Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 93
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Here we shall find a simple butquite accurate approximation for B(x0 ) whose effective classical potential V eff cl (x0 )approaches the exact expression always from above.4544555.2 Local Harmonic Trial Partition Function5.2Local Harmonic Trial Partition FunctionThe desired approximation is obtained by comparing the path integral in questionwith a solvable trial path integral. The trial path integral consists of a suitable superposition of local harmonic oscillator path integrals centered at arbitrary averagepositions x0 , each with an own frequency Ω2 (x0 ).
The coefficients of the superposition and the frequencies are chosen in such a way that the effective classicalpotential of the trial system is an optimal upper bound to the true effective classical potential. In systems with a smooth or at least not too singular potential, theaccuracy of the approximation will be very good. In Section 5.13 we show how touse this approximation as a starting point of a systematic variational perturbationexpansion which permits improving the result to any desired accuracy.As a local trial action we shall take the harmonic action (3.847), which may alsobe considered as the action of a harmonic oscillator centered around some point x0 :AxΩ0=Zh̄/kB T0(x − x0 )2ẋ22.+ Ω (x0 )dτ M22#"(5.3)However, instead of using the specific frequency Ω2 (x0 ) ≡ V 00 (x0 )/M in (3.847), weshall choose Ω(x0 ) to be an as yet unknown local trial frequency.
The effective classical Boltzmann factor B(x0 ) associated with this trial action can be taken directlyfrom (3.820). We simply replace the harmonic potential Mω 2 x2 /2 in the definingexpression (3.813) by the local trial potential Ω(x0 )(x − x0 )2 /2. Then the first termin the fluctuation expansion (3.814) of the action vanishes, and we obtain, insteadof (3.820), the local Boltzmann factorBΩ (x0 ) ≡ e−Veff cl (x0 )/kB T=h̄Ω(x0 )/2kB T≡ ZΩx0 .sinh[h̄Ω(x0 )/2kB T ](5.4)The exponential exp (−βMω 2 x20 /2) in (3.820) is absent since the local trial potentialvanishes at x = x0 . The local Boltzmann factor BΩ (x0 ) is a local partition functionof paths whose temporal average is restricted to x0 , and this fact will be emphasizedby the alternative notation ZΩx0 which we shall now find convenient to use.
Theeffective classical potential of the harmonic oscillators may also be viewed as a localfree energy associated with the local partition function ZΩx0 which we defined asFΩx0 ≡ −kB T log ZΩx0 ,such that we may identifyVΩeff cl (x0 )=FΩx0(5.5))(sinh[h̄Ω(x0 )/2kB T ].= kB T logh̄Ω(x0 )/2kB T(5.6)We now define the local expectation values of an arbitrary functional F [x(τ )] withinthe harmonic path integralZΩx0 ==Zx0Dx(τ )δ̃(x̄ − x0 )e−AΩ∞Ym=1"Z/h̄#P∞imdxre− kMT[ω 2 +Ω2 (x0 )]|xm |2m dxmm=1 mBe.2πkB T /Mωm(5.7)4565 Variational Perturbation TheoryThe definition is from (5.7)hF [x(τ )]ixΩ0 ≡ [ZΩx0 ]−1Zx0Dx δ̃(x̄ − x0 )e−AΩ/h̄F [x(τ )].(5.8)The effective classical potential can then be re-expressed as a path integral−V eff cl (x0 )/kB Te= ZZ≡x0=ZDx δ̃(x̄ − x0 )e−A/h̄x0Dx δ̃(x̄ − x0 )e−AΩx0D= ZΩx0 e−(A−AΩE)/h̄ x0Ωx/h̄ −(A−AΩ0 )/h̄e.(5.9)We now take advantage of the fact that the expectation value on the right-handside possesses an easily calculable bound given by the Jensen-Peierls inequality:Dx0e−(A−AΩE)/h̄ x0Ωx0≥ e−hA/h̄−AΩx0/h̄iΩ.(5.10)This implies that the effective classical potential has an upper boundV eff cl (x0 ) ≤ FΩx0 (x0 ) + kB T hA/h̄ − AxΩ0 /h̄ixΩ0 .(5.11)The Jensen-Peierls inequality is a consequence of the convexity of the exponentialfunction: The average of two exponentials is always larger than the exponential atthe average point (see Fig.
5.1):x1 +x2e−x1 + e−x2(5.12)≥ e− 2 .2This convexity property of the exponential function can be generalized to an exponential functional. Let O[x] be an arbitrary functional in the space of paths x(τ ),andZhO[x]i ≡ Dµ[x]O[x](5.13)an expectation value in this space. The measure of integration µ[x] is supposed tobe normalized so that h1i = 1. Then (5.12) generalizes toDEe−O ≥ e−hOi .(5.14)Figure 5.1 Illustration of convexity of exponential function e−x , satisfying he−x i ≥ e−hxieverywhere.H. Kleinert, PATH INTEGRALS4575.2 Local Harmonic Trial Partition FunctionTo prove this we first observe that the inequality (5.12) remains valid if x1 , x2 arereplaced by the values of an arbitrary function O(x):O(x1 )+O(x2 )e−O(x1 ) + e−O(x2 )2.≥ e−2(5.15)This inequality is then generalized with the help of any positive measure µ(x), withRunit normalization, dµ(x) = 1, toZand further toZRdµ(x)e−O(x),RDµ[x]e−O[x],dµ(x)e−O(x) ≥ e−O[x]Dµ[x]e≥e(5.16)(5.17)Rwhere µ[x] is any positive functional measure with the normalization Dµ[x] = 1.This shows that Eq.
