Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 76
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Kleinert, PATH INTEGRALSAppendix 3C Recursion Relations for Perturbation Coefficients357The general form of the coefficients is, now in natural units with h̄ = 1, M = 1,:c(k)m=bk/2cXλ=0k =bk/2cXλ=0v3k−2λ v4λω(k)c ,5k/2−m/2−2λ m,λv3k−2λ v4λω 5k/2−1−2λwith(k)cm,λ k,≡ 0 for m > k + 2, or λ >2k,λ .(3C.48)(3C.49)This leads to the recursion relations(k)cm,λ =k−1 m+1 λ(m+2)(m+1) (k)1 XX X(k−l)(l)n(m + 2 − n)cn,λ−λ0 cm+2−n,λ0 ,cm+2,λ +2m2mn=1 0l=1(3C.50)λ =0(k)with cm,λ ≡ 0 for m > k + 2 or λ > bk/2c. The starting values follow by comparing (3C.40) and(3C.41) with (3C.48):1(1)(1)(1)c1,0 = −1, c2,0 = 0, c3,0 = − ,373(2)(2)(2)(2)c1,0 = 0,c1,1 = 0, c2,0 = ,c2,1 = − ,8411(2)(2)(2)(2)c3,0 = 0,c3,1 = 0, c4,0 = ,c4,1 = − .84(3C.51)(3C.52)The expansion coefficients k,λ for the energy corrections k are obtained by inserting (3C.49) and(3C.48) into (3C.44) and going to natural units:(k)k,λ = −c2,λ −k−1 λ1 X X (l)(k−l)c1,λ−λ0 c1,λ0 .20(3C.53)l=1 λ =0Table 3.1 shows the nonzero even energy corrections k up to the tenth order.3C.4Ground-State Energy with External CurrentIn the presence of a constant external current j, the time-independent Schrödinger readsh̄2 00−ψ (x) +2MM 2 2324ω x + gv3 x + g v4 x − jx ψ(x) = Eψ(x).2(3C.54)For zero coupling constant g = 0, we may simply introduce the new variables x0 and E 0 :x0 = x −jj2and E 0 = E +,2Mω2M ω 2(3C.55)and the system becomes a harmonic oscillator in x0 with energy E 0 = h̄ω/2.
Thus we make theansatz for the wave functionψ(x) ∝ eφ(x)∞jMω 2 X k, with φ(x) =x−x +g φk (x),h̄ω2h̄(3C.56)k=1and for the energyE(j) =∞Xj2h̄ω−+g k k .222M ωk=1(3C.57)3583 External Sources, Correlations, and Perturbation Theorykk2−11v32 + 6v4 ω 28ω 44−465v34 − 684v32 v4 ω 2 + 84v42 ω 432ω 9−39709v36 + 91014v34v4 ω 2 − 47308v32v42 ω 4 + 2664v43 ω 6128ω 1468−3(6416935v38 − 19945048v36v4 ω 2 + 18373480v34v42 ω 4+4962400v32v43 ω 6 164720v44ω 8 )/(2048ω 19)10(−2944491879v310 + 11565716526v38v4 ω 2 − 15341262168v36v42 ω 4+7905514480v34v43 ω 6 − 1320414512v32v44 ω 8 + 29335392v45ω 10 )/(8192ω 24)Table 3.1 Expansion coefficients for the ground-state energy of the anharmonic oscillator(3C.31) up to the 10th order.The equations (3C.38) become now−k−1h̄2 X 0h̄2 00jh̄φk (x)−φk−l (x)φ0l (x) + h̄ωx −φ0k (x) + δk,1 v3 x3 + δk,2 v4 x4 = k .2M2MMωl=1(3C.58)The results are now for k = 1(1)c1 = −v3j 2 v3jv3v33h̄jv3j 3 v3(1)(1)−,c=−,c=−,=+,123M ω2M 2 h̄ω 52M h̄ω 33h̄ω2M ω 3 M 3 ω 6(3C.59)and for k = 2:4j 3 v325jv4j 3 v417jv32+−−,4M 3 ω 6M 4 h̄ω 92M 2 ω 4M 3 h̄ω 722 227v33j v33v4j v4(2)c2 =+−−,243728M ω2M h̄ω4M ω2M 2 h̄ω 5jv32jv4v32v4(2)(2)c3 =−, c4 =−,2532M h̄ω3M h̄ω8M h̄ω 34h̄ω11h̄2 v3227h̄j 2 v329j 4 v323h̄2 v43h̄j 2 v4j 4 v42 = −−−+++.8M 3 ω 44M 4 ω 72M 5 ω 104M 2 ω 2M 3ω5M 4ω8(2)c1=(3C.60)The recursive equations (3C.43) and (3C.44) becomec(k)m=k−1X m+1X(m + 2)(m + 1)h̄ (k)h̄(k−l)n(m + 2 − n)c(l)cm+2 +n cm+2−n2mM ω2mM ωn=1l=1j(m + 1) (k)+c,M mω 2 m+1(3C.61)H.
