularization andrenormalization schemes to ensure physically sensible re-sults are obtained [1]. In [1] it was advocated that non-trivial boundary conditions are a preferable alternativemethod to using delta functions.When applied to bound Coulomb states, this non-trivial boundary condition method – also known as themethod of self-adjoint extension – has been shown to pro-duce energy levels that obey Rydberg’s formula, at leastwhen the boundary condition parameter is small and pro-portional to the quantum defect [2]. In [2] it was shownthat a unique boundary condition can provide an effec-tive description of “UV” physics near the origin, such asthe effect of a finite nuclear radius or the Darwin fine-structure correction – really anything that is, or may be ∗ Electronic address: [email protected] approximated as a delta-function potential. There aretwo notable limitations to the analysis in [2]: (1) be-cause all non-trivial (cid:96) (cid:54) = 0 solutions to the Schrodingerequation are not normalizeable, the method applies onlyfor s -states ( (cid:96) = 0) and (2) it does not reproduce theRydberg-Ritz formula, the more accurate bound stateenergy formula in which the quantum defect is energy-dependent [3].The motivation of [4] was to extract a useful effec-tive theory that would apply for all angular momentumstates. In that work a finite region of space encompass-ing the origin was omitted from analysis, thereby naivelyobviating the need to discard the non-trivial (cid:96) (cid:54) = 0 solu-tions. The radius of what is referred to as the boundary ofignorance , r b was interpreted as a kind of short-distancecutoff on which the boundary conditions effectively cap-ture the omitted physics. In order to enforce unitarityconservation, however, the limit r b → averages to unity. Therefore it is not necessary to take the limit r b → x = a r X i v : . [ phy s i c s . a t o m - ph ] J a n x b , all eigenmodes must obey the same (Robin) boundarycondition in a standard analysis, i.e. ψ i ( x b ) + Z ψ (cid:48) i ( x b ) = 0 , (1)where the modes are labelled by generic index i , and Z is a real number; for example, Z = 0 corresponds tothe Dirichlet condition. The central equation of [7] iswhat results from promoting the boundary condition tobe mode-dependent, i.e. Z → Z i , or ψ i ( x b ) + Z i ψ (cid:48) i ( x b ) = 0 . (2)Because the boundary condition varies for different eigen-modes, this theory is not instantaneously hermitian orunitary; however, those standard conditions do holdwhen averaged over sufficiently long times .In this article we significantly extend the analysis of[7], demonstrating that this approach can be success-fully applied in three dimensions with coupling to thefull electromagnetic gauge field. We devote the bulk ofour effort to analysis of hydrogenic atoms and arrive atthe theory of quantum defects, albeit using a differentframework from earlier analyses (see, e.g., a well-knownreview by Seaton [8]). The theoretical framing here is inthe same spirit as that of effective field theory, howeverwe do not appeal to a Lagrangian formalism. We startwith the Schrodinger equation, using a Hamiltonian validat long distance; the conditions of the wavefunction onthe boundary of ignorance play a role analogous to theeffective action. We therefore consider this method to bea demonstration of what can be called effective quantummechanics.In Section II we derive the three-dimensional versionof the boundary condition (2) that respects electromag-netic gauge invariance and discuss its consequences. InSection III we analyze the bound states of the Coulombproblem and derive the theory of quantum defects as aconsequence of a low-energy effective theory describingthe broken SO (3) ⊗ SO (3) symmetry of the Schrodinger-Coulomb problem. In Section IV we check the effectivemethod against synthetically-generated data for a UV-complete model of an extended nucleus. In Section Vwe consider the successes and limitations of this non-relativistic theory applied to real systems. In Section VIwe consider decaying states. In Section VII we concludewith a discussion of our results, list possible applications,and mention some outstanding issues. Throughout thisarticle we use the natural unit convention (cid:126) = c = 1. Unitarity violation appears to be a consequence of only consid-ering the domain x ≥ x b , whereas a particle can in reality prop-agate in and out of the omitted region, 0 ≤ x < x b . II. BOUNDARY CONDITION ANDCONSEQUENCES
