Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 22
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Incidentally, this is alsotrue for the asymmetric top with Iξ 6= Iη 6= Iζ , where g = Iξ2 Iζ sin2 β, although themetric gµν is then much more complicated (see Appendix 1C).The canonical momenta associated with the Lagrangian (1.455) are, accordingto (1.384),Rpα = ∂L/∂ α̇ = Iξ α̇ sin2 β + Iζ cos β(α̇ cos β + γ̇),pβ = ∂L/∂ β̇ = Iξ β̇,pγ = ∂L/∂ γ̇ = Iζ (α̇ cos β + γ̇).(1.462)661 FundamentalsAfter inverting the metric tog µνµν10− cos β120sinβ0=2Iξ sin β − cos β220cos β + Iξ sin β/Iζ,(1.463)we find the classical Hamiltonian1 1 2cos2 β1H=+pβ +22 IξIξ sin β Iζ"!pγ2#2 cos β12p p .+2 pα −Iξ sin βIξ sin2 β α γ(1.464)This Hamiltonian has no apparent ordering problem.
One is therefore tempted toreplace the momenta simply by the corresponding Hermitian operators which are,according to (1.388),p̂α = −ih̄∂α ,p̂β = −ih̄(sin β)−1/2 ∂β (sin β)1/2 = −ih̄(∂β +1cot β),2p̂γ = −ih̄∂γ .(1.465)Inserting these into (1.464) gives the canonical Hamiltonian operatorĤcan = Ĥ + Ĥdiscr ,(1.466)withIh̄2Ĥ ≡ −∂β 2 + cot β∂β + ξ + cot2 β ∂γ 22IξIζ"!12 cos β2+∂ ∂2 ∂α −sin βsin2 β α γand1113Ĥdiscr ≡ (∂β cot β) + cot2 β =− .2244 sin β 4#(1.467)(1.468)The first term Ĥ agrees with the correct quantum-mechanical operator derivedabove.
Indeed, inserting the differential operators for the body-fixed angular momenta (1.438) into the Hamiltonian (1.424), we find Ĥ. The term Ĥdiscr is thediscrepancy between the canonical and the correct Hamiltonian operator. It existseven though there is no apparent ordering problem, just as in the radial coordinateexpression (1.401). The correct Hamiltonian could be obtained by replacing theclassical pβ 2 term in H by the operator g −1/4 p̂β g 1/2 p̂β g −1/4 , by analogy with thetreatment of the radial coordinates in Ĥ of Eq. (1.395).As another similarity with the two-dimensional system in radial coordinates andthe particle on the surface of the sphere, we observe that while the canonical quantization fails, the Hamiltonian operator of the symmetric spinning top is correctlygiven by the Laplace-Beltrami operator (1.378) after inserting the metric (1.460)H.
Kleinert, PATH INTEGRALS671.15 Spinning Topand the inverse (1.463). It is straightforward although tedious to verify that this isalso true for the completely asymmetric top [which has quite a complicated metricgiven in Appendix 1C, see Eqs. (1C.2), and (1C.4)]. This is an important nontrivialresult, since for a spinning top, the Lagrangian cannot be obtained by reparametrizing a particle in a Euclidean space with curvilinear coordinates. The result suggeststhat a replacementgµν (q)pµ pν → −h̄2 ∆(1.469)produces the correct Hamiltonian operator in any non-Euclidean space.19What is the characteristic non-Euclidean property of the α, β, γ space? As weshall see in detail in Chapter 10, the relevant quantity is the curvature scalar R.The exact definition will be found in Eq.
(10.42). For the asymmetric spinning topwe find (see Appendix 1C)(Iξ + Iη + Iζ )2 − 2(Iξ2 + Iη2 + Iζ2 ).R=2Iξ Iη Iζ(1.470)Thus, just like a particle on the surface of a sphere, the spinning top correspondsto a particle moving in a space with constant curvature. In this space, the correct correspondence principle can also be deduced from symmetry arguments.
Thegeometry is most easily understood by observing that the α, β, γ space may be considered as the surface of a sphere in four dimensions, as we shall see in more detailin Chapter 8.An important non-Euclidean space of physical interest is encountered in thecontext of general relativity. Originally, gravitating matter was assumed to move ina spacetime with an arbitrary local curvature.
