Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 89
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Thus we have to omit this neighborhood from all successive terms in the semiclassical(−)expansion, in particular from (4.439). According to the remarks after Eq. (4.332), the energy EeD/2+1of electrons filling all levels up to a total energy EF is found from (4.328) by replacing (−V )H. Kleinert, PATH INTEGRALS4.10 Thomas-Fermi Model of Neutral Atoms431by (1 − EF ∂/EF )(EF − V )D/2+1 .
The energy level satisfying (4.444) correspond to a Fermi levelEF , so that the energy of the electrons in these levels isEe(−) = 222 (−)Epot TF = −255M2πh̄3/2Z1(1−EF ∂EF ) d3 x[−EF − V (x)]5/2.Γ(5/2)We therefore have to subtract from the correction (4.437) a term√Z12M(−)31/2 2−3/22∆sub Ee = (1−EF ∂EF )dx(−E−V)∇V−.(∇V)(−E−V)FF12h̄π 24(4.446)(4.447)The true correction can then be decomposed into a contribution from the finite region outside thesmall sphere√Z2M(−)d3 x(−V )1/2 ∇2 V,(4.448)∆Eoutside =24h̄π 2 r≥rmaxplus a subtracted contribution from the inside√Zih2M(−)31/21/2∇2 V,∆Einside =dx(−V)−(1−E∂)(−E−V)FEFF24h̄π 2 r<rmax(4.449)plus a pure gradient term√ZZ2M 2(−)2233/233/2∆Egrad = −.dx∇(−V)−dx∇(1−E∂)(−E−V)FEFF24h̄π 2 3r<rmax(4.450)The last two volume integrals can be converted into surface integrals.
Either integrand vanisheson its outer surface [recall (4.412)]. At the inner surface, an infinitesimal sphere around the origin,the integrands coincide so that the energy (4.450) vanishes.The energy (4.449) does not vanish but can be ignored in the present approximation. Atthe δ-function at the nuclear charge in ∇2 V , the difference (−V )1/2 − (1 − EF ∂EF )(−EF − V )1/2vanishes. In the integral, we may therefore replace ∇2 V (x) by −4πe2 n(x) [dropping the δ-functionin (4.334)]. In the small neighborhood of the origin, the integral is suppressed by a power of Z −4/3as will be seen below.Thus, only the outside energy (4.448) needs to be evaluated, where the small sphere excludesthe nuclear charge, so that we may replace ∇2 V (x) as in the last integral.
Thus we obtainZ2e2 M 2 M(−)∆Eoutside = −d3 x[−V (x)]2 .(4.451)9π 3 h̄4r<rmaxExpressing V in terms of the screening function and going to reduced variables, the quantumcorrection takes the final formZ8 a(−)2∆Eoutside = − 2dξf (ξ) Z 5/39π aHe2e2,(4.452)≈ −0.07971 I2 Z 5/3 ≈ −0.04905Z 5/3aHaHwhere I2 is the integral over f 2 (ξ) calculated in Eq. (4.402).
We have re-extended the integrationover the entire space with a relative error of order Z −2/3 , due to the smallness of the sphere. Thecorrection factor to the leading Thomas-Fermi energy caused by this isCQM = 1 + 0.06381 Z −2/3.(4.453)4324 Semiclassical Time Evolution AmplitudeIn the reduced variables, the order of magnitude of the ignored energy (4.449) can most easily beestimated. It readsZ ξmax hip8 a(−)∆Einside = − 2(4.454)dξ f 2 (ξ) − f (ξ) − ξ/ξm f 3/2 (ξ) Z 5/3 .9π aH 0Since ξmax and ξm are of the order Z −2/3 , this energy is of the relative order Z −4/3 and thusnegligible since we want to find here only corrections up to Z −2/3 .Observe that the quantum correction (4.455) is of relative order Z −2/3 and precisely a fraction2/9 of the exchange energy (4.401).
Both energies together are therefore(−)(−)∆Einside + Eexch=e211 (−)Eexch ≈ −0.2699 Z 5/3.9aH(4.455)The corrections of order Z −2/3 can be collected in the expression(−)(−)∆Einside + Eexch = −0.7687 Z 7/3e2× C2 (Z),aH(4.456)where C2 (Z) is the correction factorC2 (Z) = 1 + 0.3510 Z −2/3 + . . . .(4.457)Including also the Z −1/3 -correction (4.423) from the origin we obtain the total energy(−)Etot = −0.7687 Z 7/3e2× Ctot (Z),aH(4.458)with the total correction factorCtot (Z) = 1 − 0.6504 Z −1/3 + 0.3510 Z −2/3 + . .
