Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 91
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traj.the same as in (4.539).H. Kleinert, PATH INTEGRALS4.12 Semiclassical Scattering4.12.2445Semiclassical Cross Section of Mott ScatteringIf the scattering particle is distinguishable from the target particles, the extra phasein the semiclassical formula (4.544) does not change the classical result (4.514).A quantum-mechanical effect becomes visible only if we consider electron-electronscattering, also referred to as Mott scattering. The potential is repulsive, and theabove Coulomb potential holds for the relative motion of the two identical particlesin their center-of-mass frame.
Moreover, the identity of particles requires us to addthe amplitudes for the trajectories going to pb and to −pb [see Fig. 4.6], so that thedifferential cross section isdσsc= |fpb pa − fpb ,−pa |2 .dΩ(4.545)The minus sign accounts for the Fermi statistics of the two electrons. For twoidentical bosons, we have to use a plus sign instead. Now the eikonal functionS(p, p, E) enters into the result.
According to Eq. (4.517), this is given bypbpa−pbFigure 4.6 Classical trajectories in Coulomb potential plotted in the center-of-massframe. For identical particles, trajectories which merge with a scattering angle θ and π − θare indistinguishable. Their amplitudes must be subtracted from each other, yielding thedifferential cross section (4.545).where p∞ =√√1+∆+1Mh̄cα,log √S(pb , pa ; E) = −p∞1+∆−1(4.546)2ME is the impinging momentum at infinite distance, and(p2b − p2∞ )(pa 2 − p2∞ )∆≡.p2∞ |pb − pa |2(4.547)The eikonal function is needed only for momenta pb , pa in the asymptotic regimewhere pb , pa ≈ p∞ , so that ∆ is small andS(pb , pa ; E) ≈Mh̄cαlog ∆,p∞(4.548)4464 Semiclassical Time Evolution Amplitudewhich may be rewritten asMh̄cαlog(sin2 θ/2),p∞(4.549)Mh̄cα(p2b − p2∞ )(pa 2 − p2∞ ),σ0 =log2p∞p4∞(4.550)S(p, pa E) ≈ 2σ0 −with#"and the scattering angle determined by cos θ = [pb · pa /pb pa ].
The logarithmicallydiverging constant σ0 for pa = pb p∞ does , fortunately, not depend on the scatteringangle, and is therefore the semiclassical approximation for the phase shift at angular momentum l = 0. It therefore drops, fortunately, out of the difference of theamplitudes in Eq.
(4.545). Inserting (4.549) with (4.550) into (4.544) and (4.545),we obtain the differential cross section for Mott scattering (see Fig. 4.7 for a plot)!2 (#)"1112αMclog(cot θ/2) .+±2 cos4422p∞sin θ/2 cos θ/2 sin θ/2 cos θ/2(4.551)This semiclassical result happens to be identical to the exact result. The exactness iscaused by two properties of the Coulomb motion: First there is only one trajectoryfor each scattering angle, second the motion can be mapped onto that of a harmonicoscillator in four dimensions, as we shall see in Chapter 13.dσ=dΩh̄cα4EdσdΩdσdΩfermionsθbosonsθFigure 4.7 Oscillations in differential Mott scattering cross section caused by statistics.For scattering angle θ = 900 , the cross section vanishes due to the Pauli exclusion principle.The right-hand plot shows the situation for identical bosons.Appendix 4ASemiclassical Quantization for Pure PowerPotentialsLet us calculate the local density of states (4.256) for the general pure power potential V (x) = gxp /pin D dimensions.
For D = 1 and p = 2, we shall recover the exact spectrum of the harmonicoscillator, for p = −1 that of the one-dimensional hydrogen atom. For p = 4, we shall find theenergies of the purely quartic potential which can be compared with the strong-coupling limit ofthe anharmonic oscillator with V (x) = ω 2 x2 /2 + gx4 /4 to be calculated in Section 5.16. Theintegrals on the right-hand side of Eq. (4.256) can be calculated using the formulaνZ rEZg pD−1+µDµνdr rE− rd x r [E −V (x)] = SDp0H. Kleinert, PATH INTEGRALSAppendix 4A Semiclassical Quantization for Pure Power Potentials −(D+µ)/pΓ(1 + ν)Γ((D + µ)/p) gE ν+(D+µ)/p ,= SDp Γ(1 + ν + (D + µ)/p) p447p > 0, (4A.1)where rE = (pE/g)1/p . For p < 0, the right-hand side must be replaced by−(D+µ)/pΓ(1 + ν)Γ(−ν − (D + µ)/p)g= SD(−E)ν+(D+µ)/p , p < 0.−p Γ(1 − (D + µ)/p)pRecalling (4.207), we find the total density of states for p > 0: (D 2/pD/2 −D/pΓp2gh̄2Mg 1−ρ(E) =2DDDp24MpΓ 2 p 2h̄Γ 2 + p) )DDp2 Γ D−2p +2 Γ 2 + p)−1−2/p E+ .
