arXiv:1404.2940v2 [physics.class-ph] 8 Oct 2014 omnfoiga h eeaie oriae na in the on indeed coordinates is time generalized This and the space. as configuration space four-dimensional footing treat inertial common motion given which a a of exist Hamiltonian from and equations formulations[4] observed Lagrangian Covariant as spatial frame. energy correct reference total the the which to and framework, relativistic rise a gives with compliant made be functions. energy potential ordinary H h qaino ri.TeLgaga sietfidas identified is Lagrangian The solving by orbit. followed L of written, re- equation easily this the be of can Integral system Jacobi holonomic duced corresponding this the for and one-body Lagrangian system equivalent classical an The into problem.[3] equation two- force the central convert to body is mechanics classical non-relativistic eaiesprto ewe h atce ( the of particles magnitude the the between on separation only relative depends function tential e r osat fmotion. sys- of an- the constants of total are energy the tem mechanical particular, total values the In momentum, expectation gular etc. yield observables, to physical prob- mechanics, of atom[2] quantum hydrogen in the to orbit; [1]) lem of (eg. equation mechanics the classical obtain of in solved textbook electrostatic is standard potential any the central the under The problem e.g. two-body potential. gravitational physics, po- Newtonian central the in and since Coulomb common useful particularly are mechan- is tentials quantum problem and The classical ics. in problem studied well ∗ 1 † h ug-eztno sas osato motion. of constant a also is tensor Runge-Lenz The [email protected] [email protected] h arninadHmloinfruain can formulations Hamiltonian and Lagrangian The h eta oc rbe,i hc h riaypo- ordinary the which in problem, force central The = = T T eaiitccretost h eta oc rbe nag a in problem force central the to corrections Relativistic + − V V with , n h aitna rJcb nerlas Integral Jacobi or Hamiltonian the and , hng nt eieteHmloinadetmt h correct in the power estimate fourth and the Hamiltonian to the the derive reproduce up to can on Lagrangian classical go regular then the for of potential function part energy a generalized potential generalized a a for using expression formulation, Lagrangian the n h eaiitccretost it. to students corrections undergraduate relativistic by resu the used semi-cla known and be a the can using with and value agreement atom pedagogical reasonable hydrogen give the f problems of gravitational both energy the state our under employ ground We orbit the circular system. the the of to energy corrections f potential central and the kinetic to the corrections relativistic of understanding epeetanvltcnqet banrltvsi correct relativistic obtain to technique novel a present We .INTRODUCTION I. T eateto hsc,Ida nttt fTcnlg Roor Technology of Institute Indian Physics, of Department en h iei nryand energy kinetic the being 1 h tnadapoc in approach standard The sme Singh Ashmeet | ~ v | /c efudta u oki bet rvd oecomprehensiv more a provide to able is work our that found We . r savery a is ) V ∗ approach the n io rsn Patra Krishna Binoy and eto eeosteLgaga n aitna of Hamiltonian and Lagrangian the develops the derive 4 a to function. Section define energy 3 potential section to dependent in velocity to force use generalized corrections we central which relativistic force, square add generalized inverse we attractive, 2, the section In follows. problem approach. the textbook non-trivial to standard the the corrections using to avoid relativistic able are add to and successfully is formulation able covariant approach the are in This complexities we since contributions. interesting small negligibly have oeta nrystp u arnini fteform, the of is Lagrangian Our dependent L velocity setup. generalized energy central a potential using the we by to problem paper, corrections this force Dirac In relativistic famous problem. incorporated the of- have the into for which insight by mechanics equation excellent quantum fers governed relativistic relativistic is true in that The equation[5] atom like hydrogen cases equations particle. possible the the single simplest of the a complexity for the of even that issue, seen an invari- is is Lorentz it is but which ant, formulation relativistic correct nteecss atcesed r eirltvsi such hydrogen semi-relativistic bodies. are the celestial speeds of particle that of cases, state physical orbits these ground some In and to the