. .(4.459)This large-Z approximation is surprisingly accurate. The experimental binding energy ofmercury with Z = 80 isexpEHg≈ −18130,(4.460)in units of e2 /aH = 2 Ry, whereas the large-Z formula (4.458) with the correction factor (4.459)yields the successive approximations including the first, second, and third term in (4.459):EHg ≈ −(21200, 18000, 18312) forC(80) = (1, 0.849, 0.868).(4.461)Even at the lowest value Z = 1, the binding energy of the hydrogen atomexpEH≈−12(4.462)is quite rapidly approached by the successive approximationsEH ≈ −(0.7687, 0.2687, 0.5386) for C(1) = (1, 0.350, 0.701).4.11(4.463)Classical Action of Coulomb SystemConsider an electron of mass M in an attractive Coulomb potential of a proton atthe coordinate origine2VC (x) = − .(4.464)rH.
Kleinert, PATH INTEGRALS4.11 Classical Action of Coulomb System433If the proton is substituted by a heavier nucleus, e2 has to be multiplied by thecharge Z of that nucleon. The Lagrangian of this system isL=M 2 e2ẋ − .2r(4.465)Because of rotational invariance, the orbital angular momentum is conserved andthe motion is restricted to a plane, say x − y. If φ denotes the azimuthal angle inthis plane, the constant orbital momentum isl = Mr 2 φ̇.(4.466)The conserved energy ise2M 22 2(ṙ + r φ̇ ) − .E=2rTogether with (4.466) we findṙ =1p (r);M EpE (r) =(4.467)q2M[E − Veff (r)],(4.468)where Veff (r) = VC (r) + Vl (r) is the sum of the Coulomb potential and the angularbarrier potentiall2.(4.469)Vl (r) =2Mr 2The differential equation (4.468) is solved by the integral relationM.pE (r)(4.470)ldφ= 2,drr pE (r)(4.471)t=ZdrFor the angle φ, Eq.
(4.466) implieswhich is solved by the integralφ=lZdr1.r pE (r)(4.472)2Inserting pE (r) from (4.468) this becomes explicitly!#−1/2φ=lZdre2 rl22ME r 2 +−rE2MEt=MZe2 rl2dr r 2ME r +−E2ME",(4.473)while (4.470) reads"2!#−1/2.(4.474)4344 Semiclassical Time Evolution AmplitudeConsider now the motion for negative energies. Defining√p̄E ≡ p−E (∞) = −2ME,(4.475)and introducing the parameterse2l2 p̄2Me2l2l2 va2a≡= 2 , 2 ≡ 1 − 2 E4 = 1 −=1−, va =2|E|p̄EM eaMe2e4we obtaindr1q,22r a − (r − a)2lφ=MvaZ1t=vardr q,a2 2 − (r − a)2Zsp̄e2= E,aMM(4.476)(4.477)(4.478)where va is the velocity associated with the momentum p̄E .
The ratiovaaω=(4.479)is the inverse period of the orbit, also called mean motion, which satisfies ω 2a3 =e2 /M, the third Kepler law . In the limit E → 0, the major semiaxis a becomesinfinite, and so does ω. The eccentricity vanishes and the orbit is parabolic.Introducing the variableh ≡ a(1 − 2 ) =and observing thatl2l2=,Me2p̄2E a(4.480)l= a 1 − 2 ,Mva(4.481)h= 1 + cos(φ − φ0 ).r(4.482)qthe first equation is solved byThis follows immediately from the fact thatsin(φ − φ0 ) =q1 − 2 q 2 2a − (r − a)2 .r(4.483)2Theq relation (4.482) describes an ellipse with principal axes a = h/(1 − ) , b =h/ 1 − 2 , and an eccentricity (see Fig. 4.3).
In the orbital plane, the Cartesiancoordinates of the motion arex=hcos(φ − φ0 ),1 + sin(φ − φ0 )y=hsin(φ − φ0 ).1 + sin(φ − φ0 )(4.484)H. Kleinert, PATH INTEGRALS4.11 Classical Action of Coulomb SystemyE<0y435yE>0hhφbφxxaφxa( + 1)aa( − 1)Figure 4.3 Orbits in Coulomb potential showing the parameter h and the eccentricity of ellipse (E < 0) and hyperbola (E > 0) in attractive and repulsive cases.For positive energy, we define a momentum√pE = pE (∞) = 2ME,(4.485)and the parameterse2l2 p2EMe2l2l2 va22a≡= 2 , ≡1+ 2 4 =1+= 1 + 4 , va =2|E|pEM eaMe2ese2p= E.aMM(4.486)The eccentricity is now larger than unity.
Apart from this, the solutions to theequations of motion are the same as before, and the orbits are hyperbolas as shownin Fig. 4.3. The y-coordinate above the focus is nowl2l2= 2 .h = a( − 1) =Me2pE a2(4.487)For a repulsive interaction, we change the sign of e2 in the above equations.