Kleinert, PATH INTEGRALS359Appendix 3C Recursion Relations for Perturbation Coefficientsk=−k−1h̄2 X (l) (k−l)jh̄ (k) h̄2 (k)c1 −c2 −c1 c1.MωM2M(3C.62)l=1Table 3.2 shows the energy corrections k in the presence of an external current up to the sixthorder using natural units, h̄ = 1, M = 1.kk1v3 j(2j 2 + 3ω 3 )2ω 622v4 ω 2 (4j 4 + 12j 2 ω 3 + 3ω 6 ) − v32 (36j 4 + 54j 2 ω 3 + 11ω 6)8ω 103v3 j[3v32 (36j 4 + 63j 2 ω 3 + 22ω 6 ) − 2v4 ω 2 (24j 4 + 66j 2 ω 3 + 31ω 6 )]4ω 144[36v32 v4 ω 2 (112j 6 + 324j 4ω 3 + 212j 2ω 6 + 19ω 9 )−4v42 ω 4 (64j 6 + 264j 4 ω 3 + 248j 2 ω 6 + 21ω 9 )4−3v3 (2016j 6 + 4158j 4ω 3 + 2112j 2 ω 6 + 155ω 9)]/(32ω 10 )5v3 j[27v34 (1728j 6 + 4158j 4ω 3 + 2816j 2ω 6 + 465ω 9)+4v42 ω 4 (1536j 6 + 6408j 4ω 3 + 7072j 2ω 6 + 1683ω 9)2−12v3 v4 ω 2 (3456j 6 + 10908j 4ω 3 + 9176j 2ω 6 + 1817ω 9)]/(32ω 22 )6[8v43 ω 6 (1536j 8 + 8544j 6 ω 3 + 14144j 4ω 6 + 6732j 2ω 9 + 333ω 12)2 2 4−4v3 v4 ω (103680j 8 + 454032j 6ω 3 + 584928j 4ω 6 + 221706j 2ω 9 + 11827ω 12)+6v34 v4 ω 2 (285120j 8 + 991224j 6ω 3 + 1024224j 4ω 6 + 323544j 2ω 9 + 15169ω 12)6−v3 (1539648j 8 + 4266108j 6ω 3 + 3649536j 4ω 6 + 979290j 2ω 9 + 39709ω 12)]/(128ω 26 )Table 3.2 Expansion coefficients for the ground-state energy of the anharmonic oscillator(3C.31) in the presence of an external current up to the 6th order.3C.5Recursion Relation for Effective PotentialIt is possible to derive a recursion relation directly for the zero-temperature effective potential(5.259).
To this we observe that according to Eq. (3.771), the fluctuating part of the effectivepotential is given by the Euclidean path integralZofl1nfle−βVeff (X) = Dδx exp − A [X +δx]−A [X] − AX [X]δx−Veff(X)δx.(3C.63)Xh̄This can be rewritten as [recall (3.768)]IZ1 h̄β−β[Veff (X)−V (X)]e=Dδx exp −dτh̄ 0M 20×δ ẋ (τ ) + V (X + δx) − V (X) − Veff (X)δx .2(3C.64)3603 External Sources, Correlations, and Perturbation TheoryGoing back to the integration variable x = X + δx, and taking all terms depending only on X tothe left-hand side, this becomesIZ01 h̄βM 20e−β[Veff (X)−Veff (X)X] = Dx exp −dτẋ (τ ) + V (x) − Veff(X)x .(3C.65)h̄ 02(0)In the limit of zero temperature, the right-hand side is equal to e−βE (X) , where E (0) (X) is theground state of the Schrödinger equation associated with the path integral.
Hence we obtain−h̄2 0000ψ (x) + [V (x) − Veff(X)x]ψ(x) = [Veff (X) − Veff(X)X]ψ(x).2MFor the mixed interaction of the previous subsection, this readsM 2 2h̄2 000ψ (x) +ω x + gv3 x3 + g 2 v4 x4 − Veff(X)x ψ(x)−2M20= [Veff (X) − Veff(X)X] ψ(x),(3C.66)(3C.67)and may be solved recursively. We expand the effective potential in powers of is expanded in thecoupling constant g:∞XVeff (X) =g k Vk (X) ,(3C.68)(k) mCmX .(3C.69)k=0and assume Vk (X) to be a polynomial in X:Vk (X) =k+2Xm=000Comparison with Eq.