In newer developments of the theoryone also allows for the presence of a nonvanishing torsion. In such a general situation,where the group quantization rule is inapplicable, the correspondence principle hasalways been a matter of controversy [see the references after (1.402)] to be resolved inthis text. In Chapters 10 and 8 we shall present a new quantum equivalence principlewhich is based on an application of simple geometrical principles to path integralsand which will specify a natural and unique passage from classical to quantummechanics in any coordinate frame.20 The configuration space may carry curvatureand a certain class of torsions (gradient torsion).
Several arguments suggest thatour principle is correct. For the above systems with a Hamiltonian which can beexpressed entirely in terms of generators of a group of motion in the underlyingspace, the new quantum equivalence principle will give the same results as the groupquantization rule.19If the space has curvature and no torsion, this is the correct answer. If torsion is present, thecorrect answer will be given in Chapters 10 and 8.20H. Kleinert, Mod. Phys. Lett.
A 4 , 2329 (1989) (http://www.physik.fu-berlin.de/~kleinert/199); Phys. Lett. B 236 , 315 (1990) (ibid.http/202).681.161 FundamentalsScatteringMost observations of quantum phenomena are obtained from scattering processes offundamental particles.1.16.1Scattering MatrixConsider 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 . The energy will beunchanged: E = Eb = p2b /2M. The probability amplitude for such a process isgiven by the time evolution amplitude in the momentum representation(pb tb |pa ta ) ≡ hpb |e−iĤ(tb −ta )/h̄ |pa i,(1.471)where the limit tb → ∞ and ta → −∞ has to be taken.
Long before and after thecollision, this amplitude oscillates with a frequency ω = E/h̄ characteristic for freeparticles of energy E. In order to have a time-independent limit, we remove theseoscillations, from (1.471), and define the scattering matrix (S-matrix) by the limithpb |Ŝ|pa i ≡limtb −ta →∞ei(Eb tb −Ea ta )/h̄ hpb |e−iĤ(tb −ta )/h̄ |pa i.(1.472)Most of the impinging particles will not scatter at all, so that this amplitude mustcontain a leading term, which is separated as follows:hpb |Ŝ|pa i = hpb |pa i + hpb |Ŝ|pa i0 ,(1.473)hpb |pa i = hpb |e−iĤ(tb −ta )/h̄ |pa i = (2πh̄)3 δ (3) (pb − pa )(1.474)whereshows the normalization of the states [recall (1.186)]. This leading term is commonly subtracted from (1.472) to find the true scattering amplitude.
Moreover,since potential scattering conserves energy, the remaining amplitude contains a δfunction ensuring energy conservation, and it is useful to divide this out, definingthe so-called T -matrix by the decomposition byhpb |Ŝ|pa i ≡ (2πh̄)3 δ (3) (pa − pa ) − 2πh̄iδ(Eb − Ea )hpb |T̂ |pa i.(1.475)From the definition (1.472) and the hermiticity of Ĥ it follows that scattering matrixis a unitary matrix.
This expresses the physical fact that the total probability ofan incident particle to re-emerge at some time is unity (in quantum field theory thesituation is more complicated due to emission and absorption processes).In the basis states |pm i introduced in Eq. (1.180) which satisfy the completenessrelation (1.182) and are normalized to unity in a finite volume V , the unitarity isexpressed asXm00000hpm |Ŝ † |pm ihpm |Ŝ|pm i =Xm00000hpm |Ŝ|pm ihpm |Ŝ † |pm i = 1.(1.476)H.
Kleinert, PATH INTEGRALS691.16 ScatteringRemembering the relation (1.185) between the discrete states |pm i and their continuous limits |pi, we see that0hpb m |Ŝ|pa m i ≈1hp |Ŝ|pa i,L3 b(1.477)mwhere L3 is the spatial volume, and pmb and pa are the discrete momenta closest topb and pa . In the continuous basis |pi, the unitarity relation readsZ1.16.2Zd3 pd3 p†hp|Ŝ|pihp|Ŝ|pi=hp |Ŝ|pihp|Ŝ † |pa i = 1.a(2πh̄)3 b(2πh̄)3 b(1.478)Cross SectionThe absolute square of hpb |Ŝ|pa i gives the probability Ppb ←pa for the scatteringfrom the initial momentum state pa to the final momentum state pb . Omitting theunscattered particles, we have12πh̄δ(0) 2πh̄δ(Eb − Ea )|hpb |T̂ |pa i|2 .(1.479)L6The factor δ(0) at zero energy is made finite by imagining the scattering process totake place with anincident time-independent plane wave over a finite total time T .RThen 2πh̄δ(0) = dt eiEt/h̄ |E=0 = T , and the probability is proportional to the timeT:1(1.480)Ppb ←pa = 6 T 2πh̄δ(Eb − Ea )|hpb |T̂ |pa i|2 .LBy summing this over all discrete final momenta, or equivalently, by integratingthis over the phase space of the final momenta [recall (1.184)], we find the totalprobability per unit time for the scattering to take placePpb ←pa =dP1= 6dtLZd3 pb L32πh̄δ(Eb − Ea )|hpb |T̂ |pa i|2 .(2πh̄)3(1.481)The momentum integral can be split into an integral over the final energy and thefinal solid angle.
