Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 23
Текст из файла (страница 23)
This isthe eikonal approximation to the scattering amplitude.As an example, consider2Coulomb scattering where V (r) = Ze /r and (2.727) yieldsχeib,P [v] = −21Ze2 M 1|P| h̄1dz q.−∞b2 + z 2Z∞M. Abramowitz and I. Stegun, op. cit., Formula 9.1.71.(1.499)721 FundamentalsThe integral diverges logarithmically, but in a physical sample, the potential isscreened at some distance R by opposite charges.
Performing the integral up to Ryields2χeib,P [v]Ze M 1= −|P| h̄ZRb2R+Ze M 1logdr q=−|P| h̄r 2 − b212RZe2 M 1log.≈ −2|P| h̄bqR2 − b2b(1.500)This impliesexpwhereχeib,P≈b2R!2iγ,(1.501)Ze2 M 1γ≡|P| h̄(1.502)is a dimensionless quantity since e2 = h̄cα where α is the dimensionless fine-structureconstant 22e2= 1/137.035 997 9 .
. . .(1.503)α=h̄cThe integral over the impact parameter in (1.498) can now be performed and yieldsfpeib pa ≈1Γ(1 + iγ) −2iγ log(2pR/h̄)h̄e.2+2iγ2ip sin(θ/2) Γ(−iγ)(1.504)Remarkably, this is the exact quantum mechanical amplitude of Coulomb scattering,except for the last phase factor which accounts for a finite screening length. Thisamplitude contains poles at momentum variables p = pn wheneverZe2 Mh̄= −n,iγn ≡pnn = 1, 2, 3, . .
. .(1.505)This corresponds to energiesE(n)p2nMZ 2 e4 2=−=−2n ,2Mh̄2(1.506)which are the well-known energy values of hydrogen-like atoms with nuclear chargeZe. The prefactor EH ≡ e2 /aH = Me4 /h̄2 = 4.359 × 10−11 erg = 27.210 eV, is equalto twice the Rydberg energy (see also p. 952).22The fine-structure constant is measured most precisely via the quantum Hall effect , see M.E.Cage et al., IEEE Trans. Instrum. Meas. 38 , 284 (1989).H. Kleinert, PATH INTEGRALS731.16 Scattering1.16.5Scattering Amplitude from Time Evolution AmplitudeThere exists a heuristic formula expressing the scattering amplitude as a limit ofthe time evolution amplitude.
For this we express the δ-function in the energy as alarge-time limit!1/2i tbexp −(pb − pa )2 ,h̄ 2M(1.507)where pb = |pb |. Inserting this into Eq. (1.475) and setting sloppily pb = pa forelastic scattering, the δ-function is removed and we obtain the following expressionfor the scattering amplitudeMMtbδ(Eb − Ea ) =δ(pb − pa ) =limpbpb tb →∞ 2πh̄M/ifpb pap= bMq32πh̄M/i(2πh̄)lim31tb →∞ t1/2beiEb (tb −ta )/h̄ [(pb tb |pa ta )−hpb |pa i] .(1.508)This treatment of a δ-function is certainly unsatisfactory. A satisfactory treatment will be given in the path integral formulation in Section 2.22.
