Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 19
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The cut will mostly beomitted, for brevity.1.12Free-Particle AmplitudesFor a free particle with a Hamiltonian operator Ĥ = p̂2 /2M, the spectrum is continuous. The eigenfunctions are (1.189) with energies E(p) = p2 /2M. Insertingthe completeness relation (1.187) into Eq. (1.297), we obtain for the time evolutionamplitude of a free particle the Fourier representation(xb tb |xa ta ) =Zip2dD pp(x−x)−(t − ta )expbah̄2M b(2πh̄)D("#).(1.330)The momentum integrals can easily be done. First we perform a quadratic completion in the exponent and rewrite it as!2i M (xb − xa )2.(tb − ta ) +h̄ 2 tb − ta(1.331)0Then we replace the integration variables by the shifted momenta p = p −(xb − xa )/(tb − ta )M , and the amplitude (1.330) becomesih̄ 2ih̄ip(xb − xa ) −p (tb − ta ) =2M2M1 xb − xap−M tb − tai M (xb − xa )2(xb tb |xa ta ) = F (tb − ta ) exp,h̄ 2 tb − ta"#(1.332)H.
Kleinert, PATH INTEGRALS491.12 Free-Particle Amplitudeswhere F (tb − ta ) is the integral over the shifted momentaF (tb − ta ) ≡Zi p0 2d D p0exp−(t − ta ) .h̄ 2M b(2πh̄)D()(1.333)This can be performed using the Fresnel integral formula( √Z ∞dp1a 2a > 0,√i,√ exp i p = q(1.334)a < 0.2−∞2π|a| 1/ i,√Here the square root i denotes the phase factor eiπ/4 : This follows from the GaussformulaZ ∞1dpα√ exp − p2 = √ ,Re α > 0,(1.335)2α−∞2πby continuing α analytically from positive values into the right complex half-plane.As long as Re α > 0, this is straightforward.
On the boundaries, i.e., on the positiveand negative imaginary axes, one has to be careful. At α = ±ia + η with a >0 and<infinitesimal η > 0, the integral is certainly convergent yielding (1.334). But theintegral also converges for η = 0, as can easily be seen by substituting x2 = z. SeeAppendix 1B.Note that differentiation of Eq.
(1.335) with respect to α yields the more generalGaussian integral formulaZ∞−∞dpα1 (2n − 1)!!√ p2n exp − p2 = √2ααn2πRe α > 0,(1.336)where (2n − 1)!! is defined as the product (2n − 1) · (2n − 3) · · · 1. For odd powersp2n+1 , the integral vanishes. In the Fresnel formula (1.334), an extra integrand p2nproduces a factor (i/a)n .Since the Fresnel formula is a special analytically continued case of the Gauss formula, we shall in the sequel always speak of Gaussian integrations and use Fresnel’sname only if the imaginary nature of the quadratic exponent is to be emphasized.Applying this formula to (1.333), we obtain1F (tb − ta ) = qD,2πih̄(tb − ta )/M(1.337)so that the full time evolution amplitude of a free massive point particle is1(xb tb |xa ta ) = qD2πih̄(tb − ta )/Mi M (xb − xa )2.exph̄ 2 tb − ta#"(1.338)In the limit tb → ta , the left-hand side becomes the scalar product hxb |xa i =δ (D) (xb − xa ), implying the following limiting formula for the δ-functionδ(D)i M (xb − xa )2(xb − xa ) = lim q.expDtb −ta →0h̄2t−tba2πih̄(tb − ta )/M1"#(1.339)501 FundamentalsInserting Eq.
(1.330) into (1.314), we have for the fixed-energy amplitude theintegral representationdD pp2id(tb − ta )(xb |xa )E =p(xb − xa ) + (tb − ta ) E −.exph̄2M0(2πh̄)D(1.340)Performing the time integration yields∞Z(ZZ(xb |xa )E ="!#)ih̄dD p,2D exp [ip(xb − xa )]E − p /2M + iη(2πh̄)(1.341)where we have inserted a damping factor e−η(tb −ta ) into the integral to ensure convergence at large tb − ta . For a more explicit result it is more convenient to calculatethe Fourier transform (1.338):(xb |xa )E =Z1∞0d(tb − ta ) qD2πih̄(tb − ta )/MiM (xb −xa )2E(tb − ta ) +exph̄2 tb − ta"(#).(1.342)For E < 0, we setκ≡and using the formula10Z0∞ν−1 −iγt+iβ/tdtteq−2ME/h̄2 ,β=2γ!ν/2(1.343)qe−iνπ/2 K−ν (2 βγ),(1.344)where Kν (z) = K−ν (z) is the modified Bessel function, we find2M κD−2 KD/2−1 (κR)(xb |xa )E = −i,h̄ (2π)D/2 (κR)D/2−1(1.345)where R ≡ |xb − xa |. The simplest modified Bessel function is11K1/2 (z) = K−1/2 (z) =rπ −ze ,2z(1.346)so that we find for D = 1, 2, 3, the amplitudes−iM 1 −κRe,h̄ κ−iM1K (κR),h̄ π 0−iM 1 −κRe.h̄ 2πR(1.347)10I.S.
Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press,New York, 1980, Formulas 3.471.10, 3.471.11, and 8.432.611M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965,Formula 10.2.17.H. Kleinert, PATH INTEGRALS511.12 Free-Particle AmplitudesAt R = 0, the amplitude (1.345) is finite for all D ≤ 2, where we can usesmall-argument behavior of the associated Bessel function12 −νz1Kν (z) = K−ν (z) ≈ Γ(ν)22for Re ν > 0,(1.348)to obtain2M κD−2Γ(1 − D/2).(x|x)E = −ih̄ (4π)D/2(1.349)This result can be continued analytically to D > 2, which will be needed later (forexample in Subsection 4.9.4).For E > 0 we setq(1.350)k ≡ 2ME/h̄2and use the formula13Z∞0ν−1 iγt+iβ/tdtteβ= iπγ!ν/2(1)qe−iνπ/2 H−ν (2 βγ),(1.351)where Hν(1) (z) is the Hankel function, to findMπ k D−2 HD/2−1 (kR).(xb |xa )E =h̄ (2π)D/2 (kR)D/2−1(1.352)The relation14π iνπ/2 (1)ieHν (z)(1.353)2connects the two formulas with each other when continuing the energy from negativeto positive values, which replaces κ by e−iπ/2 k = −ik.For large distances, the asymptotic behavior15Kν (−iz) =Kν (z) ≈rπ −ze ,2zHν(1) (z) ≈s2 i(z−νπ/2−π/4)eπz(1.354)shows that the fixed-energy amplitude behaves for E < 0 like(xb |xa )E ≈ −iM D−211κe−κR/h̄ ,(D−1)/2(D−1)/2h̄(2π)(κR)(1.355)and for E > 0 like(xb |xa )E ≈11M D−2eikR/h̄ .k(D−1)/2(D−1)/2h̄(2πi)(kR)For D = 1 and 3, these asymptotic expressions hold for all R.12ibid.,ibid.,14ibid.,15ibid.,13Formula 9.6.9.Formulas 3.471.11 and 8.421.7.Formula 8.407.1.Formulas 8.451.6 and 8.451.3.(1.356)521.131 FundamentalsQuantum Mechanics of General Lagrangian SystemsAn extension of the quantum-mechanical formalism to systems described by a setof completely general Lagrange coordinates q1 , .
. . , qN is not straightforward. Onlyin the special case of qi (i = 1, . . . , N) being merely a curvilinear reparametrizationof a D-dimensional Euclidean space are the above correspondence rules sufficientto quantize the system. Then N = D and a variable change from xi to qj in theSchrödinger equation leads to the correct quantum mechanics. It will be usefulto label the curvilinear coordinates by Greek superscripts and write q µ instead ofqj . This will help writing all ensuing equations in a form which is manifestly covariant under coordinate transformations. In the original definition of generalizedcoordinates in Eq. (1.1), this was unnecessary since transformation properties wereignored.
For the Cartesian coordinates we shall use Latin indices alternatively assub- or superscripts. The coordinate transformation xi = xi (q µ ) implies the relationbetween the derivatives ∂µ ≡ ∂/∂q µ and ∂i ≡ ∂/∂xi :∂µ = ei µ (q)∂i ,(1.357)ei µ (q) ≡ ∂µ xi (q)(1.358)with the transformation matrixcalled basis D-ad (in 3 dimensions triad, in 4 dimensions tetrad, etc.). Letei µ (q) = ∂q µ /∂xi be the inverse matrix (assuming it exists) called the reciprocalD-ad , satisfying with ei µ the orthogonality and completeness relationsei µ ei ν = δµ ν ,ei µ ej µ = δ i j .(1.359)Then, (1.357) is inverted to∂i = ei µ (q)∂µ(1.360)and yields the curvilinear transform of the Cartesian quantum-mechanical momentum operatorsp̂i = −ih̄∂i = −ih̄ei µ (q)∂µ .(1.361)The free-particle Hamiltonian operatorĤ0 = T̂ =1 2h̄2 2p̂ = −∇2M2M(1.362)goes over intoh̄2∆,2Mwhere ∆ is the Laplacian expressed in curvilinear coordinates:Ĥ0 = −(1.363)∆ = ∂i2 = eiµ ∂µ ei ν ∂ν= eiµ ei ν ∂µ ∂ν + (eiµ ∂µ ei ν )∂ν .(1.364)H.
