Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 26
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Kleinert, PATH INTEGRALSAppendix 1C The Asymmetric Top87From this we find the components of the Riemann connection, the Christoffel symbol defined inEq. (1.70):Γ̄11 1=Γ̄21 1=Γ̄31 1=Γ̄22 1Γ̄32 1==Γ̄33 1=2=Γ̄21 2=Γ̄31 2=Γ̄22 2Γ̄32 2==Γ̄33 2Γ̄11 3==Γ̄11Γ̄21 3=Γ̄31 3=Γ̄22 3Γ̄32 3==Γ̄33 3=[cos β cos γ sin γ(Iη2 − Iη Iζ − Iξ2 + Iξ Iζ )]/Iξ Iη ,{cos β[sin2 γ(Iξ2 − Iη2 − (Iξ − Iη )Iζ )+ Iη (Iξ + Iη − Iζ )]}/2 sin βIξ Iη ,{cos γ sin γ[Iη2 − Iξ2 + (Iξ − Iη )Iζ ]}/2Iξ Iη ,0,[sin2 γ(Iξ2 − Iη2 − (Iξ − Iη )Iζ ) − Iη (Iξ − Iη + Iζ )]/2 sin βIξ Iη ,0,{cos β sin β[sin2 γ(Iξ2 − Iη2 − Iζ (Iξ − Iη )) − Iξ (Iξ − Iζ )]}/Iξ Iη ,{cos β cos γ sin γ[Iξ2 − Iη2 − Iζ (Iξ − Iη )]}/2Iξ Iη ,{sin β[sin2 γ(Iξ2 − Iη2 − Iζ (Iξ − Iη )) − Iξ (Iξ − Iη − Iζ )]}/2Iξ Iη ,0,[cos γ sin γ(Iξ2 − Iη2 − Iζ (Iξ − Iη ))]/2Iξ Iη ,0,{cos γ sin γ[sin2 β(Iξ Iη (Iξ − Iη ) − Iζ (Iξ2 − Iη2 ) + Iζ2 (Iξ − Iη ))+ (Iξ2 − Iη2 )Iζ − Iζ2 (Iξ − Iη )]}/Iξ Iη Iζ ,{sin2 β[sin2 γ(2Iξ Iη (Iη − Iξ ) + Iζ (Iξ2 − Iη2 ) − Iζ2 (Iξ − Iη ))+Iξ Iη (Iξ − Iη ) + Iη Iζ (Iη − Iζ )] − sin2 γ((Iξ2 − Iη2 )Iζ − Iζ2 (Iξ − Iη ))− Iη Iζ (Iξ + Iη − Iζ )}/2 sin βIξ Iη Iζ ,[cos β cos γ sin γ(Iξ2 − Iη2 − Iζ (Iξ − Iη ))]/2Iξ Iη ,cos γ sin γ(Iη − Iξ )/Iζ ,{cos β[sin2 γ(Iη2 − Iξ2 + (Iξ − Iη )Iζ ) + Iη (Iξ − Iη + Iζ )]}/2 sin βIη Iξ ,0.(1C.5)The other components follow from the symmetry in the first two indices Γ̄µν λ = Γ̄λνµ .
From thisChristoffel symbol we calculate the Ricci tensor, to be defined in Eq. (10.41),R̄11R̄21R̄31R̄22R̄32R̄33= {sin2 β[sin2 γ(Iη3 − Iξ3 − (Iξ Iη − Iζ2 )(Iξ − Iη ))+ ((Iξ + Iζ )2 − Iη2 )(Iξ − Iζ )] + Iζ3 − Iζ (Iξ − Iη )2 }/2Iξ Iη Iζ ,= {sin β sin γ cos γ[Iη3 − Iξ3 + (Iξ Iη − Iζ2 )(Iη − Iξ )]}/2Iξ Iη Iζ ,= −{cos β[(Iξ − Iη )2 − Iζ2 ]}/2Iξ Iη ,= {sin2 γ[Iξ3 − Iη3 + (Iξ Iη − Iζ2 )(Iξ − Iη )] + Iη3 − (Iξ − Iζ )2 Iη }/2Iξ Iη Iζ ,= 0,= −[(Iξ − Iη )2 − Iζ2 ]/2Iξ Iη .(1C.6)Contraction with g µν gives the curvature scalarR̄ = [2(Iξ Iη + Iη Iζ + Iζ Iξ ) − Iξ2 − Iη2 − Iζ2 ]/2Iξ Iη Iζ .(1C.7)Since the space under consideration is free of torsion, the Christoffel symbol Γ̄µν λ is equal to thefull affine connection Γµν λ .
The same thing is true for the curvature scalars R̄ and R calculatedfrom Γ̄µν λ and Γµν λ , respectively.881 FundamentalsNotes and ReferencesFor more details see some standard textbooks:I. Newton, Mathematische Prinzipien der Naturlehre, Wiss. Buchgesellschaft, Darmstadt, 1963;J.L. Lagrange, Analytische Mechanik , Springer, Berlin, 1887;G. Hamel, Theoretische Mechanik , Springer, Berlin, 1949;A. Sommerfeld, Mechanik , Harri Deutsch, Frankfurt, 1977;W.