(5.14) is true, and hence also the Jensen-Peierls inequality(5.10).Since the kinetic energies in the two actions A and AxΩ0 in (5.11) are equal, theinequality (5.11) can also be written asVeff cl(x0 ) ≤FΩx0k T+ Bh̄Zh̄/kB T0dτ*+x 0iΩ2 (x0 )V (x(τ )) − M(x(τ ) − x0 )22h. (5.18)ΩThe local expectation value on the right-hand side is easily calculated. Recallingthe definition (3.853) we have to use the correlation functions of η(τ ) without thezero frequency in Eq.
(3.839):D0η(τ )η(τ )Ex0Ω≡a2τ τ 0 (x0 )1 cosh Ω(x0 )(|τ −τ 0 | − h̄β/2)1h̄, (5.19)−=M 2Ω(x0 )sinh(Ω(x0 )h̄β/2)h̄βΩ2"#valid for arguments τ, τ 0 ∈ [0, h̄β]. By analogy with (3.803) we may denote thisquantity by a2Ω(x0 ) (τ, τ 0 ), which we have shortened to a2τ τ 0 (x0 ), to avoid a pile-up ofindices.All subtracted correlation functions can be obtained from the functional derivatives of the local generating functionalZΩx0 [j]≡ZDx(τ )δ̃(x̄ − x0 ) e−(1/h̄)R h̄β0ẋ2 (τ )+Ω2 (x0 )[x(τ )−x0 ]2 ]−j(τ )[x(τ )−x0 ]}dτ [ M2 {,(5.20)whose explicit form was calculated in (3.844):)(1 Z h̄β Z h̄β 0x0x0dτdτ j(τ )GpΩ02 (x0 ),e(τ −τ 0 )j(τ 0 ) .ZΩ [j] = ZΩ exp2Mh̄ 00(5.21)If desired, the Green function can be continued analytically to the real-timeretarded Green function0GRΩ2 (t, t ) =DEMΘ(t − t0 ) x(t)x(t0 )h̄(5.22)4585 Variational Perturbation Theoryin a way to be explained later in Section 18.2.
Using the spectral decomposition ofthese correlation functions to be derived in Section 18.2, it is possible to show thatthe low-temperature value of Ω(x0 ) at the potential minimum gives an approximation to the energy difference between ground and first excited states. In Table 5.1we see that for the anharmonic oscillator, this approximation is quite good.As shown in Fig.
3.14, the local fluctuation square widthDη 2 (τ )Ex0ΩD= (x(τ ) − x0 )2with the explicit form (3.803),a2Ω(x0 )Ex0Ω= a2τ τ (x0 ) ≡ a2Ω(x0 ) ,(5.23)∞12 X1h̄βΩ(x0 )h̄βΩ(x0 )==coth− 1 , (5.24)222Mβ m=1 ωm + Ω (x0 )22MβΩ (x0 )"#goes to zero for high temperature likea2Ω(x0 ) −−−→ h̄2 /12MkB T.(5.25)T →∞DEThis is in contrast to the unrestricted expectation (x(τ ) − x0 )2of the harΩ(x0 )monic oscillator which includes the ωm = 0 -term in the spectral decomposition(3.800):DEkB T.(5.26)a2tot ≡ (x(τ ) − x0 )2= a2Ω(x0 ) +Ω(x0 )MΩ2 (x0 )This grows linearly with T following the equipartition theorem (3.800).As discussed in Section 3.25, this difference is essential for the reliability of aperturbation expansion of the effective classical potential. It will also be essentialfor the quality of the variational approach in this chapter.The local fluctuation square width a2Ω(x0 ) measures the importance of quantumfluctuations at nonzero temperatures.
These decrease with increasing temperatures.In contrast, the square width of the ω0 = 0 -term grows with the temperatureshowing the growing importance of classical fluctuations.This behavior of the fluctuation width is in accordance with our previous observation after Eq. (3.837) on the finite width of the Boltzmann factor B(x0 ) =eff cle−V (x0 )/kB T for low temperatures in comparison to the diverging width of thealternative Boltzmann factor (3.838) formed from the partition function densityeff clB̃(x0 ) ≡ le (h̄β) z(x) = e−Ṽ (x0 )/kB T .Since a2Ω(x0 ) is finite at all temperatures, the quantum fluctuations can be treatedapproximately.