semi-classically methodology as atom[6] the our such central employ of problems, that (non-relativistic) We approach the see unified force. to we a corrections the corrections, provides relativistic understanding relativistic methodology While these our using of Lagrangian prescription. physics modified standard this Hamiltonian the from the or calculated Integral then influence Jacobi is The the under force. the motion central cor- of of from the equations produces constructed relativistic function rect Lagrangian energy new potential generalized The function. and energy al poqatcpwrof power quartic upto mally re uln fteognzto fteppri as is paper the of organization the of outline brief A rerslsadpoie orcin oboth to corrections provides and results orce = l oeso h eoiy hc hnmade when which velocity, the of powers all readas h rtodrcretosto corrections first-order the also and orce | ~ v T | orc rltvsi)freeuto.We equation. force (relativistic) correct nryfnto.W eieageneral a derive We function. energy − obte nesadtecnrlforce central the understand better to ost h oa nryo h system the of energy total the to ions < ost h eta oc rbe in problem force central the to ions ehdlg ocluaerelativistic calculate to methodology t.Tu efe htti okhas work this that feel we Thus lts. sclapoc.Orpeitosin predictions Our approach. ssical U 0 U . with 1 c en h eeaie oeta energy potential generalized the being e,Ida-247667 - India kee, n hsprisu orti maxi- retain to us permits this and † T en h o-eaiitckinetic non-relativistic the being nrlzdpotential eneralized | ~ v | /c ic ihrpowers higher since , e 2 the system, using the generalized potential energy func- In the above equation, symbols have their usual meanings th tion of Section 3, and calculates the total energy of the with vi being the i component of the spatial velocity 2 −1/2 system. In section 5, we show two model applications of |~v| and γ being the Lorentz Factor γ = 1 − 2 and our methodology, to the classical gravitationally bound, c circular orbits and the other to find the first-order cor- δij the Kronecker’s delta. To understand the relative rection to the ground state energy of the hydrogen atom. importance of the two terms in the mass tensor (Eq. 4) We conclude in section 6 by summarizing our work. to the total force, we rewrite the force equation of Eq. (3) in its vector notation,
d d2~r γ3 II. RELATIVISTICALLY GENERALIZED F~ = (γm ~v)= m γ + (~v · ~a)~v , (5) dt 0 0 dt2 c2 CENTRAL FORCE where the first term in the acceleration γd2~r/dt2 is A central force acts along the position vector of a par- the contribution of the diagonal term of the mass ten- ticle drawn from the center of force and depends only on sor and the second term γ3(~v · ~a)~v/c2 represents the the scalar distance r = |~r| from the fixed center of force. off-diagonal contribution which is a product of two terms: The convenient choice of this fixed point to be the ori- γ3~v/c2 and (~v · ~a). The term γ3 ~v/c2 is much smaller gin of the coordinate system makes the central force look than unity since v/c < 1. The term (~v · ~a) is also very like, small - for elliptic orbits with small eccentricities, the velocity is nearly perpendicular to the acceleration and F~ (~r)= F (r)ˆr . (1) hence ~v ·~a is close to zero. It becomes identically zero for the circular orbits, where the velocity is perpendicular to In case of the non-relativistic central force, the force the acceleration. We can thus safely argue that the off- is derivable from an ordinary potential energy function diagonal term in the acceleration is negligibly small com- V (r) such that, pared to the diagonal term. Objects in our Solar System orbiting the Sun have very low eccentricities. For exam- F~(~r) = F (r)ˆr = −∇~ V (r) . (2) ple, the Earth’s orbit is nearly circular with a small eccen- tricity of 0.017[9] and Pluto is the body with the largest Consider two point particles of rest masses m1 and m2 having a position vector ~r and ~r , respectively, mov- mean eccentricity of 0.244[9]. A simple calculation for the 1 2 orbit of Pluto yields that the off-diagonal contribution of ing under the influence of central force field, where the −17 equation of motions can be decomposed into the relative the force is ∼ O 10 smaller than the diagonal term. Also, on the other extreme (in the microscopic length ~r(= ~r − ~r ) and center-of-mass (R~) coordinates, respec- 1 2 scale), the relativistic Sommerfeld analysis[10] predicts tively. In the absence of any external force on