Kleinert, PATH INTEGRALS1.13 Quantum Mechanics of General Lagrangian Systems53At this point one introduces the metric tensorgµν (q) ≡ eiµ (q)ei ν (q),(1.365)g µν (q) = eiµ (q)ei ν (q),(1.366)its inversedefined by g µν gνλ = δ µ λ , and the so-called affine connectionΓµν λ (q) = −ei ν (q)∂µ ei λ (q) = ei λ (q)∂µ ei ν (q).(1.367)Then the Laplacian takes the form∆ = g µν (q)∂µ ∂ν − Γµ µν (q)∂ν ,(1.368)with Γµ λν being defined as the contractionΓµ λν ≡ g λκ Γµκ ν .(1.369)The reason why (1.365) is called a metric tensor is obvious: An infinitesimal squaredistance between two points in the original Cartesian coordinatesds2 ≡ dx2(1.370)becomes in curvilinear coordinatesds2 =∂x ∂x µ νdq dq = gµν (q)dq µ dq ν .∂q µ ∂q ν(1.371)The infinitesimal volume element dD x is given bydD x =√g dD q,(1.372)whereg(q) ≡ det (gµν (q))(1.373)is the determinant of the metric tensor.
Using this determinant, we form the quantity1Γµ ≡ g −1/2 (∂µ g 1/2 ) = g λκ (∂µ gλκ )2(1.374)and see that it is equal to the once-contracted connectionΓµ = Γµλ λ .(1.375)With the inverse metric (1.366) we have furthermoreΓµ µν = −∂µ g µν − Γµ νµ .(1.376)541 FundamentalsWe now take advantage of the fact that the derivatives ∂µ , ∂ν applied to the coordinate transformation xi (q) commute causing Γµν λ to be symmetric in µν, i.e., Γµν λ= Γνµ λ and hence Γµ νµ = Γν .
Together with (1.374) we find the rotation1√Γµ µν = − √ (∂µ g µν g),g(1.377)which allows the Laplace operator ∆ to be rewritten in the more compact form1√∆ = √ ∂µ g µν g∂ν .g(1.378)This expression is called the Laplace-Beltrami operator .16Thus we have shown that for a Hamiltonian in a Euclidean spaceH(p̂, x) =1 2p̂ + V (x),2M(1.379)the Schrödinger equation in curvilinear coordinates becomesh̄2∆ + V (q) ψ(q, t) = ih̄∂t ψ(q, t),Ĥψ(q, t) ≡ −2M#"(1.380)where V (q) is short for V (x(q)). The scalar product of two wave functionsdD xψ2∗ (x, t)ψ1 (x, t), which determines the transition amplitudes of the system,transforms intoZ√dD q g ψ2∗ (q, t)ψ1 (q, t).(1.381)RIt is important to realize that this Schrödinger equation would not be obtainedby a straightforward application of the canonical formalism to the coordinatetransformed version of the Cartesian LagrangianL(x, ẋ) =M 2ẋ − V (x).2(1.382)With the velocities transforming asẋi = ei µ (q)q̇ µ ,(1.383)the Lagrangian becomesL(q, q̇) =Mg (q)q̇ µ q̇ ν − V (q).2 µν(1.384)Up to a factor M, the metric is equal to the Hessian metric of the system, whichdepends here only on q µ [recall (1.12)]:Hµν (q) = Mgµν (q).16(1.385)More details will be given later in Eqs.
(11.12)–(11.18).H. Kleinert, PATH INTEGRALS551.13 Quantum Mechanics of General Lagrangian SystemsThe canonical momenta arepµ ≡∂Lνµ = Mgµν q̇ .∂ q̇(1.386)The associated quantum-mechanical momentum operators p̂µ have to be Hermitianin the scalar product (1.381) and must satisfy the canonical commutation rules(1.269):[p̂µ , q̂ ν ] = −ih̄δµ ν ,[q̂ µ , q̂ ν ] = 0,[p̂µ , p̂ν ] = 0.(1.387)An obvious solution isp̂µ = −ih̄g −1/4 ∂µ g 1/4 ,q̂ µ = q µ .(1.388)The commutation rules are true for −ih̄g −z ∂µ g z with any power z, but only z = 1/4produces a Hermitian momentum operator:Z3√dq gΨ∗2 (q, t)[−ih̄g −1/4 ∂µ g 1/4 Ψ1 (q, t)]=Z=Zd3 q g 1/4 Ψ∗2 (q, t)[−ih̄∂µ g 1/4 Ψ1 (q, t)]√d3 q g [−ih̄g −1/4 ∂µ g 1/4 Ψ2 (q, t)]∗ Ψ1 (q, t),(1.389)as is easily verified by partial integration.In terms of the quantity (1.374), this can also be rewritten asp̂µ = −ih̄(∂µ + 12 Γµ ).(1.390)Consider now the classical Hamiltonian associated with the Lagrangian (1.384),which by (1.386) is simplyH = pµ q̇ µ − L =1g (q)pµ pν + V (q).2M µν(1.391)When trying to turn this expression into a Hamiltonian operator, we encounter theoperator-ordering problem discussed in connection with Eq.
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