For non-relativistic particles, this goes as followsZd 3 pbM13 =3(2πh̄)(2πh̄) (2πh̄)3ZdΩZ0∞dEb pb ,(1.482)where dΩ = dφb d cos θb is the element of solid angle into which the particle isscattered. The energy integral removes the δ-function in (1.481), and makes pbequal to pa .The differential scattering cross section dσ/dΩ is defined as the probability thata single impinging particle ends up in a solid angle dΩ per unit time and unit currentdensity.
From (1.481) we identifydσdṖ 11 Mp21== 3,3 2πh̄|Tpb pa |dΩdΩ jjL (2πh̄)(1.483)701 Fundamentalswhere we have sethpb |T̂ |pa i ≡ Tpb pa ,(1.484)for brevity. In a volume L3 , the current density of a single impinging particle isgiven by the velocity v = p/M asj=1 p,L3 M(1.485)so that the differential cross section becomesdσM22=2 |Tpb pa | .dΩ(2πh̄)(1.486)If the scattered particleqmoves relativistically, we have to replace the constant massM in (1.482) by E = p2 + M 2 inside the momentum integral, where p = |p|, sothatZZ ∞1 Zd3 pdp p2dΩ=0(2πh̄)3(2πh̄)3ZZ ∞1=dEE p.dΩ0(2πh̄)3(1.487)In the relativistic case, the initial current density is not proportional to p/M but tothe relativistic velocity v = p/E so that1 p.L3 E(1.488)dσE22=2 |Tpb pa | .dΩ(2πh̄)(1.489)j=Hence the cross section becomes1.16.3Born ApproximationTo lowest order in the interaction strength, the operator Ŝ in (1.472) isŜ ≈ 1 − iV̂ /h̄.(1.490)For a time-independent scattering potential, this impliesTpb pa ≈ Vpb pa /h̄,(1.491)whereVpb pa ≡ hpb |V̂ |pa i =Zd3 x ei(pb −pa )x/h̄ V (x) = Ṽ (pb − pa )(1.492)H.
Kleinert, PATH INTEGRALS711.16 Scatteringis a function of the momentum transfer q ≡ pb − pa only. Then (1.489) reduces tothe so called Born approximation (Born 1926)dσE22≈2 2 |Vpb pa | .dΩ(2πh̄) h̄(1.493)The amplitude whose square is equal to the differential cross section is usuallydenoted by fpb pa , i.e., one writesdσ= |fpb pa |2 .dΩ(1.494)By comparison with (1.493) we identifyfpb pa ≡ −MR,2πh̄ pb pa(1.495)where we have chosen the sign to agree with the convention in the textbook byLandau and Lifshitz [9].1.16.4Partial Wave Expansion and Eikonal ApproximationThe scattering amplitude is usually expanded in partial waves with the help ofLegendre polynomials Pl (z) ≡ Pl0 (z) [see (1.418)] asfpb pa =∞h̄ X(2l + 1)Pl (cos θ) e2i∂l (p) − 12ip l=0(1.496)where p ≡ |p| = |pb | = |pa | and θ is the scattering defined by cos θ ≡ pb pb /|pb ||pa |.In terms of θ, the momentum transfer q = pb − pa has the size |q| = 2p sin(θ/2).For small θ, we can use the asymptotic form of the Legendre polynomials21Pl−m (cos θ) ≈1J (lθ),lm m(1.497)to rewrite (1.496) approximately as an integralfpeib pap=ih̄Znhiodb b J0 (qb) exp 2iδpb/h̄ (p) − 1 ,(1.498)where b ≡ lh̄/p is the so called impact parameter of the scattering process.
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