Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 79
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The bound-state wave functions must satisfy the boundary conditionto vanish exponentially fast at x = ±∞. Imposing these, the connection formulaslead to the semiclassical or Bohr-Sommerfeld quantization rule√Zbadx k(x) = (n + 1/2)π,−1 −Rxbdx0 κ−iπ/4n = 0, ±1, ±2, . . . .(4.25)For the harmonic oscillator, the semiclassical quantization rule (4.25) gives the exactenergy levels.
Indeed, for an energy E, the classical crossover points with V (xE ) = Eares2E,(4.26)xc = ±Mω 2to be identified in (4.25) with a and b, respectively. Inserting furtherp(x)=k(x) =h̄s2M1E − Mω 2 x2 ,22h̄(4.27)we obtain the WKB approximation for the energy levelsZxE−xEdx0 k(x) =Eπ = (n + 1/2)π,h̄ω(4.28)which indeed coincides with the exact ones. Only nonnegative values n = 0, 1, 2, . . .lead to oscillatory waves.As an example consider the quartic potential V (x) = gx4 /2 for which theSchrödinger equation cannot be solved exactly.
Inserting this into equation (4.28),we obtainZq1 xEdx 2M(E − gx4 /4) = π(n + 1/2),(4.29)h̄ −xEwith the turning points ±xE = ±(4E/g)1/4 . The integral is done using the formula3Z10dttµ−1 (1 − tλ )ν−1 =1B(µ/λ, ν)λ(4.30)2See for example E. Merzbacher, Quantum Theory, John Wiley & Sons, New York, 1970,p. 122; M.L. Goldberger and K.M. Watson, Collision Theory, John Wiley & Sons, New York,1964, p. 324. The analytical argument is given in L.D. Landau and E.M.
Lifshitz, QuantumMechanics, Pergamon, London, 1965, p. 158.3I.S. Gradshteyn and I.M. Ryzhik, op. cit., Formula 3.251.1.H. Kleinert, PATH INTEGRALS4.2 Saddle Point Approximation373for µ = 1, λ = 4 and ν = 3/2. The right-hand√side has the√value (1/4)B(1/4, 3/2)which can be written with Γ(1/4) = 2π/Γ(3/4) 2 as π 3/2 /3 2Γ2 (3/4). In this way,(4.29) leads to the energy in the Bohr-Sommerfeld approximation(n)EBS=gh̄4M 2 ω 3(n)h̄ωκBS!1/3,(4.31)with(n)κBS=1π2/3 4/332Γ8/3 (3/4) n1/3 ≈ 0.688 253 702 × 2(n + 1/2)4/3 .(4.32)whose numerical value follows by inserting Γ(3/4) ≈ 1.2254167024.
This large-nresult may be compared with the precise of κ(0) = 0.667986 . . . to be derived inSection 5.19 (see Table 5.9).4.2Saddle Point ApproximationLet us now look at the semiclassical expansion within the path integral approach toquantum mechanics. Consider the time evolution amplitude(xb tb |xa ta ) =ZDx eiA[x]/h̄ ,(4.33)imagining Planck’s constant h̄ to be again a free parameter which is very small compared to the typical fluctuations of the action. With h̄ appearing in the denominatorof an imaginary exponent we see that in the limit h̄ → 0, the path integral becomesa sum of rapidly oscillating terms which will approximately cancel each other. Thisphenomenon is known from ordinary integralsZ√dx ia(x)/h̄e,2πih̄(4.34)which converge to zero for h̄ → 0 according to the Riemann-Lebesgue lemma.
Theprecise behavior is given by the saddle point expansion of integrals which we shallfirst recapitulate.4.2.1Ordinary IntegralsThe evaluation of an integral of the type (4.34) proceeds for small h̄ via the so-calledsaddle point approximation. In the limit h̄ → 0, the integral is dominated by theextremum of the function a(x) with the smallest absolute value, call it xcl (assumingit to be unique, for simplicity), wherea0 (xcl ) = 0.(4.35)In the path integral, the point xcl in this example corresponds to the classical orbit forwhich the functional derivative vanishes. This is the reason for using the subscript3744 Semiclassical Time Evolution Amplitudecl.
For x near the extremum, the oscillations of the integrand are weakest. Theleading oscillatory behavior of the integral is given byZ∞−∞√dx ia(x)/h̄ h̄→0e−−−→ const × eia(xcl )/h̄ ,2πih̄(4.36)with a constant proportionality factor independent of h̄. This can be calculated byexpanding a(x) around its extremum as11a(x) = a(xcl ) + a00 (xcl )(δx)2 + a(3) (xcl )(δx)3 + . . . ,23!(4.37)where δx ≡ x − xcl is the deviation from xcl .