this system, ~ the low eccentricity for the elliptical orbits of the elec- the center-of-mass (R) of the system moves as a free par- tron in the hydrogen atom. Thus, for an attractive cen- ticle. Thus in classical, non-relativistic mechanics, the tral force, two-body problem can be reduced to the dynamics of an equivalent one-body of a hypothetical mass point of F~ (r)= −k/r2 rˆ k> 0 , (6) reduced mass µ = (m1m2)/(m1 + m2) with the relative position vector ~r from the origin, moving under the ac- the force equation (5) for the reduced one-body problem, tion of the internal interaction force2. In general, the after neglecting the non-diagonal term, can therefore be relativistic force equation for a particle of rest mass m0 written as in classical mechanics can be written as (Einstein sum- mation implied), µ~r¨ = Q rˆ , (7)
d2x where Q is the generalized force, and its magnitude is F = m j , (3) i ij dt2 given by 1/2 with mij being the relativistic mass tensor,[8] k |~r˙|2 Q = − 1 − , (8) r2 c2 δ v v ! m = m γ3 ij + i j . (4) ij 0 γ2 c2 which depends on both the relative separation (r) and the velocity (~r˙) of the particle. This re-arrangement al- lows us to write the force equation as typical Newtonian equation of motion and use the familiar non-relativistic 2 Reduced mass does not have a simple definition in relativistic theory. However, we could treat the heavier mass as a source of Lagrangian Formulation with L = T − U, (U being static field and also the motion of the heavier mass can be taken the generalized velocity dependent potential energy func- as non-relativistic to a very good approximation[7]. This justifies tion) such that the Euler-Lagrange differential equation the use of an equivalent one-body having a reduced mass µ. of motion[11] can reproduce the correct force equation. 3
Under the central force, the motion is planar and we contribution of higher powers to the central force. Thus, can use two-dimensional polar coordinates (r, θ) to de- up to fourth power in |~r˙|/c, the generalized force becomes scribe the motion of the reduced particle. Thus, the k kL2 kL4 kr˙2 kL2r˙2 1 k velocity of the reduced particle can be resolved into its Q = − + + + + + r˙4, radial and tangential components as, r2 2µ2c2r4 8µ4c4r6 2c2r2 4µ2c4r4 8 c4r2 (17) |~r˙|2 =r ˙2 + r2θ˙2 . (9) where the first term is the usual inverse square attractive force. Both the second and third terms represent the rel- Therefore the force equation (7) can be be decomposed ativistic corrections to the (non-relativistic) central force into radial and tangential components: and allow for the coupling of the angular momentum component with the central force component. However, µr¨ − µrθ˙2 = Q (10) in the third term higher powers of angular momentum couples with the central force. Later we will see that these two terms in the generalized force will contribute ¨ ˙ µrθ +2µr˙θ =0 . (11) to the (relativistic) corrections of the potential energy of the system. The fourth and fifth terms which depend We can clearly see from Eq. (11) that the total angular on the square of the radial speed and also couple the momentum L is a constant of motion such that, angular momentum of the system with the central force, dL d will represent quadratic and quartic power corrections to = (µr2θ˙)=0 . (12) the kinetic energy of the system respectively. The sixth dt dt term is a higher order kinetic correction to the central Hence, we can eliminate θ˙ in terms of L and r, so that force of the system depending on the fourth power of the planar problem again be reduced to the radial (r) the radial speed. problem only. For |~v|/c < 1, (~v = ~r˙), we can write the generalized We add a further simplification here - building on force by expanding the Lorentz factor in powers of |~v|/c the fact that (~v · ~a) (in Eq. 5) is negligibly small for using a binomial expansion, closed orbits of smaller eccentricity. It can be shown that the term (~v · ~a), which is a measure of the eccentricity, k ∞ 1 |~v| 2p is proportional to the radial velocity,r ˙, so for a circular Q = − 2 , (13) orbit,r ˙ = 0 identically, and for nearly circular orbits,r ˙ r2 p c "p=0 # is small. Out of the three terms in the expansion having X fourth power in |~r˙|/c in the generalized force, we neglect 1 2 where p is the binomial coefficient of 1/2 and p. Due only the term having the fourth power of the radial to the conservation of angular momentum, the velocity speed ∼ r˙4/c4 but retain the contribution of the fourth |~v| becomes power