Weizel, Lehrbuch der Theoretischen Physik , Springer, Berlin, 1963;H. Goldstein, Classical Mechanics, Addison-Wesley, Reading, 1950;L.D. Landau and E.M. Lifshitz, Mechanics, Pergamon, London, 1965;R. Abraham and J.E. Marsden, Foundations of Mechanics, Benjamin, New York, 1967;C.L. Siegel and J.K. Moser, Lectures on Celestial Mechanics, Springer, Berlin, 1971;P.A.M. Dirac, The Principles of Quantum Mechanics, Clarendon, Oxford, 1958;L.D.
Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965;A. Messiah, Quantum Mechanics, Vols. I and II, North-Holland , Amsterdam, 1961;L.I. Schiff, Quantum Mechanics, 3rd ed., McGraw-Hill, New York, 1968;E. Merzbacher, Quantum Mechanics, 2nd ed, Wiley, New York, 1970;L.D. Landau and E.M. Lifshitz, Statistical Physics, Pergamon, London, 1958;E.M. Lifshitz and L.P. Pitaevski, Statistical Physics, Vol. 2, Pergamon, London, 1987.The particular citations in this chapter refer to the publications[1] For an elementary introduction see the bookH.B. Callen, Classical Thermodynamics, John Wiley and Sons, New York, 1960.
More detailsare also found later in Eqs. (4.49) and (4.50).[2] The integrability conditions are named after the mathematician of complex analysis H.A.Schwarz, a student of K. Weierstrass, who taught at the Humboldt-University of Berlin from1892–1921.[3] L. Schwartz, Théorie des distributions, Vols.I-II, Hermann & Cie, Paris, 1950-51;I.M. Gelfand and G.E.
Shilov, Generalized functions, Vols.I-II, Academic Press, New YorkLondon, 1964-68.[4] An exception occurs in the theory of Bose-Einstein condensation where the single statep = 0 requires a separate treatment since it collects a large number of particles in what iscalled a Bose-Einstein condensate.
See p. 169 in the above-cited textbook by L.D. Landauand E.M. Lifshitz on Statistical Mechanics. Bose-Einstein condensation will be discussed inSections 7.2.1 and 7.2.3.[5] This was first observed byB. Podolsky, Phys. Rev. 32, 812 (1928).[6] B.S. DeWitt, Rev. Mod. Phys. 29, 377 (1957);K.S. Cheng, J.Math. Phys.
13, 1723 (1972);H. Kamo and T. Kawai, Prog. Theor. Phys. 50, 680, (1973);T. Kawai, Found. Phys. 5, 143 (1975);H. Dekker, Physica A 103, 586 (1980);G.M. Gavazzi, Nuovo Cimento 101A, 241 (1981).See also the alternative approach byN.M.J. Woodhouse, Geometric Quantization, Oxford University Press, Oxford, 1992.[7] C. van Winter, Physica 20, 274 (1954).[8] For detailed properties of the representation matrices of the rotation group, seeA.R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press,1960.H. Kleinert, PATH INTEGRALSNotes and References[9] L.D.
Landau and E.M. Lifshitz, Quantum Mechanics, Pergamon, London, 1965.89H. Kleinert, PATH INTEGRALSNovember 20, 2006 ( /home/kleinert/kleinert/books/pathis3/pthic2.tex)A dancing shape, an image gay,To haunt, to startle, and waylay.John Milton, Phantom of Delight (1804)2Path Integrals — Elementary Properties andSimple SolutionsThe operator formalism of quantum mechanics and quantum statistics may notalways lead to the most transparent understanding of quantum phenomena.
Thereexists another, equivalent formalism in which operators are avoided by the use ofinfinite products of integrals, called path integrals. In contrast to the Schrödingerequation, which is a differential equation determining the properties of a state at atime from their knowledge at an infinitesimally earlier time, path integrals yield thequantum-mechanical amplitudes in a global approach involving the properties of asystem at all times.2.1Path Integral Representation of Time EvolutionAmplitudesThe path integral approach to quantum mechanics was developed by Feynman1 in1942.
In its original form, it applies to a point particle moving in a Cartesian coordinate system and yields the transition amplitudes of the time evolution operatorbetween the localized states of the particle (recall Section 1.7)(xb tb |xa ta ) = hxb |Û(tb , ta )|xa i,tb > ta .(2.1)For simplicity, we shall at first assume the space to be one-dimensional. The extension to D Cartesian dimensions will be given later. The introduction of curvilinearcoordinates will require a little more work. A further generalization to spaces witha nontrivial geometry, in which curvature and torsion are present, will be describedin Chapters 10–11.2.1.1Sliced Time Evolution AmplitudeWe shall be interested mainly in the causal or retarded time evolution amplitudes[see Eq.