(3C.54) shows that we may set j = Veff(X) and calculate Veff (X) − Veff(X)Xby analogy to the energy in (3C.57). Inserting the ansatz (3C.68), (3C.69) into (3C.67) we find allequations for Vk (X) by comparing coefficients of g k and X m . It turns out that for even or odd k,also Vk (X) is even or odd in X, respectively. Table 3.3 shows the first six orders of the effectivepotential, which have been obtained in this way.The equations for Vk (X) are obtained as follows. We insert into (3C.67) the ansatz for thewave function ψ(x) ∝ eφ(x) with∞M ωXMω 2 X kφ(x) =x−x +g φk (x),h̄2h̄(3C.70)k=1and expandVeff (X) =∞h̄ω M 2 2 X k+ω X +g Vk (X),22(3C.71)k=1to obtain the set of equations−k−1h̄2 00h̄2 X 0φk (x) −φk−l (x)φ0l (x) + h̄ω(x−X)φ0k (x)−xVk0 (X) + δk,1 v3 x3 + δk,2 v4 x42M2Ml=1= Vk (X) − Vk0 (X)X.(3C.72)From these we find for k = 1:(1)c1 =2v3 X 2 (1)v3 X (1)v33v3 h̄v3+, c2 = −, c =−, V1 (X) =+ v3 X 3 ,22M ωh̄ω2h̄ω 33h̄ω2M ω(3C.73)H.
Kleinert, PATH INTEGRALSAppendix 3C Recursion Relations for Perturbation CoefficientskVk (X)0ω ω2 2+X22v3 X 3 +13613v3X2ωv32 (1 + 9ωX 2 ) + v4 ω 2 (3 + 12ωX 2)4ω 42v4 X 4 −3v3 X[3v32 (4 + 9ωX 2) − 2v4 ω 2 (13 + 18ωX 2)]4ω 64−[4v42 ω 4 (21 + 104ωX 2 + 72ω 2 X 4 ) − 12v32 v4 ω 2 (13 + 152ωX 2 + 108ω 2 X 4 )+v34 (51 + 864ωX 2 + 810ω 2X 4 )]/(32ω 9 )53v3 X[9v34 (51 + 256ωX 2 + 126ω 2X 4 ) + 4v42 ω 4 (209 + 544ωX 2 + 216ω 2 X 4 )−4v32 v4 ω 2 (341 + 1296ωX 2 + 540ω 2 X 4 )]/(32ω 11 )6[24v43 ω 6 (111 + 836ωX 2 + 1088ω 2X 4 + 288ω 3 X 6 )−36v32 v42 ω 4 (365 + 5654ωX 2 + 8448ω 2X 4 + 2160ω 3X 6 )+6v34 v4 ω 2 (2129 + 46008ωX 2 + 85248ω 2X 4 + 22680ω 3X 6 )−v36 (3331 + 90882ωX 2 + 207360ω 2X 4 + 61236ω 3X 6 )]/(128ω 14)Table 3.3 Effective potential of the anharmonic oscillator (3C.31) up to the 6th order,expanded in the coupling constant g (in natural units with h̄ = 1 and M = 1).
The lowestterms agree, of course, with the two-loop result (3.767).and for k = 2:13v32 X2v 2 X 37v4 X3v4 X 3v323v4v4 X 2(2)− 3 3 ++, c2 =−−,2422424M ωM h̄ω2M ωh̄ω8M ω4M ω2h̄ωv4 X (2)v4v32v32 X(2)−,c−,c3 ==42M h̄ω 33h̄ω8M h̄ω 34h̄ω2 22h̄ v39h̄v32 X 23h̄ v43h̄v4 X 2V2 (X) = −−+++ v4 X 4 .3423224M ω4M ω4M ωMω(2)c1= −(3C.74)(3C.75)For k ≥ 3, we must solve recursivelyc(k)m=k−1X m+1X(m + 2)(m + 1)h̄ (k)h̄(k−l)n(m + 2 − n)c(l)cm+2 +n cm+2−n2mM ω2mM ωn=1l=1(k)c1=X(m + 1) (k)+cm+1 for m ≥ 2 and with c(k)m ≡ 0 for m > k + 2,mk−1h̄ X (k−l) (l)3h̄ (k)1 0(k)(k−l) (l)c3 + 2Xc2 +V (X),c2c1 + c1c2 +MωMωh̄ω kl=1(3C.76)(3C.77)3623 External Sources, Correlations, and Perturbation Theoryk−1Vk (X) = −−h̄2 X (k−l) (l)h̄2 (k) 3h̄2(k)(k)(k−l) (l)c2 −Xc3 − 2h̄ωX 2 c2 −Xc2c1 + c1c2MMMl=12 k−1Xh̄2M(l) (k−l)c1 c1.(3C.78)l=1The results are listed in Table 3.3.Interaction r4 in D -Dimensional Radial Oscillator3C.6It is easy to generalize these relations further to find the perturbation expansions for the eigenvaluesof the radial Schrödinger equation of an anharmonic oscillator in D dimensions1 d21D−1 dl(l + D − 2) 1 2 g 4−−++ r + r Rn (r) = E (n) Rn (r).(3C.79)2 dr22 r dr2r224The case g = 0 will be solved in Section 9.2, with the energy eigenvaluesE (n) = 2n0 + l + D/2 = n + D/2,n = 0, 1, 2, 3, .