. . E (D/2)(1+2/p)−1, p > 0,×D−22DΓ2 (1 + p ) Γ p(4A.2)(4A.3)and for p < 0:−D/p Γ 1 − D − D (2/pD/2 2ph̄2gg1+−−ρ(E) = − Dp24MpΓ( 2 ) pΓ 1− Dp )2Dp2 Γ 1− D−22 (1+ p ) Γ 1− p (−E)−1−2/p + . . . (−E)(D/2)(1+2/p)−1, p < 0.×DDΓ1−Γ −1− D−2−)p2p2M2h̄2For a harmonic oscillator with p = 2 and g = M ω 2 , we obtain1Dh̄2 ω 21D−1D−3E−E+ ...
.ρ(E) =Γ(D)24 Γ(D−2)(h̄ω)D(4A.4)(4A.5)In one two dimension, only the first term survives and ρ(E) = 1/h̄ω or ρ(E) = E/(h̄ω)2 . InsertingREthis into Eq. (1.582), we find the number of states N (E) = 0 dE 0 ρ(E 0 ) = E/h̄ω or E 2 /2(h̄ω)2 .According to the exact quantization condition (1.584), we set N (E) = n + 1/2 the exact energiesEn = (n + 1/2)h̄ω. Since the semiclassical expansion (4A.3) contains only the first term, the exactquantization condition (1.584) agrees with the Bohr-Sommerfeld quantization condition (4.184).In two dimensions we obtain ρ(E) = E/(h̄ω)2 and N (E) = E 2 /2(h̄ω)2 . Here the exactquantization condition N (E) = n + 1/2 cannot be used to find the energies En , due to thedegeneracies of the energy eigenvalues En . In order to see that it is nevertheless a true equation,let us expanding the partition function of the two-dimensional oscillator [recall (2.399)]Z=∞X11−h̄βω=e=(n + 1)e−(n+1)h̄βω .[2 sinh(h̄βω/2)]2(1 − e−h̄βω )2n=0(4A.6)This shows that the the energies are En = (n + 1)h̄ω withy n + 1 -fold degeneracy.
The density ofstates is found from this by the Fourier transform (1.581):ρ(E) =∞X(n0 + 1)δ(E − h̄ω(n0 + 1)).(4A.7)n0 =0Integrating this over E yields the number of statesZ E∞XN (E) =dE 0 ρ(E 0 ) =(n0 + 1)Θ(E − En ).0n0 =0(4A.8)4484 Semiclassical Time Evolution AmplitudeFor E = En this becomes [recall (1.310)]N (En ) =n−1X(n0 + 1) + (n + 1)n0 =011= (n + 1)2 .22(4A.9)This shows that the exact energies En = (n + 1)h̄ω of the two-dimensional oscillator satisfy thequantization condition N (En ) = (n + 1)2 /2 rather than (1.584).For a quartic potential gx4 /4, Eq. (4A.3) becomesD 4 12 3DD 4 − 4 D3D3− 32Γ( 4 ) gh̄Γ( 2 + 4 )Γ( 4 )E1gh̄ + .
. . E 4 −1. (4A.10)ρ(E) =1−3D3DD22M2 Γ( 2 ) Γ( 4 ) M3Γ( 4 )Γ 4 (D − 2)Integrating this over E yields N (E). Setting N (E) = n + 1/2 in one dimension, we obtain theBohr-Sommerfeld energies (4.31) plus a first quantum correction. Since we have studied thesecorrections to high order in Subsection 4.9.6 (see Fig. 4.1), we do not write the result down here.A physically important case is p = −1, g = h̄cα, with α of Eq. (1.503), where V (x) = −h̄cα/rbecomes the Coulomb potential. Here we obtain from (4A.10):ρ(E) =×D/2Γ 1+ D2Γ (1 + D))Γ(1+D)p2 Γ D−2h̄22 E + .
. . (−E)(D/2)(1+2/p)−1 .1−24M g −2 Γ (D − 3) Γ 1 + D2)2)Γ( D( 2M g22πh̄2For d = 1, only the leading term survives andrM g2(−E)−3/2 ,ρ(E) =2h̄2(4A.11)(4A.12)implyingrM g2(−E)−1/2 .(4A.13)2h̄2In order to find the bound-state energies, we must watch out for a subtlety in one dimension:only the positive half-space is accessible to the particle in a Coulomb potential, due to the strongsingularity at the origin. For this reason, the “surface of a sphere” SD for D = 1 , which is equalto 2, must be replaced by 1, so that we must equate N (E)/2 to n + 1/2. This yields the spectrumEn = −α2 M c2 /2(n + 1/2)2 . As in the harmonic oscillator, the Bohr-Sommerfeld approximationgives the exact energies.N (E) = 2Appendix 4BDerivation of Semiclassical Time EvolutionAmplitudeHere we derive the semiclassical approximation to the time evolution amplitude (4.259).