The dynamics of a point charge of mass m and charge Q coupled electromagnetically is described by the Hamil-tonian H = (cid:16) (cid:126)p − Q (cid:126)A (cid:17) m + Qϕ . (3)We expect that the boundary condition (2) can be pro-moted to a spherically symmetric and gauge-invariantversion, namely R i ( r b ) + Z i ( r b ) D r R i ( r b ) = 0 , (4)where D r = ∂ r − iQA r , (5)and A r is the radial component of the vector potential, (cid:126)A . It would then follow that, under a local U (1) trans-formation of the wave function and the electromagneticfield characterized by the function Ω,Ψ = e iQ Ω Ψ (cid:48) (cid:126)A = (cid:126)A (cid:48) + (cid:126) ∇ Ω ϕ = ϕ (cid:48) − ˙Ω , (6)equation (4) will be invariant.We derive (4) by mandating conservation of probabilityassociated with a single eigenmode where, for simplernotation, we note that Ψ represents such a mode: ddt (Ψ , Ψ) = − (cid:90) dV ∇ · (cid:126)J = 0 , (7)where the probability current density (cid:126)J following fromthe Hamiltonian (3) is (cid:126)J = i m (cid:104) ( ∇ Ψ ∗ ) Ψ − Ψ ∗ ( ∇ Ψ) + 2 iQ (cid:126)A | Ψ | (cid:105) . (8)Given the spherically symmetric boundary at r = r b , andseparability of the eigenmode asΨ = e − iωt R ( r ) Y (cid:96)m ( θ, φ ) , (9)the divergence theorem may be used to demonstrate that( D r R ) ∗ R − R ∗ D r R (cid:12)(cid:12)(cid:12)(cid:12) r = r b = 0 , (10)where D r is as defined in (5). Following [9], one maythen multiply by an arbitrary constant w with units oflength and define the dimensionless complex quantities x ≡ R ( r b ) (11) y ≡ wD r R ( r ) | r = r b , (12)so that equation (10) is then equivalent to | x + iy | − | x − iy | = 0 . (13)The argument of the two terms above have an equal mag-nitude and differ only by an arbitrary phase factor e iθ ; itfollows that R ( r b ) − cot θ wD r R ( r ) (cid:12)(cid:12)(cid:12)(cid:12) r = r b = 0 , (14)which is boundary condition (4) once we make the iden-tification Z = − cot θ w ; (15)again, we note that θ , w and, therefore, Z are unique tothe specific eigenmode in question.There are consequences of promoting the boundarycondition to be mode-dependent. For example, theHamiltonian is not Hermitian, which is observed by com-puting (cid:104) H Ψ i , Ψ j (cid:105) − (cid:104) Ψ i , H Ψ j (cid:105) = − m (cid:90) dV ∇ · (cid:104)(cid:16) (cid:126)D Ψ i (cid:17) ∗ Ψ j − Ψ ∗ i (cid:16) (cid:126)D Ψ j (cid:17)(cid:105) = 12 m r b (cid:90) d Ω (cid:2) ( D r Ψ i ) ∗ Ψ j − Ψ ∗ i ( D r Ψ j ) (cid:3) r = r b , (16)which is not generally zero for two distinct eigenstateslabeled by i and j . In particular, because of the orthog-onality of the spherical harmonics, this term is nonzerowhen states i and j have the same angular momentumquantum numbers .If the angular quantum numbers are the same, e.g. (cid:96) i = (cid:96) j , etc., then (cid:104) H Ψ i , Ψ j (cid:105) − (cid:104) Ψ i , H Ψ j (cid:105) = 12 m r b ( Z i − Z j ) (cid:2) ( D r R i ) ∗ ( D r R j ) (cid:3) r = r b e − i ( ω j − ω i ) t , (17)Equation (17) is gauge invariant and never zero when i (cid:54) = j , but does average to zero over a period 2 π/ | ω i − ω j | ;the same is true about the inner product of two distincteigenmodes. It may therefore be said that hermiticityand orthogonality do not generally hold at each instant,but they do in a time-averaged sense.Unitarity is also temporarily violated. By construc-tion, the norm of each eigenmode is equal to unity for all The outward normal to the boundary points in the inward radialdirection, hence the change of sign in the last line of (16). This would also include the spin quantum number if it were con-sidered in this analysis. time, but the same cannot be said for composite state.Following [7], consider the stateΥ = c i Ψ i + c j Ψ j , (18)for two complex coefficients, c i and c j . We assume thestandard normalization condition | c | + | c | = 1 , (19)which will be justified below. The inner product of thecomposite state with itself is therefore (cid:104) Υ , Υ (cid:105) = 1 + c ∗ i c j (cid:104) Ψ i , Ψ j (cid:105) + c ∗ j c i (cid:104) Ψ j , Ψ i (cid:105) . (20)The last two offending terms do not vanish because theeigenmodes are not instantaneously orthogonal. Becausethe time derivative of the inner product is ddt (cid:104) Ψ i , Ψ j (cid:105) = i (cid:104) H Ψ i , Ψ j (cid:105) + cx. conj. (21)the time derivative of the composite state is ddt (cid:104) Υ , Υ (cid:105) = c ∗ i c j ddt (cid:104) Ψ i , Ψ j (cid:105) + cx. conj.= − ( Z i − Z j ) | ρ ij | sin [( ω i − ω j ) t + θ ij ] , (22)where ρ ij = c ∗ i c j m r b (cid:2) ( D r R i ) ∗ ( D r R j ) (cid:3) (cid:12)(cid:12)(cid:12)(cid:12) r = r b (23)and θ ij = arg ρ ij . (24)Apparently, the norm of the composite state is (cid:104) Υ , Υ (cid:105) = 1 + ( Z i − Z j ) ω i − ω j | ρ ij | cos [( ω i − ω j ) t + θ ij ] , (25)which averages to unity over a period 2 π/ | ω i − ω j | .The above analysis suggests that the best way to nor-malize a composite state is to demand (cid:104) Υ , Υ (cid:105) T ≡ , (26)where the subscript indicates an averaging over a time, T of appropriate length. A similar condition is apparentlyobeyed for the orthogonality of modes (cid:104) Ψ i , Ψ j (cid:105) T = 0 , (27)which justifies (19), as well as the hermiticity of theHamiltonian, (cid:104) H Ψ i , Ψ j (cid:105) T = (cid:104) Ψ i , H Ψ j (cid:105) T . (28)Precisely what is considered appropriately long de-pends on the physical system being studied. Becausewe focus almost exclusively on Coulomb bound states inthe sections below, let us consider a composite state builtfrom two such eigenmodes with the same angular quan-tum numbers. Clearly, averaging over a time longer than2 π/ | ω i − ω j | is sufficient according to the above analysis,and this is always larger than the time scale associatedwith any omitted short-distance physics. T must alsobe much shorter than any processes not included in thisanalysis that occur over relatively long times, such asthe spontaneous transition time between the two states,which is also true . Although this by no means consti-tutes a proof that this method will work for all systems,it does suggest that its success as an effective theory de-pends on a clear hierarchy of time scales, in addition tolength scales. III. COULOMB STATES