Theequation (4.482) for r becomes nowh= −1 + cos(φ − φ0 ),r(4.488)and yields the right-hand hyperbola shown in Fig. 4.3.For later discussions in Chapter 13, where we shall solve the path integral of theCoulomb system exactly, we also note that by introducing a new variable in termsof the variable ξ, to so-called eccentric anomalyr = a(1 − cos ξ),(4.489)we can immediately perform the integral (4.478) to findt=avaZdξ (1 − cos ξ) =1(ξ − sin ξ),ω(4.490)4364 Semiclassical Time Evolution Amplitudewhere we have chosen the integration constant to zero.
Using Eq. (4.488) we seethatx = r cos φ =y = r sin φ =h−r= a(cos ξ − ),qq(4.491)r 2 − x2 = a 1 − 2 sin ξ = b sin ξ.(4.492)Equations (4.489) and (4.490) represent a parametric representation of the orbit.From Eqs. (4.490) and (4.489) we see that1dt= d(ξ/ω),ra(4.493)exhibiting ξ/ω is a path-dependent pseudotime. As a function of this pseudotime,the coordinates x and y oscillate harmonically.The pseudotime facilitates the calculation of the classical action A(xb , xa ; tb −ta ),as done first in the eighteenth century by Lambert.12 If we denote the derivativewith respect to ξ by a prime, the classical action readsA=Zξbξae2M0202.aω x + y +dξ2raω#"(4.494)For an elliptic orbit with principal axes a, b, this becomesA=ZξbξaM a2 sin2 ξ + b2 cos2 ξ+ Mωa2 (ξb − ξa ).dξω21 − cos ξ#(4.495)q(4.496)"After performing the integral using the formulaZdξ2=arctan1 − cos ξ1 − 21 − 2 tan(ξ/2),1−we findA =M 2a ω [3(ξb − ξa ) + (sin ξb − sin ξa )] .2(4.497)Introducing the parameters α, β, γ, and δ by the relationscos α ≡ cos[(ξb + ξa )/2],β ≡ (ξb − ξa )/2,γ ≡ α + β,δ ≡ α − β,(4.498)the action becomesA =M 2a ω [(3γ + sin γ) − (3δ + sin δ)] .2(4.499)12Johann Heinrich Lambert (1728–1777) was an ingenious autodidactic taylor’s son who with16 years found Lambert’s law for the apparent motion of comets (and planets) on the sky: If thesun lies on the concave (convex) side of the apparent orbit, comet is closer to (farther from) thesun than the earth.
In addition, he laid the foundations to photometry.H. Kleinert, PATH INTEGRALS4.11 Classical Action of Coulomb System437Using (4.490), we find for the elapsed time tb − ta the relationtb − ta =1[(γ − sin γ) ∓ (δ − sin δ)] .ω(4.500)The ∓-signs apply to an ellipse whose short or long arc connects the two endpoints,respectively. The parameters γ and δ in the action and in the elapsed time arerelated to the endpoints xb and xa byrb + ra + R = 4a sin2 (γ/2) ≡ 4aρ+ ,rb + ra − R = 4a sin2 (δ/2) ≡ 4aρ− , (4.501)where rb ≡ |xb |, ra ≡ |xa |, R ≡ |xb − xa |, and ρ± ∈ [0, 1]. Expressing the semimajoraxis in (4.499) in terms of ω as a = (e2 /Mω 2)1/3 , and ω in terms of tb − ta we obtainthe desired classical action A(xb , xa ; tb − ta ).The elapsed time depends on the endpoints via a transcendental equation whichcan only be solved by a convergent power series (Lambert’s series)∞(2j)!11 Xj+1/2j+1/2.j−1 ρ+± ρ−tb − ta = √2j 2 22ω j=1 2 j! j − 1/4(4.502)In the limit of a parabolic orbit the series has only the first term, yielding√2 3/23/2(ρ+ ± ρ− ).(4.503)tb − ta =3ωWe can also express a and ω in terms of the energy E as e2 /(−2E) and−2E/Ma2 , respectively, and go over to the eikonal S(xb , xa ; E) via the Legendre transformation (4.80).
Substituting for tb − ta the relation (4.500), we obtainfor the short arc of the ellipseqS(xb , xa ; E) = A(xb , xa ; E) + (tb − ta ) E =M 2a ω 4(γ − δ).2(4.504)For a complete orbit, the expression (4.497) yields a total actionA =M 2M 2 2πp̄ 2πa ω 2π(1 + 2) =va(1 + 2) = E(1 + 2),22ω2M ω(4.505)where the numbers 1 and 2 indicate the source of the contributions from the kineticand potential parts of the action (4.494), respectively. Since the action is the difference between kinetic and potential energy, the average potential energy is minustwice as big as the kinetic energy, which is the single-particle version of the virialtheorem observed in the Thomas-Fermi approximation (4.388).For positive energies E, the eccentricity is > 1 and the orbit is a hyperbola(see Fig.
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