It is the analog of the quantumfluctuation introduced in Section 2.2. Due to (4.35), the linear term in δx is absent.If a00 (xcl ) 6= 0 and the higher derivatives are neglected, the integral is of the Fresneltype and can be done, yieldingZ∞−∞dx ia(x)/h̄√e→ eia(xcl )/h̄2πih̄Z∞−∞dδx ia00 (xcl )(δx)2 /2h̄eia(xcl )/h̄√.e=q2πih̄a00 (xcl )(4.38)The right-hand side is the saddle point approximation to the integral (4.34).The saddle point approximation may be viewed as the consequence of the classical limit of the exponential function:√2πih̄ia(x)/h̄ h̄→0e−−−→ qδ(x − xcl ) .(4.39)a00 (xcl )Corrections can be calculated perturbatively by expanding the integral in powersof h̄, leading to what is called the saddle point expansion. For this we expand theremaining exponent in powers of δx:i 1 (3)1a (xcl )(δx)3 + a(4) (xcl )(δx)4 + .
. .exph̄ 3!4!i 1 (3)1=1+a (xcl )(δx)3 + a(4) (xcl )(δx)4 + . . .h̄ 3!4!1 1 (3)a (xcl )2 (δx)6 + . . . + . . .− 2h̄ 72(4.40)and perform the resulting integrals of the typeZ∞−∞(n − 1)!!dδx ia00 (xcl )(δx)2 /2h̄(ih̄)n/2 , n = even,n00(1+n)/2√e(δx) =[a (xcl )]2πih̄0,n = odd.(4.41)qEach factor δx in (4.40) introduces a power h̄/a00 (xcl ). This is the average relativesize of the quantum fluctuations. The increasing powers of h̄ ensure the decreasingH. Kleinert, PATH INTEGRALS4.2 Saddle Point Approximation375importance of the higher terms for small h̄.
For instance, the fourth-order terma(4) (xcl )(δx)4 /4! is accompanied by h̄, and the lowest correction amounts to a factor1 − ia(4) (xcl )h̄3!!.004! [a (xcl )]2(4.42)The cubic term a(3) (xcl )(δx)3 /3! yields a factor1 + i[a(3) (xcl )]25!!h̄.0072 [a (xcl )]3(4.43)Thus we obtain the saddle point expansion to the integral isdx ia(x)/h̄1 a(4) (xcl )5 [a(3) (xcl )]2eia(xcl )/h̄√1 − ih̄−+ O(h̄2 ) .e=q00200300824−∞[a (xcl )][a (xcl )]2πih̄a (xcl )(4.44)Expectation values in this integral can also be expanded in powers of h̄, forinstance hxi = xcl + hδxi whereZ(∞hδxi = −ih̄2− h̄""#)1 a(3) (xcl )2 [a00 (xcl )]22 a(3) (xcl )a(4) (xcl ) 5 [a(3) (xcl )]3 1 a(5) (xcl )3−004005 −003 + O(h̄ ).38 [a (xcl )]8 [a (xcl )][a (xcl )]#(4.45)Since the saddle point expansion is organized in powers of h̄, it corresponds preciselyto the semiclassical expansion of the eikonal in the previous section.The saddle point expansion can be used for very small h̄ to calculate an integralwith increasing accuracy.
It is impossible, however, to achieve arbitrary accuracysince the resulting series is divergent for all physically interesting systems. It ismerely an asymptotic series whose usefulness decreases rapidly with an increasingsize of the expansion parameter. A variational expansion must be used to achieveconvergence. For more details, see Sections 5.15 and 17.9.An important property of the semiclassical approximation is that Fourier transformations become very simple. Consider the Fourier integralZ∞−∞dx e−ipx/h̄ eia(x)/h̄ .(4.46)For small h̄, this can be done in the saddle point approximation according to therule (4.39), and obtainZ∞−∞dx e−ipx/h̄ eia(x)/h̄ →√2πih̄ei[a(xcl )−pxcl ]/h̄qa00 (xcl ),(4.47)3764 Semiclassical Time Evolution Amplitudewhere xcl is now the extremum of the action with a source term p, i.e., it is determined by the equation p = a0 (xcl ).