of tangential speed and a mixed biquadratic term in radial and tangential velocities. Thus the generalized L2 µ2r2 force of Eq. (17) is simplified into |~v| = 1+ r˙2 , (14) µ2r2 L2 k kL2 kL4 kr˙2 kL2r˙2 Q = − 2 + 2 2 4 + 4 4 6 + 2 2 + 2 4 4 , (18) so we further expand (|~v|/c)2p using Eq. (14) in Eq. (13) r 2µ c r 8µ c r 2c r 4µ c r in terms of a power series in the radial speedr ˙ only. which will make the radial momentum (Eq. 33) linear in Therefore the generalized force becomes radial velocity,r ˙ and presents a welcome simplification to construct the Hamiltonian from the Lagrangian without ∞ p 2(q−p−i) 2q loosing much on the physics of the problem. Q = k Bp,q(L,µ)r (r ˙ ) , (15) "p=0 q=0 # X X III. THE RELATIVISTIC GENERALIZED where we have defined Bp,q(L,µ) as the expansion coef- POTENTIAL ficient depending on L and µ given by, ˙ (−1)p+1 1 p L 2(p−q) We now construct a Lagrangian function L(r, ~r, t) of B (L,µ) = 2 . (16) the non-relativistic, standard form of L = T − U, such p,q c2p p q µ that Eq. (10) can be reproduced through the Euler- It is proper to mention here that the generalized force for Lagrange equation of motion satisfying, a relativistic particle of rest mass µ contains all powers d ∂L ∂L − =0 , (19) in |~v|/c. In this paper, we have limited ourselves upto dt ∂r˙ ∂r the quartic power in |~v|/c since higher order terms will have negligibly small contribution in the semi-relativistic and the right hand side of the above equation is zero regime. Interested readers are encouraged to explore the since there are no non-potential forces acting on the sys- 4 tem. The kinetic energy function T is chosen to have the gives us a general expression for h2n(r), Newtonian, non-relativistic form, k ∞ r2(n−p)−1 1 h2n(r)= Bp,n(L,µ) +A2n, (26) ˙ 2 2n − 1 2(n − p) − 1 T (r, r,t˙ )= µ|~r| , (20) p=n 2 X and U ≡ U(~r, ~r,˙ t) is the generalized potential energy with A2n being the constants of integration, to be de- function. The potential energy function U will be chosen termined by boundary conditions. Thus, we have con- such that the generalized force Q should be derivable structed a general expression for the generalized potential from U as, U(r, v) which when plugged in Eq. (21) will give the ex- act expansion for the corrected force. To determine the d ∂U ∂U integration constants A2n, we use the usual boundary Q = − . (21) dt ∂r˙ ∂r condition as employed in standard analysis of the central force. As r → ∞, the force field dies down to zero. In Since the corrected force also depends on the radial ve- this limit, the potential goes to zero and the first term locityr ˙ in addition to r, hence an ordinary potential (r-dependent) in the expansion coefficients h2n(r)=0 V (r)= −k/r will not be sufficient to describe the gener- vanishes, we get, alized force. At this stage, we can expect that the gener- ∞ alized potential U to have a dominant contribution from 2n lim U(r, v)=0= A2nv . (27) the ordinary potential term V and other additive, correc- r→∞ n=0 tive terms which depend on r and ~r˙. Since the problem X is entirely in one-dimension in the r coordinate, we label Since this holds for any arbitrary radial velocity v of the without confusionr ˙ ≡ v. Since the relativistically gen- particle, we find all A2n to be zero identically, for all n, eralized force expansion of Eq. (15) contains only even powers ofr ˙, we choose a power series for the generalized A2n =0 . (28) 2 potential U(r,v,t) in powers of v , with the expansion By comparing with the relativistic force equation Eq. coefficients depending on r and t, (18), we obtain the expansion coefficients as ∞ 2n k kL2 kL4 U(r,v,t)= h2n(r, t) v , (22) h0(r)= − + 2 2 3 + 4 4 5 + A0 , (29) n=0 r 6µ c r 40µ c r X where in the above equation, h2n(r, t) is the 2n-th order k kL2 expansion coefficient. This power series for U is used to h2(r)= − − + A2 . (30) find an expression for the generalized force, 2c2r 12µ2c4r3