Here we consider an electron of charge − e bound to apositive nucleus of charge Ze , so that the long-distance Hamiltonian is given by equation (3) with (cid:126)A = 0 andscalar potential ϕ = Zer . (29)The time-independent radial Schrodinger equation is − r ∂ r (cid:0) r R (cid:48) ( r ) (cid:1) + (cid:18) (cid:96) ( (cid:96) + 1) r − κr + q (cid:19) R ( r ) = 0(30)where κ = Zmα , (31) α (cid:39) /
137 is the fine structure constant, and the energyeigenvalues are defined by E = − q m . (32)As described in [4] there is one independent solution tothis differential equation that is guaranteed to be square-integrable in the r → ∞ limit . We write this as, up toa normalization constant, R ( r ) = e − qr (2 qr ) (cid:96) U (cid:18) (cid:96) − κq (cid:12)(cid:12)(cid:12) (cid:96) + 1) (cid:12)(cid:12)(cid:12) qr (cid:19) , (33)where U is Tricomi’s confluent hypergeometric function. The transitions between states with the same angular momentumare accompanied by the emission of at least two photons. Suchprocesses occur at a rate that does not exceed ∼ | ω i − ω j | α (see, e.g., [10]). One of us (D.M.J.) would like to acknowledge Harsh Mathurfor explaining the importance of this particular (decaying) linearcombination of the two solutions to the confluent hypergeometricequation.
Quantization of the energies comes from application ofthe boundary condition (4). It must be obeyed in such away that any observables, such as the energy or, equiva-lently, q are independent of the location of the boundary.Because this is a long-distance effective theory, we expectthe spatial scale of the wavefunction to be much largerthan the boundary radius, or qr b (cid:28)
1. This means that,in principle, equation (4) could be expanded to arbitraryorder in qr b and would then provide an arbitrarily pre-cise analysis. Any function of q could then be solved for;in particular, we find it best to solve for ψ (cid:16) (cid:96) − κq (cid:17) ,where ψ ( z ) ≡ Γ (cid:48) ( z )Γ( z ) (34)is the digamma function.The digamma function is readily seen to appear in theseries form of the Tricomi function, given in AppendixA. By using the digamma identity, ψ (1 + z ) = ψ ( z ) + 1 z , (35)one could, in principle, solve equation (4) to write ψ (cid:18) (cid:96) − κq (cid:19) = F (cid:96) [ Z ( r b ) , r b ] , (36)where F (cid:96) is a function of both the boundary function Z ( r b ) and r b , and accurate up to a particular order inthe expansion parameter, qr b . One could differentiatethis equation with respect to r b and demand that it beequal to zero, resulting in a first order differential equa-tion for the boundary function Z ( r b ). It could then besaid that Z ( r b ) runs, in the sense of a renormalizationgroup, with the boundary radius in way that ensures thatthe eigenvalues do not depend on where the boundary is;this was first described in [4], but a similar proceduremay be found in [5].However, at this point we simply integrate with respectto r b to implicitly solve for Z ( r b ), i.e. F (cid:96) [ Z ( r b ) , r b ] = χ (cid:96) ( q ) , (37)where χ (cid:96) ( q ) is an arbitrary integration function that mustonly be constant with respect to r b . As in [7], we positthat χ (cid:96) ( q ) captures the unspecified interactions behindthe boundary, r < r b . This establishes the result ψ (cid:18) (cid:96) − κq (cid:19) = χ (cid:96) ( q ) . (38)We follow this apparently tautological procedure be-cause, in practice, solving explicitly for Z ( r b ) is cumber-some even at lowest order in the qr b expansion and for (cid:96) = 0; it becomes increasingly challenging at higher orderand at higher values of (cid:96) . Instead, given the series formof the Tricomi function (see Appendix A) and boundarycondition (4) that must be satisfied for arbitrary r b , wemake the generic ansatz for the boundary function, Z ( r b ) = κ − ∞ (cid:88) j =1 ( c j + d j ln 2 κr b ) (2 κr b ) j . (39)By solving (4) for each term proportional to r jb andln r b r jb we can determine the dimensionless coefficients c j and d j uniquely, up to the arbitrary integration func-tion χ (cid:96) ( q ) which must appear in any equation containing ψ (cid:16) (cid:96) − κq (cid:17) . It may be verified that χ (cid:96) ( q ) only appearsin the c j and does not first appear in the series until c (cid:96) +2 ;it then appears in all subsequent c j which also may beunderstood through use of the digamma identity (35).Below we will not explicitly refer to χ (cid:96) ( q ), but its pres-ence is implied in any discussion of c (cid:96) +2 , which we sim-ply refer to as the integration function.We do not, at present, have an analysis valid for ar-bitrary (cid:96) . However, we have checked that the followingprocedure works at least up to (cid:96) = 3; it therefore seemsimplausible that it would not work to arbitrarily high (cid:96) .We explicitly show the procedure for (cid:96) = 0 and (cid:96) = 1 be-low and the (cid:96) = 2 analysis may be found in Appendix B.The summary is that at each (cid:96) we may write the solutionsas deviations from their canonical form as q = κn − δ (cid:96) (40)where n is an integer and δ (cid:96) is called the quantum defect (see, e.g., [8]). For each (cid:96) -state we have considered it ispossible to write the defect in the form δ (cid:96) = δ (cid:96) (0) + λ (cid:96) (1) E Λ + λ (cid:96) (2) (cid:18) E Λ (cid:19) + . . . , (41)in other words, a low energy expansion in E/ Λ, where Λis a high energy (UV) scale.The connection between non-trivial boundary condi-tions and the quantum defect ansatz of (40) was firstmade in [2]. The ansatz for q , equation (40), is a devia-tion from the canonical solutions, q = κn , (42)and is motivated by two distinct considerations. The firstreason is obvious: from an experimental point of view,hydrogenic atoms and highly-excited (Rydberg) states oflarge atoms are known to display spectra that are largelyin agreement with (42) – this was, of course, one of theearliest successes of quantum mechanics. This canoni-cal case apparently corresponds to the limit χ (cid:96) → ±∞ , An arbitrary choice of length of (2 κ ) − was put into the argu-ment of the logarithm; any other choice can be made with acorresponding redefinition of the c j ’s. suggesting that it or, equivalently, the c (cid:96) +2 will actuallytake on very large (but finite) values when this methodis applied to real systems.The second reason to use the canonical solutions asa point of departure is that they are special from thetheoretical point of view; they are the unique solutionsfor which there is exists an n -fold degeneracy at eachenergy level, n . This can be traced to presence of a“hidden” SO (3) ⊗ SO (3) symmetry, the result of a con-served Runge-Lenz vector, in addition to angular mo-mentum (see, e.g., [11]). As deviations are made fromthe canonical solution (42), therefore, one could say thatthe SO (3) ⊗ SO (3) symmetry is broken to the usual SO (3) symmetry associated with 3-dimensional rotations[2]. Although the Runge-Lenz vector operator continuesto be conserved, the (cid:96) -dependent boundary conditionsmean that it acts on a different domain than that of theHamiltonian, making it an unphysical operator . A. Effective description of (cid:96) = 0 bound states
For the (cid:96) = 0 solutions, the term-by-term considerationof the boundary condition (4) with the ansatz of (39)yields c = − γ − q κ − (cid:20) ln qκ + ψ (cid:18) − κq (cid:19)(cid:21) , (43)where γ is the Euler-Mascheroni constant. Performing anasymptotic expansion of the digamma function yields c = − π π κq − γ − EE Ry − (cid:18) EE Ry (cid:19) + O (cid:18) EE Ry (cid:19) . (44)where E Ry ≡ κ m . (45)We make the ansatz q = κn − δ (46)where, as argued in [7], it should always be possible todefine | δ | ≤ /
2. It appears that this is the only depar-ture from the original quantum defect model, whereinthere is no such restriction on the size of the defect (see, See, e.g., [12] for a discussion about an analogous problem on aconical space in two dimensions. Although this series does not converge, any truncation will beincreasingly accurate as | E | decreases. In [7] the definition n − δ ≡ ˜ n − ˜ δ was made, where ˜ n is theclosest integer to n − δ , therefore (cid:12)(cid:12)(cid:12) ˜ δ (cid:12)(cid:12)(cid:12) < / | δ | < / e.g., [3]). We expand equation (44) in small δ and findthat the defect can be solved for implicitly as δ = (cid:16) − π δ − π δ + O (cid:0) δ (cid:1)(cid:17) c + 2 γ + EE Ry + (cid:16) EE Ry (cid:17) + O (cid:16) EE Ry (cid:17) . (47)We do not know what functional form the integrationfunction – or c – should have, but two comments are war-ranted. Firstly, deviations from the canonical Coulombspectrum are assumed here to be the result of short-ranged/high-energy physics not included explicitly in theCoulomb potential, and therefore we expect those de-viations not to depend explicitly on the ratio E/E Ry .Secondly, a series form for c as an expansion in E oversome high energy scale is arguably the simplest guess,and is also consistent with the well-known and successfulapproach taken when writing down an effective action inthe context of effective field theory. Not knowing a pri-ori what the coefficients of this expansion should be, weparametrize the denominator of equation (47) to be inthe series form A + A E Λ + A (cid:18) E Λ (cid:19) + . . . , (48)where Λ is a high energy scale and, in the parlance offield theory, we call the A i renormalized expansion co-efficients. Equivalently, the integration function couldapparently be written c = B + B E Λ + B E Λ + . . . , (49)where the bare expansion coefficients, B i are related totheir renormalized counterparts by B = A − γB = A −
124 Λ E Ry B = A − E , (50)and so on. Summarizing, we have δ = (cid:16) − π δ − π δ + O (cid:0) δ (cid:1)(cid:17) A + A E Λ + A (cid:0) E Λ (cid:1) + . . . , (51)which can be iteratively solved for δ . Without any lossof generality, we therefore write δ = δ + λ E Λ + λ (cid:18) E Λ (cid:19) + . . . , (52)where δ and the λ i are dimensionless coefficients. Onecould speculate that, because δ → δ is proportional to E Ry / Λ, pos-sibly raised to a positive power. We demonstrate below that this is indeed the case, at least when applied to thehydrogen atom.
