Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 90
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4.3). Then equations (4.489)–(4.492) becomer = a( cosh ξ − 1),x = −a(cosh ξ − ),t=a(ξ − sin ξ),va(4.506)q(4.507)y = a 2 − 1 sinh ξ.4384 Semiclassical Time Evolution AmplitudeThe orbits take a simple form in momentum space. Using Eq. (4.466), we findfrom (4.484):lpx = − [ + sin(φ − φ0 )] ,hpy =lcos(φ − φ0 ).h(4.508)As a function of time, the momenta describe a circle of radiuslMe2p̄p0 = ==q Ehl1 − 2(4.509)around a center on the py -axis with (see Fig. 4.4)lpcx = − = − q.h1 − 2(4.510)py√ 11−2py√ 12 −11pxE<0E>0√ 1−21px√ 2 −1Figure 4.4 Circular orbits in momentum space in units p̄E for E < 0 and pE for E > 0.For positive energies, the above solutions can be used to describe the scatteringof electrons or ions on a central atom. For helium nuclei obtained by α-decay ofradioactive atoms, this is the famous Rutherford scattering process.
The potentialis then repulsive, and e2 in the potential (4.464) must be replaced by −2Ze2 , where2e is the charge of the projectiles and Ze the charge of the central atom. As wecan see on Fig. 4.5, the trajectories in an attractive potential are simply related tothose in a repulsive potential. The momentum p̄E is the asymptotic momentum ofthe projectile, and may be called p∞ . The impact parameter b of the projectile fixesthe angular momentum vial = bp∞ = bp̄E .(4.511)Inserting l and p̄E we see that b coincides with the previous parameter b.The relation of the impact parameter b with the scattering angle θ may be takenfrom Fig. 4.5, which shows that (see also Fig.
4.4)tanThus we haveθp1= 0 = 2.2p∞ −1θb = a cot .2(4.512)(4.513)H. Kleinert, PATH INTEGRALS4.11 Classical Action of Coulomb System439The particles impinging into a circular annulus of radii b and b + db come outbetween the angles θ and θ + dθ, with db/dθ = −a/2 sin2 (θ/2). The area of theannulus dσ = 2πbdb is the differential cross section for this scattering process. Theabsolute ratio with respect to the associated solid angle dΩ ≡ 2π sin θdθ is thena2Z 2 α2 M 2dσ==.dΩ4 sin4 (θ/2)4p4∞ sin4 (θ/2)(4.514)The right-hand side is the famous Rutherford formula, which arises after expressing a in terms of the incoming momentum p∞ [recall (4.486)] as a = Ze2 /2E =Zαh̄cM/p2∞ .Let us also calculate the classical eikonal in momentum space.
We shall do thisfor the attractive interaction at positive energy. Inserting (4.484) and (4.508) wehaveS(pb , pa ; E) = −lZφbφadφ.1 + (φ − φ0 )(4.515)pbpbθpa(pb )p0pb(pa )φabpascatteringcenterp∞γaE = const.p∞paFigure 4.5 Geometry of scattering in momentum space. The solid curves are for attractive Coulomb potential, the dashed curves for repulsive (Rutherford scattering). Theright-hand part of the figure shows the circle on which the momentum moves from pa topb as the angle φ runs from φa to φb . The distance b is the impact parameter, and θ isthe scattering angle.4404 Semiclassical Time Evolution AmplitudeUsing the formula13Zq1 + + 2 − 1 tan(ξ/2)dξ2q=qlog.1 + cos ξ2 − 11 − − 2 − 1 tan(ξ/2)(4.516)qInserting l/ 1 − 2 = Mva a = Me2 /p∞ , we find after some algebraS(pb , pa ; E) = −withζ≡Me2ζ +1log,p∞ζ −1vuut1 +p∞ =√(p2b − p2∞ )(p2b − p2∞ ).p2∞ |pb − pa |22ME,(4.517)(4.518)The expression (4.517) has no definite limit if the impinging particle comes infrom spatial infinity where pa becomes equal to p∞ .
There is a logarithmic divergencewhich is due to the infinite range of the Coulomb potential. In nature, the chargesare always screened at some finite radius R, after which the logarithmic divergencedisappears. This was discussed before when deriving the eikonal approximation(1.500) to Coulomb scattering.There is a simple geometric meaning to the quantity ζ. Since the force is central,the change in momentum along a classical orbit is always in the direction of thecenter, so that (4.515) can also be written asS(pb , pa ; E) = −Zpbr dp.(4.519)paExpressing r in terms of momentum and total energy, this becomes for an attractive(repulsive) potentialS(pb , pa ; E) = ±2Me2Zdp.p − 2MEpb2pa(4.520)Now we observe, that for E < 0, the integrand is the arc length on a sphere ofradius 1/p̄E in a four-dimensional momentum space.