∞ ∂h Thus, upto fourth power in |~v|/c, the generalized poten- Q = (2n − 1) 2n v2n +2n(2n−1)h v2n−2v,˙ (23) tial energy can be written as, ∂r 2n n=0 X k kL2 kL4 kv2 kL2v2 U(r, v)= − + + − − . where we have assumed that the expansion coefficients r 6µ2c2r3 40µ4c4r5 2c2r 12µ2c4r3 are independent of time explicitly. This allows welcome (31) simplification and is justified since there is no explicit We see an ordinary potential correction coming from time dependence in the generalized force equation as well. h0(r) which tends to couple the angular momentum with The generalized force Q of Eq. (15) is independent of the the central force constant k. There are additional “ki- particle’s radial accelerationv ˙, as is the case with most netic” corrections proportional to the square of the ra- naturally occurring forces in nature and hence the sum dial velocity arising from h2(r) and these will modify the of the second term in Eq. (23) must result up to zero, kinetic energy of the system, as we shall see later. The we get, contribution of these terms will be discussed in detail in ∞ the next section. 2n−2 (2n)(2n − 1)h2n(r) v =0 , (24) n=0 X IV. LAGRANGIAN AND HAMILTONIAN and thus, the generalized force can be expressed in a FORMULATIONS power series form as follows, We now proceed to build a Lagrangian and following ∞ ∂h (r) Q = (2n − 1) 2n v2n . (25) which, a Hamiltonian of the relativistic system under cen- ∂r n=0 tral force. As suggested earlier, we will use a Lagrangian X function L(~r, ~r,˙ t)(= T −U), with the relativistically cor- We now compare the coefficients in Eq. (25) with the rected generalized potential U found in the previous sec- terms of the force of Eq. (15) in equal powers of v, which tion. With such a form, we see that the Euler-Lagrange 5 equation will reproduce the relativistic force equation. the Hamiltonian in terms of the radial coordinate and its Thus, we now construct the Lagrangian as, conjugate momentum exclusively,
1 L2 p2 L2 k ΓL2 ΓL4 L(~r, ~r˙)= µv2 + − U(r, v) (32) H = r + − + + . (36) 2 2µr2 2µχ(r) 2µr2 r 6µr3 40µ3c2r5 where U(r, v) can be substituted from Eq. (22) and the Since the Hamiltonian does not depend on time explicitly, coefficients from Eq. (26). Working only till fourth power so it can be identified as the total energy E of the system, in |~v|/c, we can utilize the potential from Eq. (31). Once which is a constant of motion. Thus, the total mechanical the Lagrangian has been constructed, the canonical mo- energy of a system is seen to have relativistic correction mentum conjugate to the radial coordinate r can be cal- to both the kinetic and potential energy terms. It can culated as follows, readily be verified that in the Γ → 0 limit, we recover the non-relativistic results. In the next section, we present ∂L ∂L k kL2 p = = = µ + + v . (33) two model applications of this relativistically corrected r ∂r˙ ∂v c2r 6µ2c4r3 central force formulation to physical systems under the It is here we see the usefulness of neglecting the term influence of the central force. proportional to fourth power in radial speed in Eq. (18). It allows us to keep the radial momentum linear in the V. MODEL APPLICATIONS radial speed. Relativistic corrections can be clubbed into a single pre-factor which corrects the non-relativistic mo- mentum µv. This is in line with our motivation to use a We now present model applications of the relativistic Newtonian framework for formulating relativistic correc- central force formulation using a generalized potential to tions to the central force. Thus the canonical momentum the classical circular orbit in celestial mechanics under can be expressed in a succinct form by allowing compar- the gravitational force and the first order energy correc- ison with the classical, non-relativistic momentum as, tion to the ground state of the hydrogen atom in quantum mechanics. These applications can be used as a pedagog- Γ L2 ical tool in the undergraduate classroom to understand p = 1+ 1+ µv = χ(r)µv . (34) r r 6µ2c2r2 the implications of relativistic corrections to the central force. These can be seen as approximations that bring In the above equation, we have substituted Γ for k/(µc2), out the physics of the problem without having to indulge which is an interaction parameter, useful in better under- non-trivial mathematics. standing the physics of the problem. The parameter Γ is an indicative of the relative strength of the central force coupling to the rest mass energy µc2 of the system, and A. Relativistic Correction to the Classical Circular has dimensions of length. It allows us to compare the Orbit rest mass energy with the binding energy of the system. In the limit Γ << 1, the problem can be treated non- We shall now be considering a gravitationally bound relativistically since the rest mass energy is much larger circular orbit and calculate a relativistic correction to the binding energy of the system, while in the opposite it. Before we calculate the correction, we assert that limit, the treatment is required to be relativistic, since in case of planetary orbits in the