B. Effective description of (cid:96) = 1 bound states
Following the procedure used in the previous section,for (cid:96) = 1 we discover c = 9 − γ − qκ − (1 − γ )16 q κ + 164 q κ + q − κ κ (cid:20) ln qκ + ψ (cid:18) − κq (cid:19)(cid:21) . (53)After expanding the digamma function in small q/κ andwriting this expression in terms of energies we find δ = (cid:16) EE Ry (cid:17) (cid:16) − π δ − π δ + O (cid:0) δ (cid:1)(cid:17) c − +2 γ − − γ EE Ry + (cid:16) EE Ry (cid:17) + O (cid:16) EE Ry (cid:17) . (54)Here a convenient parametrization of the denominator inequation (54) is the series form (cid:18) EE Ry (cid:19) (cid:32) A + A E Λ + A (cid:18) E Λ (cid:19) + . . . (cid:33) , (55)or, equivalently, c = B + B E Λ + B E Λ + . . . (56)where B = A
32 + 9256 − γ B = A
32 + (cid:18) − γ (cid:19) Λ E Ry B = A − (cid:18) Λ E Ry (cid:19) . (57)The result is the same as that of the (cid:96) = 0 case, namely δ may be put in a form identical to equations (51) and(52). The (cid:96) = 2 analysis follows similarly and may befound in Appendix B. C. Brief comments about scattering
Because of the apparent equivalence between this effec-tive approach and that of quantum defect theory we donot dwell on the analysis of scattering states. We simplynote that, whereas in bound state calculations the defi-nition q = − mE is made, for scattering one defines thewave number k by k = 2 mE , (58)where the energy, E >
0. This suggests an analytic con-tinuation of the integration function χ (cid:96) ( q ) in the vari-able to q → − k , in other words an analytic continua-tion of the defects δ (cid:96) ( E ) from E <
E >
0. This isprecisely what is known to occur within quantum defecttheory and we direct the interested reader toward therelevant literature (see, e.g., [8] and references therein).
IV. FITS TO SYNTHETIC DATA
Here we consider the long-range Coulomb potentialmodified at short distance with a specific UV-completion,namely one in which there is a constant “nuclear” chargedensity. The scalar potential is therefore ϕ = (cid:40) Zer R , (0 ≤ r ≤ R nuc ) Zer , ( r > R nuc ) . (59)The time-independent radial Schrodinger equation inthe nuclear interior is − r ∂ r (cid:0) r R (cid:48) ( r ) (cid:1) + (cid:18) (cid:96) ( (cid:96) + 1) r − r b + q (cid:19) R ( r ) = 0(60)where b ≡ R Zmα . (61)Imposing regularity at the origin, the solution to (60)may be written, up to a normalization constant, as R ( r ) = e − i ( rb ) (cid:16) rb (cid:17) (cid:96) × M (cid:18) (cid:96) + 34 − i qb ) (cid:12)(cid:12)(cid:12) (cid:96) + 32 (cid:12)(cid:12)(cid:12) i (cid:16) rb (cid:17) (cid:19) , (62)where M is Kummer’s hypergeometric function.It is important to separately consider two differenttypes of hydrogenic systems, namely those in which de-viations from the Coulomb potential occur at radii thatare smaller and larger than the Bohr radius. For thisreason we consider two examples in which the nuclearradius, R nuc satisfies either or κR nuc < κR nuc > r = R nuc .To display the robustness of the effective theory we ap-ply it to bound states with a leading order (LO) fit usingonly δ (cid:96) (0) , next-to-leading order (NLO) by fitting for δ (cid:96) (0) and λ (cid:96) (1) , and next-to-next-to leading order (NNLO) byfitting for δ (cid:96) (0) , λ (cid:96) (1) , and λ (cid:96) (2) . We assume both m and α are perfectly known by some independent means, andutilize equations (32), (40), and (41) to fit to the lowestenergy levels, i.e. those with the largest | E | , so as tomake predictions for the higher energy levels. In Figures 1 and 2 we display the relative error in thepredicted energy levels for κR nuc = 0 .
31 when (cid:96) = 0 and (cid:96) = 1, respectively; in those figures we normalize theenergies to the ground state, E . - - - E ( E ) - - - - Δ E E ℓ = Canonical LO NLONNLO
FIG. 1: Relative errors in the (cid:96) = 0 binding energies, com-pared to the UV-complete model, wherein κR nuc = 0 . - - - - E ( E ) - - - - Δ E E ℓ = Canonical LO NLONNLO
FIG. 2: Relative errors in the (cid:96) = 1 binding energies, com-pared to the UV-complete model wherein κR nuc = 0 .