We shalldo this for imaginary times τ = it. Decomposing the path x(τ ) into path average of the endspoints x = (xb + xa )/2 and fluctuations (τ ), we calculate the imaginary-time amplitudeZ (τb )=∆x/2Z1 τbM 2˙ (τ ) + V x + (τ )(xb τb |xa τa ) =D exp −,dτ (τa )=−∆x/2h̄ τa2where ∆x ≡ xb − xa .
For smooth potentials we expand1V x + (τ ) = V (x) + ∂i V (x) ηi (τ ) + ∂i ∂j V (x) ηi (τ ) ηj (τ ) + . . . ,2(4B.1)(4B.2)H. Kleinert, PATH INTEGRALSAppendix 4B Derivation of Semiclassical Time Evolution Amplitude449where Vij... (x) ≡ ∂i ∂j · · · V (x), and rewrite the path integral (4B.1) asZZ (τb )=∆x/2M 21 τb−βV (x)˙ (τ )dτ(xb τb |xa τa ) = eD exp − (τa )=−∆x/2h̄ τa2(Zi1 τb h1× 1−dτ Vi (x) ηi (τ ) + Vij (x) ηi (τ ) ηj (τ ) + . .
.h̄ τa2)Z τb Z τb hi100dτ Vi (x)Vj (x) ηi (τ )ηj (τ )+. . . +. . . .dτ+ 22h̄ τaτaAt this point it is useful to introduce an auxiliary harmonic imaginary-time amplitudeZ (τb )=∆x/2ZM 21 τb˙ (τ )dτ(∆x/2 τb |−∆x/2 τa ) =D exp − (τa )=−∆x/2h̄ τa2and the harmonic expectation values Z τbZ (τb )=∆x/211M 2˙ (τ ) ,hF [ ]i ≡D F [ ] exp −dτ(∆x/2 τb |−∆x/2 τa) (τa )=−∆x/2h̄ τa2(4B.3)(4B.4)(4B.5)which allows us to rewrite (4B.3) more concisely asZ1 τbdτ Vi (x) hηi (τ )i(xb τb |xa τa ) = e(∆x/2 τb |−∆x/2 τa ) 1 −h̄ τZ τbZ τb Za τb1100dτ hηi (τ )ηj (τ )i+.
. . .dτ hηi (τ )ηj (τ )i+ 2 Vi (x)Vj (x)dτ− Vij (x)2h̄2h̄τaτaτa−βV (x)(4B.6)The amplitudes (4B.4) reads explicitly, with ∆τ ≡ τb − τa :(∆x/2 τb | − ∆x/2 τa ) =M2πh̄∆τD/2M2exp −(∆x) ,2h̄∆τ(4B.7)and (4B.5) can be calculated from the generating functional Z τb Z (τb )=∆x/21M 2˙ (τ ) − j(τ ) (τ ) ,dτ(∆x/2 τb |−∆x/2 τa )[j] =D − (τa )=−∆x/2h̄ τa2(4B.8)whose explicit solution isD/2 Z τbMM1dτ (τ − τ̄ ) ∆x j(τ )(∆x/2 τb |−∆x/2 τa )[j] =exp −(∆x)2 +2πh̄∆τ2h̄∆τh̄∆τ τaZ τb Z τb1Θ(τ − τ 0 )(∆τ − τ )τ 0 + Θ(τ 0 − τ )(∆τ − τ 0 )τ+dτj(τ ) j(τ 0 ) .dτ 02h̄ τaM ∆ττaThe expectation values in (4B.6) and (4B.5) are obtained from the functional derivativesh̄δ1(4B.9)hηi (τ )i =(∆x/2 τb |−∆x/2 τa )[j] ,(∆x/2 τb | − ∆x/2 τa ) δji (τ )j=01h̄δh̄δ .(∆x/2τ|−∆x/2τ)[j](4B.10)hηi (τ )ηj (τ 0 )i =ba0(∆x/2 τb |−∆x/2 τa ) δji (τ ) δjj (τ )j=0This yields the ∆x-dependent expectation valuehηi (τ )i = (τ − τ̄ )∆xi,∆τ(4B.11)4504 Semiclassical Time Evolution Amplitudeand the ∆x-dependent correlation functionh̄[Θ(τ − τ 0 )(∆τ − τ )τ 0 + Θ(τ 0 − τ )(∆τ − τ 0 )τ ] δijM ∆τ∆xi ∆xj+ (τ − τ̄ ) (τ 0 − τ̄ )∆τ ∆τhηi (τ )ηj (τ 0 )i =≡A2 (τ, τ 0 )δij + B 2 (τ, τ 0 )∆xi ∆xj ≡ Gij (τ, τ 0 ),suppressing the argument ∆x in Gij (τ, τ 0 ), for brevity.
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