Note that the formula holds also if the exponential carries an x-dependent prefactor, since the x-dependence gives only correctionsof the order of h̄ in the exponent:Z∞−∞dx e−ipx/h̄ c(x)eia(x)/h̄ →√2πih̄ c(xcl )ei[a(xcl )−pxcl ]/h̄qa00 (xcl ).(4.48)If the equation p = a0 (xcl ) is inverted to find xcl as a function xcl (p), the exponenta(xcl ) − pxcl may be considered as a function of p:b(p) = a(xcl ) − pxcl ,p = a0 (xcl ).(4.49)This function is recognized as being the Legendre transform of the function a(x)[recall (1.9)].The original function a(x) can be recovered from b(p) via an inverse Legendretransformationa(x) = b(pcl ) + xpcl ,x = −b0 (pcl ).(4.50)This formalism is the basis for many thermodynamic calculations.
For large statistical systems, fluctuations of global properties such as the volume and the totalinternal energy are very small so that the saddle point approximation is very good.In this chapter, the formalism will be applied on many occasions.4.2.2Path IntegralsA similar saddle point expansion exists for the path integral (4.33). For small h̄, theamplitudes eiA/h̄ from the various paths will mostly cancel each other by interference.The dominant contribution comes from the functional regime where the oscillationsare weakest, which is the extremum of the actionδA[x] = 0.(4.51)This gives the classical Euler-Lagrange equation of motion. For a point particle withthe actionZ tbM 2dtA[x] =ẋ − V (x) ,(4.52)2tait readsM ẍ = −V 0 (x).(4.53)Let xcl (t) denote the classical orbit.
After multiplying (4.53) by ẋ, an integration int yields the law of energy conservationE=M 2ẋ + V (xcl ) = const .2 cl(4.54)This implies that the classical momentumpcl (t) ≡ M ẋcl (t)(4.55)H. Kleinert, PATH INTEGRALS4.2 Saddle Point Approximation377can be written aspcl (t) = p(xcl (t)),(4.56)where p(x) is the local classical momentum defined in (4.3). From (4.54), the timedependence of the classical orbit xcl (t) is given byt − t0 =Z0xcldxM=p(x)Z0xcldx qM2M[E − V (x)].(4.57)When solving the integral on the right-hand side we find for a given time intervalt = tb − ta the energy for which a pair of positions xa , xb can be connected by aclassical orbit:E = E(xb , xa ; tb − ta ).(4.58)The classical action is given byZA[xcl ] =tbtatbZ=tZ axb=xaM 2dtx˙ − V (xcl )2 cldt [pcl (t)ẋcl − H(pcl , xcl )]dxp(x) − (tb − ta )E.(4.59)Just like E, the classical action is a function of xb , xa and tb − ta , to be denoted byA(xb , xa ; tb − ta ), for which (4.59) reads more explicitlyA(xb , xa ; tb − ta ) ≡Zxbxadx p(x) − (tb − ta )E(xb , xa ; tb − ta ).(4.60)Recalling (4.11), the first term on the right-hand side is seen to be the classicaleikonalZ xbS(xb , xa ; E) =dx p(x),(4.61)xawhere E is the energy function (4.58) and p(x) is given by (4.3),The eikonal may be viewed as a functional SE [x] of paths x(t) of a fixed energy,in which case it is extremal on the classical orbits.
This was observed as early as1744 by Maupertius. The proof for this is quite simple: We insert the classicalmomentum (4.3) into SE [x] and writeSE [x] ≡Zp(x)dx =Zdt p(x)ẋ =Zdt LE (x, ẋ) =Zqdt 2M[E − V (x)]ẋ, (4.62)thus introducing a Lagrangian LE (x, ẋ) for this problem.Lagrange equation readsd ∂LE∂LE=.dt ∂ ẋ∂xThe associated Euler-(4.63)3784 Semiclassical Time Evolution AmplitudeInserting LE (x, ẋ) = p(x)ẋ we find the correct equation of motion ṗ = −V 0 (x).There is an interesting geometrical aspect to this variational procedure. In orderto see this let us go to D dimensions and write the eikonal (4.62) asSE [x] =Zdt LE (x, ẋ) =Zdtqgij (x)ẋi (t)ẋj (t),(4.64)with an energy-dependent metricgij (x) = p2E (x)δij .(4.65)Then the Euler-Lagrange equations for x(t) coincides with the equation (1.72) forthe geodesics in a Riemannian space with a metric gij (x).
In this way, the dynamicalproblem has been reduced to a geometric problem. The metric gij (x) may be calleddynamical metric of the space with respect to the potential V (x). This geometricview is further enhanced by the fact that the eikonal (4.62) is, in fact, independentof the parametrization of the trajectory.
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