Indeed, the three-dimensionalmomentum space can be mapped onto the surface of a four-dimensional unit sphereby the following transformationp2 − p̄2En4 ≡ 2,p + p̄2EThen we find that2p̄E p.p + p̄2Edpdϑ,2 =2p̄Ep + p̄E213n≡2(4.521)(4.522)See the previous footnote.H. Kleinert, PATH INTEGRALS4.12 Semiclassical Scattering441where dϑ is the infinitesimal arc length on the unit sphere. But then the eikonalbecomes simply2Me2S(pb , pa ; E) = ±ϑ ,(4.523)p̄E bawhere ϑ is the angular difference between the images of the momenta pb and pa .This is easily calculated.
From (4.521) we find directlycos ϑba =4p̄2E pb · pa + (p2b − p̄2E )(p2a − p̄2E )p̄2E (pb − pa )2=1−2.(p2b + p̄2E )(p2a + p̄2E )(p2b + p̄2E )(p2a + p̄2E )(4.524)Continuing E analytically to positive energies, we may replace p̄E by ip∞ = 2ME,and obtainp2∞ (pb − pa )2cos ϑba = 1 + 2 2.(4.525)(pb − p2∞ )(p2a − p2∞ )Hence ϑ becomes imaginary, ϑ = iϑ̄, withsinp2 (p − p )2ϑ̄= 2 ∞ 2b 2 a 2 ,2(pb − p∞ )(pa − p∞ )(4.526)and the eikonal function (4.527) takes the formS(pb , pa ; E) = ±2Me2ϑ̄ .p∞ ba(4.527)This is precisely the expression (4.517) with ζ = 1/ tanh(ϑ̄ab /2).4.12Semiclassical ScatteringLet us also derive the semiclassical limit for the scattering amplitude.4.12.1General FormulationConsider a particle impinging with a momentum pa and energy E = Ea = p2a /2Mupon a nonzero potential concentrated around the origin.
After a long time, itwill be found far from the potential with some momentum pb and the same energyE = Eb = p2b /2M. Let us derive the scattering amplitude for such a process fromthe heuristic formula (1.508):fpb pap= bMq32πh̄M/i(2πh̄)3lim1tb →∞ t1/2beiEb (tb −ta )/h̄ [(pb tb |pa ta )−hpb |pa i] .(4.528)In the semiclassical approximation we replace the exact propagator in the momentum representation by a sum over all classical trajectories and associated phases,4424 Semiclassical Time Evolution Amplitudeconnecting pa to pb in the time tb − ta .
According to formula (4.166) we have inthree dimensions(2πh̄)3(pb tb |pa ta )−hpb|pa i =(2πh̄/i)3/2X0class. traj.!∂x 1/2− a eiA(pb ,pa ;tb −ta )/h̄−iνπh̄/2 .∂pbdet(4.529)The sum carries a prime to indicate that unscattered trajectories are omitted. Theclassical action in momentum space isA(pb , pa ; tb − ta ) =Zpapbx · ṗ −tbZtaHdt = S(pb , pa ; E) − E(tb − ta ),(4.530)where S(pb , pa ; E) is the eikonal function introduced in Eqs. (4.233) and (4.61).Inserting (4.529) into (1.508) we obtain the semiclassical scattering transitionamplitudefpb pa = limtb →∞pbX0class.
traj.qtb Mdet∂xa 1/2 iS(pb ,pa ;E)/h̄−iνπ/2e.∂pb (4.531)The determinant has a simple physical meaning. To see this we rewriteso that (4.533) becomesfpb pa = limtb →∞−1∂pb ∂xa = ,∂pb p∂xa pX0pclass. traj.We now note that for large tbqtb Madet(4.532)a∂pb −1/2 iS(pb ,pa ;E)/h̄−iνπ/2e.∂xa papb = p(tb ) = Mxb (tb )/tb(4.533)(4.534)along any trajectory. Thus we findfpb pa = limtb →∞X0class. traj.rb det∂xb −1/2 iS(pb ,pa ;E)/h̄−iνπ/2e,∂xa pb(4.535)where rb = |xb |.From the definition of the scattering amplitude (1.495) we expect the prefactor ofthe exponential to be equal to the square root of the classical differential cross sectiondσcl /dΩ.