31, as inFigure 1.
For the large nuclear radius we choose κR nuc = 2 . long-distance effective theory; thusone should only expect it to provide accurate predictionswhen the characteristic length scale of the wavefunctionis large compared to the nuclear radius, or qR nuc < q = κ/
21 and can therefore makepredictions beginning near q = κ/
22. In Figures 3 and4 we display our results for (cid:96) = 0 and (cid:96) = 1, respec-tively, normalizing the energies to the 21st excited state, E . Although the errors initially grow marginally ashigher energy levels are considered, eventually there is aturnover and the errors begin to decrease. In any case,at any given energy level, the effective method gives pre-dictions that are always more accurate at higher order. E ( E ) - - - Δ E E ℓ = Canonical LO NLONNLO
FIG. 3: Relative errors in the (cid:96) = 0 binding energies, com-pared to the UV-complete model, wherein κR nuc = 2 . E ( E ) - - - Δ E E ℓ = Canonical LO NLONNLO
FIG. 4: Relative errors in the (cid:96) = 1 binding energies, com-pared to the UV-complete model, wherein κR nuc = 2 .
73, asin Figure 3.
V. APPLICATION TO PHYSICAL SYSTEMS
In so far as Rydberg atoms are concerned, the δ ansatz of (41) is equivalent to the usual quantum defectmethod(s), wherein the modified Rydberg-Ritz expres-sion is often written δ n(cid:96)j = δ + δ ( n − δ ) + δ ( n − δ ) + . . . , (63)for some experimentally determined constants δ , δ , etc.[3]. The only difference with our approach, as mentionedin Section III A, is that here we restrict the size of thedefect to obey | δ | <
1. For example, the measured tran-sition frequencies of the alkalis Na, K, and Rb arefit with the original defect model to give a leading order s -state defect, δ QDT0 (cid:39) . .
180 and 3 . δ = 0 . . . n = 1, rather thanthe principal quantum number corresponding to its rowin the periodic table.Fitting the hydrogen spectrum, for example, one canachieve reasonably accurate results; however, the predic-tions become only marginally more accurate at higherorder in the effective theory, and this is likely because ofrelativistic effects that are not accounted for. Considerthe effective theory applied to a particular state of a hy-drogenic atom in which the nucleus has a charge + e . Atleading order, δ (cid:96) = δ (cid:96) (0) so that, expanding in small δ (cid:96) (0) ,the energy levels are E = − mα n − mα n δ (cid:96) (0) − mα n δ (cid:96) (0) + . . . (64)The first term corresponds to the canonical eigenval-ues, whereas the second term is proportional to the cor-rections that are usually obtained using perturbationtheory; in particular, short-ranged corrections to theCoulomb potential proportional to 1 /r , 1 /r , etc., aswell as a delta-function centered about r = 0 give cor-rections proportional to n − (see, e.g., [13]). Let us callthose potential corrections U rel . Perturbation theory isused to correct the canonical energy levels by an amount (cid:104) Ψ | U rel | Ψ (cid:105) = − mα (2 j + 1) n , (65)where j = (cid:96) ± / δ (cid:96) (0) = O ( α ) , (66)or δ (cid:96) (0) ∝ E Ry / Λ, which is true when we set the highenergy scale Λ = m , the mass of the electron.At the next-to-leading order, we apparently have δ (cid:96) = δ (cid:96) (0) − λ (cid:96) (1) α n . (67)With an additional parameter there is, of course, animproved fit to the hydrogen spectrum; however, it isonly a marginal improvement. Although equation (64)is modified, the effect of the parameter λ (cid:96) (1) only ap-pears at order n − , whereas there is already trouble withthe order n − term. This is because there is a remain-ing fine-structure effect not captured by U rel , but insteadcomes from the relativistic correction to the kinetic en-ergy. That kinetic correction amounts to+ 3 mα n (68)for all (cid:96) states of Hydrogen, which cannot be accountedfor simultaneously with the order n − correction in equa-tion (65). In any case it would not be appropriate; therelativistic correction to the kinetic energy is not a short-ranged effect that should be hidden behind the boundaryof ignorance.Although this and the preceding sections demonstratethe utility of the non-relativistic effective quantum me-chanics, a relativistically corrected version of the theoryis clearly warranted. Those results will appear in forth-coming work [15]. VI. DECAYS DUE TO UV EFFECTS
Here we consider if and how the analysis must be mod-ified if the eigenmode in question decays at a rate, Γ, viasome interaction(s) near the origin . In other words,normalizing the state at t = 0, (cid:104) Ψ , Ψ (cid:105) = e − Γ t , (69)where, for simpler notation, we note that Ψ represents asingle time-dependent eigenmode. We assume the eigen-modes may be written in the variable-separated form asΨ = e − iωt − Γ2 t R ( r ) Y (cid:96)m ( θ, φ ) , (70)where ω is real and any normalization constant is ab-sorbed into R ( r ). In the case of a decaying state wemust modify equation (7) to ddt (cid:104) Ψ , Ψ (cid:105) = − (cid:90) dV ∇ · (cid:126)J = − Γ e − Γ t . (71)The probability current density is still given by equation(8), but here the application of the divergence theoremresults in a modification to equation (10), namely( D r R ) ∗ R − R ∗ D r R (cid:12)(cid:12)(cid:12)(cid:12) r = r b = 2 imr b Γ . (72)Multiplying this equation by w , an arbitrary constantwith units of length and making the same definitions for x and y as in equation (11), it may be verified that equation(72) is equivalent to (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + iy (cid:32) − mw Γ r b | y | (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − iy (cid:32) mw Γ r b | y | (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . (73)Because there is an equivalence of the arguments up to aphase factor e iθ , one may write R − (cid:32) cot θ w + i m Γ r b | D r R | (cid:33) D r R = 0 . (74) This effective method can describe decays of states due to short-ranged effects, such as annihilation; the analysis presented heredoes not describe, e.g., transitions from one state to another.