solar system, |~r˙|/c < the binding energy is comparable to the rest mass en- 10−4, so we retain only in quadratic powers in |~r˙|/c. This 2 ergy of the system. In many physical situations, µc is would also allow us to give a closed form to the relativistic very large compared to the binding energy of the system, correction to the radius of the circular orbit. In this case, for example, an electron in the hydrogen atom, the rest the central force constant k = Gm1m2, where G is the mass energy (∼ 511keV) is much larger than its bind- universal gravitational constant. As is customary in non- ing energy (∼ 10eV) so the ground state properties can relativistic central force problems, we define an effective be understood through the non-relativistic Schr¨odinger potential Veff,NR (NR stands for non-relativistic), which wave equation. The entire pre-factor has been labeled as compares the effect of the (attractive) central potential χ(r) to allow expressions to have a condensed form. The and the (repulsive) centrifugal terms[13], as angular momentum pθ, canonically conjugate to the θ coordinate is a constant of motion as we had seen earlier −Gm m L2 V (r)= 1 2 + . (37) and labeled it as L. The Hamiltonian H of the system eff,NR r 2µr2 can now be calculated as a Legendre dual transform of the Lagrangian,[12] On similar lines, we can define a relativistic effective po- tential for the central force which has an additional cor- H(~r, ~p, t)= prr˙ + pθθ˙ − L . (35) rective term,
2 2 Again eliminating θ˙ using the fact that the angular mo- −Gm1m2 L ΓL Veff,rel(r)= + + , (38) mentum pθ ≡ L is a constant of motion, we can express r 2µr2 6µr3 6
ΓL2 V (r)= V (r)+ , (39) of the circular orbit. It can easily be calculated that the eff,rel eff,NR 6µr3 energy of the circular orbit with the relativistic correction E∗ is, such that the energy of the system can now be compactly written as (where ‘rel’ stands for relativistic), in terms of E µk2 1 k2 E = 0 + + − 1 , (47) effective potential ∗ δ 2L2δ δ 3δ2L2c2 1 Γ where E is the energy of the non-relativistic circular E = µ 1+ v2 + V . (40) 0 2 r eff,rel orbit with the same angular momentum, 2 We can now solve for the radial speedr ˙ by re-arranging −µk E0 = Veff,NR(r0)= . (48) terms in Eq. (40), 2L2
−1 Thus, we have calculated the radius and energy of a circu- 2 2 Γ lar orbit under the given central force with the relativistic r˙ = 1+ (E − Veff,rel(r)) . (41) µ r correction incorporated and have presented our results in a comparative way with the non-relativistic ones for a It is known that that bound orbits, under the classical given value of L. central force problem of an inverse square attractive po- tential, occur when min(Veff ) ≤ E < 0.[13] A special case occurs when E = min(Veff ) such thatr ˙ = 0 and the re- B. Relativistic Correction to the Ground State sultant orbit is circular. Non-relativistically, a circular Energy of the Hydrogen Atom in Vacuum orbit occurs when E = min(Veff,NR) at a radius r0, In this section, we present a first order correction to L2 r0 = ≡ 2η , (42) the energy of the ground state of the hydrogen atom µk in vacuum due to relativistic corrections in the Hamil- where, we have defined η for mathematical convenience. tonian. We assume that the proton is infinitely heavy Relativistically, we will now give an estimate of the radius as compared to the electron and can therefore be as- of the circular orbit for a given value of L. We observe sumed to be stationary. Thus, the electron feels an that electrostatic Coulomb force of attraction and in vacuum, −1 2 1+ Γ 6= 0, hence,r ˙ = 0 for a circular orbit occurs k = e /(4πǫ0) with e being the electronic charge and ǫ0 r the permittivity of free space. Contrasted to the analy- only when E = min(Veff,rel). Let the minimum of Veff,rel sis of celestial circular orbits in the previous section V A, occur at r = r∗, such that, where we retained only quadratic powers in |~r˙|/c, here we 2 dVeff,rel d Veff,rel consider corrections till fourth power in |~r˙|/c. Thus the = 0 and > 0 . (43) 2 Hamiltonian of the hydrogen atom can be written from dr r∗ dr r∗ Eq.(36), The stationarity condition of Eq.