At the presentstage, we may proceed with more care with the following operator calculation. Werewrite the limit (1.472) with the help of the time evolution operator (2.5) as follows:hpb |Ŝ|pa i ≡=limei(Eb tb −Ea ta )/h̄ (pb tb |pa ta )limhpb |ÛI (tb , ta )|pa i,tb −ta →∞tb ,−ta →∞(1.509)where ÛI (tb , ta ) is the time evolution operator in Dirac’s interaction picture (1.287).1.16.6Lippmann-Schwinger EquationFrom the definition (1.287) it follows that the operator ÛI (tb , ta ) satisfies the samecomposition law (1.255) as the ordinary time evolution operator Û (t, ta ):ÛI (t, ta ) = ÛI (t, tb )ÛI (tb , ta ).(1.510)Now we observe thate−iH0 t/h̄ ÛI (t, ta ) = e−iHt/h̄ ÛI (0, ta ) = ÛI (0, ta − t)e−iH0 t/h̄ ,(1.511)so that in the limit ta → −∞e−iH0 t/h̄ ÛI (t, ta ) = e−iHt/h̄ ÛI (0, ta ) −−−→ ÛI (0, ta )e−iH0 t/h̄ ,(1.512)and thereforelim Û (t , t ) = lim eiH0 tb /h̄ e−iHtb /h̄ ÛI (0, ta ) = lim eiH0 tb /h̄ ÛI (0, ta )e−iH0 tb /h̄,ta →−∞ I b ata →−∞ta →−∞(1.513)741 Fundamentalswhich allows us to rewrite the scattering matrix (1.509) ashpb |Ŝ|pa i ≡limtb ,−ta →∞ei(Eb −Ea )tb /h̄ hpb |ÛI (0, ta )|pa i.(1.514)Note that in contrast to (1.472), the time evolution of the initial state goes now onlyover the negative time axis rather than the full one.Taking the matrix elements of Eq.
(1.292) between free-particle states hpb | and|pb i, and using Eqs. (1.292) and (1.512), we obtain at tb = 0ihpb |ÛI (0, ta )|pb i = hpb |pb i −h̄Z0−∞dt ei(Eb −Ea −iη)t/h̄ hpb |V̂ ÛI (0, ta )|pb i. (1.515)A small damping factor eηt/h̄ is inserted to ensure convergence at t = −∞. For atime-independent potential, the integral can be done and yieldshpb |ÛI (0, ta )|pb i = hpb |pb i −1hp |V̂ ÛI (0, ta )|pb i.Eb − Ea − iη b(1.516)This is the famous Lippmann-Schwinger equation. Inserting this into (1.514), weobtain the equation for the scattering matrixhpb |Ŝ|pa i =limtb ,−ta →∞i(Eb −Ea )tbe"#1hpb |pa i −hp |V̂ ÛI (0, ta )|pb i .
(1.517)Eb − Ea − iη bThe first term in brackets is nonzero only if the momenta pa and pb are equal, inwhich case also the energies are equal, Eb = Ea , so that the prefactor can be setequal to one. In front of the second term, the prefactor oscillates rapidly as thetime tb grows large, making any finite function of Eb vanish, as a consequence of theRiemann-Lebesgue lemma.
The second term contains, however, a pole at Eb = Eafor which the limit has to be done more carefully. The prefactor has the propertyei(Eb −Ea )tb /h̄=limtb →∞ E − E − iηba(0,i/η,Eb =6 Ea ,Eb = Ea .(1.518)It is easy to see that this property defines a δ-function in the energy:ei(Eb −Ea )tb /h̄= 2πiδ(Eb − Ea ).tb →∞ E − E − iηbalim(1.519)Indeed, let us integrate the left-hand side together with a smooth function f (Eb ),and setEb ≡ Ea + ξ/tb .(1.520)Then the Eb -integral is rewritten asZeiξf (Ea + ξ/ta ) .dξξ + iη−∞∞(1.521)H. Kleinert, PATH INTEGRALS751.16 ScatteringIn the limit of large ta , the function f (Ea ) can be taken out of the integral and thecontour of integration can then be closed in the upper half of the complex energyplane, yielding 2πi. Thus we obtain from (1.517) the formula (1.475), with theT -matrix1(1.522)hpb |T̂ |pa i = hpb |V̂ ÛI (0, ta )|pb i.h̄For a small potential V̂ , we approximate ÛI (0, ta ) ≈ 1, and find the Born approximation (1.491).The the Lippmann-Schwinger equation can be recast as an integral equation forthe T -matrix.
Multiplying the original equation (1.516) by the matrix hpb |V̂ |pa i =Vpb pc from the left, we obtainTpb pa = Vpb pa −Zd 3 pc1T.3 Vpb pcEc − Ea − iη pc pa(2πh̄)(1.523)To extract physical information from the T -matrix (1.522) it is useful to analyzethe behavior of the interacting state ÛI (0, ta )|pa i in x-space. From Eq. (1.512),we see that it is an eigenstate of the full Hamiltonian operator Ĥ with the initialenergy Ea .