It follows that the boundary condition is a complexifiedform of (4), namely R ( r b ) + ( Z re ( r b ) + i Z im ( r b )) D r R ( r b ) = 0 , (75)where Z re ( r b ) = − cot θ w , (76)and Z im ( r b ) = − m Γ | r b D r R | . (77)The presence of D r R ( r b ) in the definition Z im ( r b )may seem strange because it suggests a non-linearly real-ized boundary condition; however, we remind the readerthat this analysis is meant to be approximate. We haveshown that if the eigenmode is exactly proportional to e − Γ2 t , then it follows that equation (77) must hold ex-actly . However, such a rigid expectation is inappropriate.Clearly, this method is not capable of describing, e.g.,deviations from a pure exponential decay law which isknown to occur at very short times after an unstable statehas been established (see, e.g., [16]). The method de-scribed here is intended as a long-distance – hence a long-time – effective theory. We therefore suggest that equa-tion (77) gives only a qualitative relationship: Z im ( r b ) isboth proportional to the decay rate of the system andbounded according to Z im ( r b ) ≤
0. Beyond these qual-itative features, we only assume that Z im ( r b ) has some dependence on r b that can be determined in the analysisof a particular system.Consider positronium, a system described at long dis-tance by a Coulomb potential with a reduced mass of m/
2. The analysis from Section III follows in a nearlyidentical fashion, but the energy eigenvalues and quan-tum defects are complex, i.e. E = − q m = ω − i Γ2 , (78)where q = mα n − δ ) , (79)and, at lowest order, δ = δ , re + iδ , im . (80)We will further assume that δ , im (cid:28) δ is still possible and thereforethe analysis of Section III is equally valid; we confirmthis below. From equations (78), (79), and (80), it isapparent that the real part of the energy ω = − m ( Zα ) n − m ( Zα ) n δ , re + . . . , (81)whereas the decay rate is given byΓ = m ( Zα ) n δ , im + . . . , (82)0which displays the standard n − dependence expectedfrom equation (65). At higher order, δ , im would affect ω as well; however, we have already established in SectionV that this analysis is limited because it is missing rela-tivistic corrections and therefore only these lowest-orderresults are worth reporting here.Within Quantum Electrodynamics, the lowest orderdecay rate of positronium is predicted to be (see, e.g.,[17]) Γ QED = (cid:40) mα n (singlet) π (cid:0) π − (cid:1) mα n (triplet) (83)which means that matching to that UV-complete theorywould yield δ , im = (cid:40) O ( α ) (singlet) O ( α ) (triplet) . (84) VII. DISCUSSION
We have shown how to construct a nonrelativistic effec-tive quantum mechanics in three dimensions for systemspossessing spherical symmetry. The short-distance cutofflength, r b is a conceptual and calculational crutch usedto derive our results and ultimately vanishes from any fi-nal result. The role of the boundary function is that of acoupling constant; each mode “feels” a different couplingconstant that varies with energy. A high energy scale,Λ appears in the low-energy expansion of physical quan-tities, such as bound states. We focused primarily onthe Coulomb interaction and have found non-trivial re-sults for all angular momentum states, ultimately show-ing an equivalence to quantum defect theory. We havealso shown the method provide a means of describingdecays due to effects at short distance.The most pressing question is how to apply thisapproach relativistically for application to high precisionspectroscopy of atoms and molecules. Rydberg atoms,are of particular importance because they have potentialapplications in quantum computing and electromagneticfield sensing, for example [18]. There also appears tobe a pertinent application to positronium, in particularbecause of a recently discovered discrepancy between ameasured transition frequency in that system and thepredictions from QED [19]. These ideas presumablyalso have applications in the areas of condensed matter,particle physics, and possibly gravitation. Like theCoulomb interaction, blackholes provide a 1 /r potentialat long distances and exhibit a kind of boundary, theevent horizon, behind which information is obscured. Acknowledgements
Thanks are owed to Harsh Mathur, with whom manydiscussions were had during early stages of this work.Thanks are also owed to the late Bryan Lynn for raisingthe question about decaying systems and positronium, inparticular. One of us (MJ) would like to thank Hamil-ton College for funding during a portion of this workand another of us (DMJ) would like to thank the Hamil-ton College Physics Department for its hospitality duringearly stages of this work.