The approximation improves with growing temperatures wherea2Ω(x0 ) tends to zero. The thermal fluctuations, on the other hand, diverge at hightemperatures. Their evaluation requires a numeric integration over x0 in the finaleffective classical partition function (3.808).Having determined a2Ω(x0 ) , the calculation of the local expectation valuehV (x(τ ))ixΩ0 is quite easy following the steps in Subsection 3.25.8. The result isthe smearing formula analogous to (3.872): We write V (x(τ )) as a Fourier integralV (x(τ )) =Z∞−∞dk ikx(τ )eṼ (k),2π(5.27)H. Kleinert, PATH INTEGRALS4595.2 Local Harmonic Trial Partition Functionand obtain with the help of Wick’s rule (3.304) the expectation valuehV (x(τ ))ixΩ0 = Va2 (x0 ) ≡Z∞−∞22dkṼ (k)eikx0 −a (x0 )k /2 .2π(5.28)For brevity, we have used the shorter notation a2 (x0 ) for a2Ω(x0 ) , and shall do so inthe remainder of this chapter.
Reinsert the Fourier coefficients of the potentialṼ (k) =Z∞−∞dx V (x)e−ikx ,(5.29)we may perform the integral over k and obtain the convolution integralhV (x(τ ))ixΩ0 = Va2 (x0 ) ≡Zdx00∞−∞q202 /2a2 (xe−(x0 −x0 )0)2πa (x0 )V (x00 ).(5.30)As in (3.872), the convolution integral smears the original potential V (x0 ) out overa length scale a(x0 ), thus accounting for the effects of quantum-statistical pathfluctuations.Ex0Din Eq.
(5.26) is, of course, a special caseThe expectation value (x(τ ) − x0 )2Ωof this general smearing rule:(x − x0 )2a2 =Z∞−∞dx0 −(1/2a2 )(x0 −x0 )2 02√e(x − x0 ) = a2 (x0 ).22πa(5.31)Hence we obtain for the effective classical potential the approximationW1Ω (x0 ) ≡ FΩx0 + Va2 (x0 ) −M 2Ω (x0 )a2 (x0 ),2(5.32)which by the Jensen-Peierls inequality lies always above the true result:W1Ω (x0 ) ≥ V eff cl (x0 ).(5.33)A minimization of W1Ω (x0 ) in Ω(x0 ) produces an optimal variational approximationto be denoted by W1 (x0 ).For the harmonic potential V (x) = Mω 2 x2 /2, the smearing process leads toVa2 (x0 ) = Mω 2 (x20 + a2 )/2.
The extremum of W1Ω (x0 ) lies at Ω(x0 ) ≡ ω, so that theoptimal upper bound isx2W1 (x0 ) = Fωx0 + Mω 2 0 .(5.34)2Thus, for the harmonic oscillator, W1Ω (x0 ) happens to coincide with the exact effective classical potential Vωeff cl (x0 ) found in (3.827).4605.35 Variational Perturbation TheoryThe Optimal Upper BoundWe now determine the frequency Ω(x0 ) of the local trial oscillator which optimizesthe upper bound in Eq. (5.33). The derivative of W1Ω (x0 ) with respect to Ω2 (x0 )has two terms:∂W1Ω (x0 )∂W1Ω (x0 ) dW1Ω (x0 )∂a2 (x0 )=+.dΩ2 (x0 )∂Ω2 (x0 )∂a2 (x0 ) Ω(x0 ) ∂Ω2 (x0 )The first term is"(!#)∂W1 (x0 )h̄Ωh̄ΩkB TM=coth− 1 − a2 (x0 ) .2 MΩ2 (x0 ) 2kB T2kB T∂Ω2 (x0 )(5.35)It vanishes automatically due to (5.24).
Thus we only have to minimize W1Ω (x0 )with respect to a2 (x0 ) by satisfying the condition∂W1Ω (x0 )= 0.∂a2 (x0 )(5.36)Inserting (5.32), this determines the trial frequencyΩ2 (x0 ) =2 ∂Va2 (x0 ).M ∂a2 (x0 )(5.37)In the Fourier integral (5.28) for Va2 (x0 ), the derivative 2(∂/∂a2 )Va2 is representedby a factor −k 2 which, in turn, is equivalent to ∂/∂x20 . This leads to the alternativeequation:"#1 ∂22Ω (x0 ) =V 2 (x ).(5.38)M ∂x20 a 0 a2 =a2 (x0 )Note that the partial derivatives must be taken at fixed a2 which is to be set equalto a2 (x0 ) at the end.The potential W1Ω (x0 ) with the extremal Ω2 (x0 ) and the associated a2 (x0 ) of(5.24) constitutes the Feynman-Kleinert approximation W1 (x0 ) to the effective classical potential V eff cl (x0 ).It is worth noting that due to the vanishing of the partial derivative∂W1Ω (x0 )/∂Ω2 (x0 ) in (5.35) we may consider Ω2 (x0 ) and a2 (x0 ) as arbitrary variational parameters in the expression (5.32) for W1Ω (x0 ).
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