(43) can be solved to p2 L2 k ΓL2 ΓL4 yield the following quadratic equation in r, H = r + − + + , (49) rel 2µχ(r) 2µr2 r 6µr3 40µc2r5 r2 − 2ηr − Γη =0 , (44) It is worth mentioning at this stage that the usual rela- The radius of the relativistic circular orbit is found to be, tivistic kinetic correction in standard texts involves ex- panding the relativistic kinetic energy in powers of p/(µc) r Γ 1/2 and retaining terms till the fourth power of p[14]. We r = 0 1+ 1+ ≡ r δ , (45) ∗ 2 η 0 later argue as to why both approaches, even though re- " # taining the same power in their respective expansions do where we have defined δ as a parameter to compare not result in the same numerical value of the corrections. the circular radii for the non-relativistic and relativistic They, however match in order of magnitude, as expected. cases, for a given L and coupling k, The r dependent function χ(r) can be re-written as, Γ L2 Γα(r) 1 2k2 1/2 χ(r)= 1+ 1+ = 1+ , (50) δ = 1+ 1+ . (46) r 6µ2c2r2 r 2 L2c2 " # where we have defined α(r) for convenience. Readers’ It is readily seen that δ > 1, thus, the radius of the attention is brought to the fact that the fourth term circular orbit with the relativistic correction is more in the relativistic Hamiltonian, kL2/(6µ2c2r3) resembles than without it, r∗ > r0. The presence of the addi- the spin-orbit[15] coupling term in the hydrogen atom tional positive term in the relativistic effective potential, problem. It is very interesting to note that a unified gen- ΓL2/6µr3 tends to flatten the potential curve at larger eralized potential correction to the central force Hamilto- radii and can be understood as the reason for the swelling nian automatically provides a fore-runner for spin-orbit 7 like terms, which essentially work to couple the central We use the ground state of the hydrogen atom to obtain force with the angular momentum of the system. In par- the expectation value of the first order energy correction ticular, the spin-orbit term in the hydrogen atom involves to the system as, the coupling of the electron intrinsic spin angular momen- Γ l(l + 1)~2 e2 tum with the orbital magnetic field of the proton seen by E(1) = − 1+ E(0) + r a 6µ2c2a 2 4πǫ a the electron. Our generalized potential approach pre- 0 0 0 0 dicts a similar form of the coupling of the central force Γl(l + 1)~2 1 Γl2(l + 1)2~4 1 + + (55). with the total angular momentum of the system as a rel- 6µ rˆ3 40µ3c2 rˆ5 ativistic phenomenon, and may be understood as a ped- agogical explanation for coupling of the spin and orbital The expectation values of powers of (1/rˆ) can be com- angular momentum of the system with the central force. puted from the famous Kramer’s relation (or the sec- Our treatment here would be semi-classical, whereby we ond Pasternack relation)[17], [18]. A numerical compar- would use the classical Hamiltonian derived in the ear- ison is in place here. The unperturbed ground state en- lier sections, to find a quantum mechanical expectation ergy E(0) = −13.6eV, while standard texts[19] quote value of the first order correction in the ground state en- a first order correction its kinetic energy solely to be ergy. The non-relativistic, unperturbed Hamiltonian for −8.99 × 10−4eV by expanding the relativistic kinetic en- the hydrogen atom HNR (NR stands for non-relativistic) ergy in powers of p/(µc) and retaining upto fourth power can be easily written as, in p. The correction due to the fine-structure which in- cludes both kinetic and spin-orbit corrections is quoted 2 2 2 −4 pr L e to be −1.81 × 10 eV. The relativistic correction due to HNR(r, pr)= + 2 − . (51) 2µ 2µr 4πǫ0r our Hamiltonian, using the generalized potential is found to be from Eq. (55) to be −2.41 × 10−4eV. Thus, we As expected, Eq. (51) corresponds to substituting Γ = 0 see that our prediction matches more closely to the fine- in Eq. (49). In case of the hydrogen atom, we see that structure value which includes corrections to the kinetic Γ ≃ 2.81×10−15m, while the most probable radius in the energy and also incorporates corrections due to angular −11 ground state is the Bohr radius a0(≃ 5.29 × 10 m).[16] momenta coupling to the central force. On utilizing the As a first approximation for the ground state, we replace Schr¨odinger’s Equation to find out the expectation value 4 3 r in the kinetic term in Eq. (49) by the Bohr radius a0, of p in the standard approach , we notice that it splits to ease the calculation. This is justified, since for the into terms containing 1/r and 1/r2. This can also be seen ground state of the hydrogen atom, the variance in the in the kinetic correction term (first term) in our Hamil- observable corresponding to r is small. Since Γ/a0 ≪ 1, tonian of Eq. (49), which when expanded in powers of we can expand χ(r) in the kinetic term binomially and Γ/r yields similar expressions as in the