Appendix A: Series form of the Tricomi function
The solutions to the Schrodinger-Coulomb problem in-volve solutions to the confluent hypergeometric equation yg (cid:48)(cid:48) ( y ) + ( B − y ) g (cid:48) ( y ) − Ag ( y ) = 0 . (A1)A standard textbook analysis, e.g., [20], involves a seriesansatz one may show that gives two independent solu-tions M ( A, B, y ) = ∞ (cid:88) n =0 n ! A ( n ) B ( n ) y n , (A2)known as Kummer’s function, and M ( A, B, y ) = y − B M ( A + 1 − B, − B, y ) (A3)The Tricomi function, U ( A, B, y ) is the special linearcombination of the two that is guaranteed to decay as y → ∞ , usually defined as U ( A, B, y ) ≡ Γ(1 − B )Γ( A − B + 1) M ( A, B, y )+ Γ( B − A ) M ( A, B, y ) ; (A4)this along with, e.g., M ( A, B, y ) may be chosen as a lin-early independent set of solutions as long as B is not aninteger greater than 1.In the Coulomb problem, however, B = 2 + 2 (cid:96) , socare must be taken to understand the series form of U ( A, B, y ). One can, for example, let B = 2 + 2 (cid:96) + (cid:15) ,where (cid:15) is treated as perturbatively small; in the end onecan let (cid:15) → U ( A, (cid:96), y ) = 1Γ ( A ) Γ ( A − (cid:96) − × (cid:32) − (cid:96) (cid:88) n =0 ( − y ) n − (cid:96) − n ! Γ ( A − (cid:96) − n ) Γ (2 (cid:96) + 1 − n )+ ∞ (cid:88) n =0 y n n ! Γ( A + n )Γ(2 (cid:96) + 2 + n ) (cid:20) ψ ( A + n ) − ψ (2 (cid:96) + 2 + n ) − ψ ( n + 1) + ln y (cid:21)(cid:33) (A5)1 Appendix B: Effective description of (cid:96) = 2 boundstates
Here we find c = 59 − γ
373 248 −
520 736 qκ + (180 γ − q κ + 56912 q κ + (35 − γ )31 104 q κ − q κ − (cid:0) q − q κ + κ (cid:1) κ (cid:20) ln qκ + ψ (cid:18) − κq (cid:19)(cid:21) . (B1)After expanding the digamma function in small q/κ andwriting this in terms of energies we find c = − π (cid:32) EE Ry + 4 E E (cid:33) cot π κq + 59 − γ
373 248 − γ − (cid:18) EE Ry (cid:19) − − γ (cid:18) EE Ry (cid:19) + O (cid:18) EE Ry (cid:19) . (B2)We make the defect ansatz in equation (46) to find δ = (cid:32) EE Ry + 4 E E (cid:33) (cid:18) − π δ − π δ + O (cid:0) δ (cid:1)(cid:19) × (cid:20) c − γ − (cid:18) − γ (cid:19) EE Ry + (cid:18) − γ (cid:19) (cid:18) EE Ry (cid:19) + O (cid:18) EE Ry (cid:19) (cid:35) − . (B3) We may parametrize the denominator of equation (B3)to be in the series form (cid:32) EE Ry + 4 E E (cid:33) (cid:32) A + A E Λ + A (cid:18) E Λ (cid:19) + . . . (cid:33) , (B4)or, equivalently, c = B + B E Λ + B E Λ + . . . (B5)where B = A
10 368 + 59373 248 − γ B = A
10 368 + (cid:18) − γ (cid:19) Λ E Ry B = A
10 368 + (cid:18) − γ (cid:19) (cid:18) Λ E Ry (cid:19) . (B6)It follows that δ may be put in a form identical to equa-tions (51) and (52). [1] R. Jackiw, Diverse topics in theoretical and mathematicalphysics (World Scientific, 1995).[2] S. Beck, Ph.D. thesis, Case Western Reserve Uni-versity (2016), URL http://rave.ohiolink.edu/etdc/view?acc_num=case1465577450 .[3] T. F. Gallagher,
Rydberg Atoms , Cambridge Monographson Atomic, Molecular and Chemical Physics (CambridgeUniversity Press, 1994).[4] D. M. Jacobs, J. Phys.
A49 , 295203 (2016), 1511.03954.[5] C. P. Burgess, P. Hayman, M. Williams, and L. Zalavari,JHEP , 106 (2017), 1612.07313.[6] C. P. Burgess, P. Hayman, M. Rummel, M. Williams,and L. Zalavari, JHEP , 072 (2017), 1612.07334.[7] D. M. Jacobs, Phys. Rev. A , 062122 (2019),1909.13407.[8] M. J. Seaton, Reports on Progress in Physics , 167 (1983), URL https://doi.org/10.1088%2F0034-4885%2F46%2F2%2F002 .[9] G. Bonneau, J. Faraut, and G. Valent, Am.J.Phys. ,322 (2001), quant-ph/0103153.[10] R. Fitzpatrick, Quantum Mechan-ics .[11] S. Weinberg,