. ., l = 0, 1, 2, 3, . . . .(3C.80)For a fixed principal quantum number n = 2nr + l, the angular momentum runs through l =0, 2, . . . , n for even, and l = 1, 3, . . . , n for odd n. There are (n + 1)(n + 2)/2 degenerate levels.Removing a factor rl from Rn (r), and defining Rn (r) = rl wn (r), the Schrödinger equation becomes1 2l + D − 1 d1 2 g 41 d2wn (r) = E (n) wn (r).(3C.81)−+r+r−2 dr22rdr 24The second term modifies the differential equation (3C.6) tokX0(2l + D − 1) 01(n)Φk (r)+r4 Φk−1 (r)+(−1)k Ek0 Φk−k0 (r). (3C.82)rΦ0k (r)−2n0 Φk (r) = Φ00k (r)+22r0k =1The extra terms change the recursion relation (3C.15) intok(p − 2n0 )Apk =X01(n)p−4[(p + 2)(p + 1) + (p + 2)(2l + D−1)]Ap+2+ Ak−1+(−1)k Ek0 Apk−k0 .(3C.83)k20k =10For even n = 2n + l with l = 0, 2, 4, .
. . , n, we normalize the wave functions by settingCk0 = (2l + D)δ0k ,(3C.84)rather than (3C.12), and obtain02(p − n00)Ckp00= [(2p + 1)(p + 1) + (p00+ 1)(l + D/2 − 1/2)]Ckp +1+p0 −2Ck−1−kX0pCk10 Ck−k0 , (3C.85)k0 =1instead of (3C.20).For odd n = 2n0 + l with l = 1, 3, 5, . . . , n, the equations analogous to (3C.13) and (3C.22) areCk1 = 3(2l + D)δ0k(3C.86)and000p −22(p0 − n0 )Ckp = [(2p0 + 3)(p0 + 1) + (p0 + 3/2)(l + D/2−1/2)]Ckp +1 + Ck−1−In either case, the expansion coefficients of the energy are given by(n)Ek=−(−1)k 2l + D + 1 1Ck .22l + DkX0pCk10 Ck−k0 . (3C.87)k0 =1(3C.88)H.
Kleinert, PATH INTEGRALSAppendix 3D Feynman Integrals for T 6= 03C.7363Interaction r2q in D DimensionsA further extension of the recursion relation applies to interactions gx2q /4. Then Eqs. (3C.20) and0(3C.22) are changed in the second terms on the right-hand side which become Ckp −q . In a first0step, these equations are now solved for Ckp for 1 ≤ p ≤ qk + n, starting with p0 = qk + n0 andlowering p0 down to p0 = n0 + 1. As before, the knowledge of the coefficients Ck1 (which determinethe yet unknown energies and are contained in the last term of the recursion relations) is notrequired for p0 > n0 .
The second and third steps are completely analogous to the case q = 2.The same generalization applies to the D-dimensional case.3C.8Polynomial Interaction in D DimensionsIf the Schrödinger equation has the general form1 d21D−1 dl(l + D − 2) 1 2−−++ r22 dr2 r dr2r22ig46+ (a4 r + a6 r + . . . + a2q x2q ) Rn (r) = E (n) Rn (r),4(3C.89)we simply have to replace in the recursion relations (3C.85) and (3C.87) the second term on theright-hand side as follows0000p −2p −2p −3p −qCk−1→ a4 Ck−1+ a6 Ck−1+ .
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