Multiplying this state by hx| from the left, and inserting a complete setof momentum eigenstates, we calculatehx|ÛI (0, ta )|pa i =Zd3 phx|pihp|ÛI (0, ta )|pa i =(2πh̄)3d3 phx|pihp|ÛI (0, ta )|pa i.(2πh̄)3ZUsing Eq. (1.516), this becomes0hx|ÛI (0, ta )|pa i = hx|pa i +ZZ3 0dxThe function0(x|x )Ea =Zeipb (x−x )/h̄d 3 pbV (x0 )hx0 |ÛI (0, ta )|pa i.32(2πh̄) Ea −pb /2M +iη(1.524)ih̄d3 pb ipb (x−x0 )/h̄3e2(2πh̄)Ea − p /2M + iη(1.525)is recognized as the fixed-energy amplitude (1.341) of the free particle. In threedimensions it reads [see (1.356)]00(x|x )Ea2Mi eipa |x−x |/h̄,=−h̄ 4π|x − x0 |pa =q2MEa .(1.526)In order to find the scattering amplitude, we consider the wave function (1.524) faraway from the scattering center, i.e., at large |x|.
Under the assumption that V (x0 )is nonzero only for small x0 , we approximate |x − x0 | ≈ r − x̂x0 , where x̂ is the unitvector in the direction of x, and (1.524) becomesipa x/h̄hx|ÛI (0, ta )|pa i ≈ eeipa r−4πrZ0d4 x0 e−ipa x̂x2M002 V (x )hx |ÛI (0, ta )|pa i.h̄(1.527)761 FundamentalsIn the limit ta → −∞, the factor multiplying the spherical wave factor eipa r/h̄ /ris the scattering amplitude f (x̂)pa , whose absolute square gives the cross section.For scattering to a final momentum pb , the outgoing particles are detected far awayfrom the scattering center in the direction x̂ = p̂b .
Because of energy conservation,we may set pa x̂ = pb and obtain the formulafpb paM Z 4 −ipb xb= lim −d xb eV (xb )hxb |ÛI (0, ta )|pa i.ta →−∞2πh̄2(1.528)By studying the interacting state ÛI (0, ta )|pa i in x-space, we have avoided the singular δ-function of energy conservation.We are now prepared to derive formula (1.508) for the scattering amplitude. Weobserve that in the limit ta → −∞, the amplitude hxb |ÛI (0, ta )|pa i can be obtainedfrom the time evolution amplitude (xb tb |xa ta ) as follows:hxb |ÛI (0, ta )|pa i = hxb |Û (0, ta )|pa ie−iEa ta /h̄=limta →−∞−2πih̄taM!3/2(1.529)2(xb tb |xa ta )ei(pa xa −pa ta /2M )/h̄ This follows directly from the Fourier transformation−iEa ta /h̄hxb |Û(0, ta )|pa ie=Z2d3 xa (xb tb |xa ta )ei(pa xa −pa ta /2M )/h̄ ,xa =pa ta /M.(1.530)by substituting the dummy integration variable xa by pta /M. Then the right-handside becomes Z−ta 32d3 p (xb 0|pta ta )ei(pa p−pa )ta /2M h̄ .(1.531)MNow, for large −ta , the momentum integration is squeezed to p = pa , and we obtain(1.529).
The appropriate limiting formula for the δ-function(−ta )D/2i ta(pb − pa )2(1.532)δ (pb − pa ) = lim √D exp −ta →−∞h̄ 2M2πih̄Mis easily obtained from Eq. (1.339) by an obvious substitution of variables. Itscomplex conjugate for D = 1 was written down before in Eq. (1.507) with ta replacedby −tb . The exponential on the right-hand side can just as well be multiplied by a22 2factor ei(pb −pa ) /2M h̄ which is unity when both sides are nonzero, so that it becomes2e−i(pa p−pa )ta /2M h̄ . In this way we obtain a representation of the δ-function by which2the Fourier integral (1.531) goes over into (1.529).
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