standard textbook retain terms upto first order in Γ/a0. The idea is to approach. Thus, we stress on the similarity of structure treat the relativistic correction to the Hamiltonian as a of the kinetic energy corrections in both the approaches. small perturbation since Γα(a0)/a0 ≪ 1, and calculate Our methodology of building a generalized potential en- the expectation value of the first order perturbation to ergy is able to give both a kinetic correction and in addi- the ground state wave energy of the hydrogen atom by tion generates terms which correct the potential energy treating r and pr as corresponding quantum mechanical of the system as well. Corrections to the potential energy operators. The relativistic correction is treated as a per- of a system due to relativistic effects are not very surpris- ′ turbation (H ) to the non-relativistic Hamiltonian HNR, ing. A famous motivation for their existence is the Dar- win term[20] which occurs naturally in Dirac equation ′ ′ Hrel = HNR + H (r, pr); H << HNR, (52) and involves corrections to the effective potential at the nucleus in the hydrogen atom. Our Hamiltonian splits and correspondingly the relativistic energy correction into terms independent ofr ˙ and also terms depending on may be added to E(0) (the unperturbed, non-relativistic various powers ofr ˙. It allows for coupling of the central ground state energy of the hydrogen atom), force with the angular momentum of the system (spin and orbital) and produces corresponding corrections to (0) (1) the system, which match closely to the predicted value of E = E + Er , (53)
(1) where Er is the first order correction to the energy due to relativistic effects. After replacement of r by a0 in the 3 In the standard approach, the first-order energy correction is kinetic term and retaining upto first power in Γα(a0)/a0, proportional to
In our novel approach to introduce relativistic cor- ACKNOWLEDGEMENTS rections to the central force problem, we have derived a generalized potential energy function which depends The authors would like to thank the anonymous re- both on the position and velocity of the two-body sys- viewers and the Associate Editor of the journal for their tem. The Lagrangian constructed has a familiar classical constructive comments to help improve the manuscript.
[1] N. C. Rana and P.S. Joag, Classical Mechanics, (Tata Bohr Model of Atomic Structure 19131925 (Cambridge McGraw Hill, 2011), p. 118. University Press, 1998), p. 228. [2] David J. Griffiths, Introduction to Quantum Mechanics, [11] N. C. Rana and P.S. Joag, Classical Mechanics, (Tata 2nd edition (Pearson Education, 2011), p. 157. McGraw Hill, 2011), p. 61. [3] R. G. Takwale and P.S. Puranik, Introduction to Classical [12] N. C. Rana and P.S. Joag, Classical Mechanics, (Tata Mechanics, (Tata McGraw Hill, 2012), p. 134. McGraw Hill, 2011), p. 183. [4] Herbert Holdstein, Charles P. Poole and John Safko, [13] R. G. Takwale and P.S. Puranik, Introduction to Classical Classical Mechanics, 3rd edition (Pearson Education, Mechanics, (Tata McGraw Hill, 2012), p. 141 . Inc., 2012), p. 318. [14] David J. Griffiths, Introduction to Quantum Mechanics, [5] Leonard I. Schiff, Quantum Mechanics, 3rd edition (Tata 2nd edition (Pearson Education,Inc., 2011), p. 280. McGraw Hill, 2010), p. 472. [15] David J. Griffiths, Introduction to Quantum Mechanics, [6] Robert Eisberg and Robert Resnick, Quantum Physics 2nd edition (Pearson Education,Inc., 2011), p. 285. of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd [16] Robert Eisberg and Robert Resnick, Quantum Physics edition (John Wiley & Sons, Inc., 2011), p. 100. of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd [7] Paul Strange, Relativistic Quantum Mechanics: With edition (John Wiley & Sons, Inc., 2011), p. 246. Applications in Condensed Matter and Atomic Physics [17] H. Beker, Am. J. Phy., 65, 1118 (1997). (Cambridge University Press, 1998), p. 228. [18] S. Balasubramanian, Am. J. Phy., 68, 959 (2000). [8] N. C. Rana and P.S. Joag, Classical Mechanics, (Tata [19] David J. Griffiths, Introduction to Quantum Mechanics, McGraw Hill, 2011), p. 192. 2nd edition (Pearson Education,Inc., 2011), p. 179. [9] Dr. David R. Williams, NSSDC, [20] Wolfgang Demtrder, Atoms, Molecules and Photons: NASA Goddard Space Flight Center, An Introduction to Atomic-, Molecular- and Quantum http://nssdc.gsfc.nasa.gov/planetary/factsheet/index.html Physics, 2nd edition (Springer, 2010), p. 174. [10] Helge Kragh, Niels Bohr and the Quantum Atom: The