Let us choose convenient coordinates in which the particle trajectories startout at a point with cartesian coordinates xa = (xa , ya , za ) with a large negativeza and a momentum pa ≈ pa ẑ, where ẑ is the direction of the z-axis. The finalpoints xb of the trajectories will be described in spherical coordinates. If p̂b =H. Kleinert, PATH INTEGRALS4.12 Semiclassical Scattering443(sin θb cos φb , sin θ sin φb , cos θb ) denotes the direction of the final momentum pb =pb p̂b , then xb = rb p̂. Let us introduce an auxiliary triplet of spherical coordinatessb ≡ (rb , θb , φb ).
Then we factorize the determinant in (4.535) asdet∂xb∂x∂s∂s= det b × det b = rb2 det b .∂xa∂sa∂xa∂xaWe further calculatedet∂sb=∂xa∂rb∂ya∂θb∂ya∂φb∂ya∂rb∂xa∂θb∂xa∂φb∂xa∂rb∂za∂θb∂za∂φb∂za.(4.536)Long after the collision, for tb → ∞, a small change of the starting point along thetrajectory dza will not affect the scattering angle. Thus we may approximate thematrix elements in the third column by ∂za ≈ ∂φ/∂za ≈ 0. After the same amountof time the particle will only wind up at a slightly more distant rb , where drb ≈ dzb .Thus we may replace the matrix element in the right upper corner by 1, so that thedeterminant (4.536) becomes in the limitlim dettb →∞∂sb≈ det∂xa∂θb∂ya∂φb∂ya∂θb∂xa∂φb∂xa=dΩdθb dφb=,dxb dybdσ(4.537)where dΩ = sin θb dθdφb is the element of the solid angle of the emerging trajectories,and dσ the area element in the x − y -plane, for which the trajectories arrive in anelement of the final solid angle dΩ.
Thus we obtain"∂xdet b∂xa#−1≈1 dσ.rb2 dΩ(4.538)The ratio dσ/dΩ is precisely the classical differential cross section of the scatteringprocess.Combining (4.535) and (4.537), we see that the contribution of an individualtrajectory to the semiclassical amplitude is of the expected form [3]fpb pa =sdσcldΩ×X0eiS(pb ,pa ;E)/h̄−iνπ/2 .(4.539)class. traj.Note that this equation is also valid for some potentials which are not restricted toa finite regime around the origin, such as the Coulomb potentials. In the operatortheory of quantum-mechanical scattering processes, such potentials always causeconsiderable problems since the outgoing wave functions remain distorted even atlarge distances from the scattering center.4444 Semiclassical Time Evolution AmplitudeUsually, there are only a few trajectories contributing to a process with a givenscattering angle.
If the actions of these trajectories differ by less than h̄, the semiclassical approximation fails since the fluctuation integrals overlap. Examples arethe light scattering causing the ordinary rainbow in nature, and glory effects seenat night around the moonlight.We now turn to a derivation of the amplitude (4.539) from the more reliableformula (1.529) for the interacting wave functionhxb |ÛI (0, ta )|pa i =−2πih̄taMlimta →−∞!3/22(xb tb |xa ta )ei(pa xa −pa ta /2M )/h̄ xa =pa ta /M,by isolating the factor of eipa rb /rb for large rb , as discussed at the end of Section 1.16.On the right-hand side we now insert the x-space form (4.121) of the semiclassicalamplitude, and use (4.80) to writehxb |ÛI (0, ta )|pa i =limta →−∞−taM3/2 "det3∂p− a∂xb!#1/2× ei[S(xb ,xa ;Ea )+ipa xa −iνπ/2]/h̄ xa =pa ta /MNow we observe that−taM3/2 "det3∂p− a∂xb!#1/2"= det3∂xa∂xb!#1/2In Eq.
(4.538) we have found that this determinant is equal toEq. (4.540) to the form1hxb |ÛI (0, ta )|pa i = limta →−∞ rbs..(4.540)(4.541)qdσ/dΩ/rb , bringingdσcl i[S(xb ,xa ;Ea )+ipa xa −iνπ/2]/h̄ edΩ. (4.542)xa =pa ta /MFor large xb in the direction of the final momentum pb , we can rewrite the exponentas [recalling (4.233)]S(xb , xa ; Ea ) + ipa xa = pb rb + S(pb , pa ; Ea )(4.543)so that (4.542) consists of an outgoing spherical wave function eipb rb /h̄ /rb multipliedby the scattering amplitudefpb pa =sdσcldΩ×X0eiS(pb ,pa ;E)/h̄−iνπ/2 ,(4.544)class.
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