(1.300)]. These contain all relevant quantum-mechanical information and1For the historical development, see Notes and References at the end of this chapter.902.1 Path Integral Representation of Time Evolution Amplitudes91possess, in addition, pleasant analytic properties in the complex energy plane [seethe remarks after Eq. (1.307)]. This is why we shall always assume, from now on,the causal sequence of time arguments tb > ta .Feynman realized that due to the fundamental composition law of the time evolution operator (see Section 1.7), the amplitude (2.1) could be sliced into a largenumber, say N + 1, of time evolution operators, each acting across an infinitesimaltime slice of width ≡ tn − tn−1 = (tb − ta )/(N + 1)> 0:(xb tb |xa ta ) = hxb |Û(tb , tN )Û(tN , tN −1 ) · · · Û (tn , tn−1 ) · · · Û(t2 , t1 )Û (t1 , ta )|xa i.
(2.2)When inserting a complete set of states between each pair of Û ’s,Z∞−∞dxn |xn ihxn | = 1,n = 1, . . . , N,(2.3)the amplitude becomes a product of N-integrals(xb tb |xa ta ) =N ZYn=1∞−∞dxn NY+1n=1(xn tn |xn−1 tn−1 ),(2.4)where we have set xb ≡ xN +1 , xa ≡ x0 , tb ≡ tN +1 , ta ≡ t0 . The symbol Π[· · ·] denotesthe product of the integrals within the brackets. The integrand is the product ofthe amplitudes for the infinitesimal time intervals(xn tn |xn−1 tn−1 ) = hxn |e−iĤ(tn )/h̄ |xn−1 i,(2.5)with the Hamiltonian operatorĤ(t) ≡ H(p̂, x̂, t).(2.6)The further development becomes simplest under the assumption that the Hamiltonian has the standard form, being the sum of a kinetic and a potential energy:H(p, x, t) = T (p, t) + V (x, t).(2.7)For a sufficiently small slice thickness, the time evolution operatore−iĤ/h̄ = e−i(T̂ +V̂ )/h̄(2.8)is factorizable as a consequence of the Baker-Campbell-Hausdorff formula (to beproved in Appendix 2A)e−i(T̂ +V̂ )/h̄ = e−iV̂ /h̄ e−iT̂ /h̄ e−i2 X̂/h̄2,(2.9)where the operator X̂ has the expansion 11i[V̂ , [V̂ , T̂ ]] − [[V̂ , T̂ ], T̂ ] + O(2 ) .X̂ ≡ [V̂ , T̂ ] −2h̄ 63(2.10)922 Path Integrals — Elementary Properties and Simple SolutionsThe omitted terms of order 4 , 5 , .
. . contain higher commutators of V̂ and T̂ . If weneglect, for the moment, the X̂-term which is suppressed by a factor 2 , we calculatefor the local matrix elements of e−iĤ/h̄ the following simple expression:−iH(p̂,x̂,tn )/h̄hxn |e=Z∞−∞Z|xn−1 i ≈∞−∞dxhxn |e−iV (x̂,tn )/h̄ |xihx|e−iT (p̂,tn )/h̄ |xn−1 idxhxn |e−iV (x̂,tn )/h̄ |xiZdpn ipn (x−xn−1 )/h̄ −iT (pn ,tn )/h̄ee.2πh̄∞−∞(2.11)Evaluating the local matrix elements,hxn |e−iV (x̂,tn )/h̄ |xi = δ(xn − x)e−iV (xn ,tn )/h̄ ,(2.12)this becomeshxn |e−iH(p̂,x̂,tn )/h̄ |xn−1 i ≈Z∞−∞nodpnexp ipn (xn − xn−1 )/h̄ − i[T (pn , tn ) + V (xn , tn )]/h̄ .2πh̄(2.13)Inserting this back into (2.4), we obtain Feynman’s path integral formula, consistingof the multiple integralN ZY(xb tb |xa ta ) ≈dxn" NY+1 Z∞−∞n=1#i NdpnexpA ,2πh̄h̄N+1XNA =2.1.2−∞n=1where AN is the sum∞n=1[pn (xn − xn−1 ) − H(pn , xn , tn )].(2.14)(2.15)Zero-Hamiltonian Path IntegralNote that the path integral (2.14) with zero Hamiltonian produces the Hilbert spacestructure of the theory via a chain of scalar products:(xb tb |xa ta ) ≈N ZY∞−∞n=1dxn" NY+1 Zn=1∞−∞#dpn i PN+1 pn (xn −xn−1 )/h̄e n=1,2πh̄(2.16)which is equal to(xb tb |xa ta ) ≈N ZYn=1∞−∞dxn NY+1n=1= δ(xb − xa ),hxn |xn−1 i =N ZYn=1∞−∞dxn NY+1n=1δ(xn − xn−1 )(2.17)whose continuum limit is(xb tb |xa ta ) =ZDxZDp i R dtp(t)ẋ(t)/h̄e= hxb |xa i = δ(xb − xa ).2πh̄(2.18)H.