Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 98
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The advantage of our variational approachis that it yields good results for all magnetic field strengths and temperatures, and4845 Variational Perturbation Theorythat it can be improved systematically by methods to be developed in Section 5.13,with rapid convergence. The figure shows also the energy of Landau and Lifshitzwhich grossly overestimates the binding energies even at very large magnetic fields,such as 2000B0 ∝ 1012 G. Obviously, the nonleading terms in Eq. (5.132) giveimportant contributions to the asymptotic behavior even at such large magneticfields.
As an peculiar property of the asymptotic behavior, the absolute value ofthe difference between the Landau-Lifshitz result and our approximation (5.132)diverges with increasing magnetic field strengths B. Only the relative differencedecreases.5.11.1Weak-Field BehaviorLet us also calculate the weak-field behavior of the variational energy (5.127). Setting η ≡ Ωk /Ω⊥ , we rewrite W1 (0) asηB2Ω⊥1+++W1 (0) =228Ω⊥s√1− 1−ηηΩ⊥1√√ln.π1−η 1+ 1−η(5.136)This is minimized in η and Ω⊥ by expanding η(B) and Ω(B) in powers of B 2 withunknown coefficients, and inserting these expansions into extremality equations.
Theexpansion coefficients are then determined order by order. The optimal expansionsare inserted into (5.136), yielding the optimized binding energy ε(1) (B) as a powerseriesW1 (0) =∞Xεn B 2n .(5.137)n=0The coefficients εn are listed in Table 5.3 and compared with the exact ones. Ofcourse, the higher-order coefficients of this first-order variational approximation become rapidly inaccurate, but the results can be improved, if desired, by going tohigher orders in variational perturbation theory of Section 5.13.Table 5.3 Perturbation coefficients up to order B 6 in weak-field expansions of variationalparameters, and binding energy in comparison to exact ones (from J.E. Avron et al. andB.G.
Adams et al. quoted in Notes and References).nηnΩnεnεexn01.032≈ 1.13189π4−≈ −0.42443π−0.5123405π 2−≈ −0.5576716899π≈ 1.38852249π≈ 0.220912816828965π 4≈ 1.302312588154881293975π 3−≈ −2.03982196689928019π 3−≈ −0.1355183500853−≈ −0.27601923886999332075π 6−≈ −4.2260884272562962432524431667187π 5≈ 5.807727633517592576256449807π 5≈ 0.24353222567649285581≈ 1.211246080.25H. Kleinert, PATH INTEGRALS5.11 Hydrogen Atom in Strong Magnetic Field5.11.2485Effective Classical HamiltonianThe quantum statistical properties of the system at an arbitrary temperature arecontained in the effective classical potential H eff cl (p0 , x0 ) defined by the threedimensional version of Eq.
(3.821):D 3 p (3)3−A[p,x]/h̄,3 δ (x0 − x)(2πh̄) δ(p0 − p) e(2πh̄)(5.138)where Ae [p, x] is the Euclidean actionB(p0 , x0 ) ≡ e−βH≡ID3xAe [p, x] =Zh̄βeff cl (p0 ,x0 )0Idτ [−ip(τ )ẋ(τ ) + H(p(τ ), x(τ ))],(5.139)and x = 0h̄β dτ x(τ )/h̄β and p = 0h̄β dτ p(τ )/h̄β are the temporal averages ofposition and momentum. Note that the deviations of p(τ ) from the average p0 sharewith x(τ )−x0 the property that the averages of the squares go to zero with increasingtemperatures like 1/T , and remains finite for T → 0. while the expectation of p2grows linearly with T (Dulong-Petit law).
For T → 0, the averages of the squares ofp(τ ) remain finite. This property is the basis for a reliable accuracy of the variationaltreatment.Thus we separate the action (5.139) (omitting the subscript e) asRRAe [p, x] = βH(p0, x0 ) + ApΩ0 ,x0 [p, x] + Aint [p, x],(5.140)p0 ,x0where AΩ[p, x] is the most general harmonic trial action containing the magneticfield. It has the form (3.825), except that we use capital frequencies to emphasizethat they are now variational parameters:1[p(τ ) − p0 ]22M0oM 2M+ΩB lz (p(τ )−p0 , x(τ )−x0 ) + Ω⊥ [x⊥ (τ )−x0⊥ ]2 + Ω2k [z(τ ) − z0 ]2 .
(5.141)22ApΩ0 ,x0 [p, x]=Zh̄βdτn− i[p(τ ) − p0 ] · ẋ(τ ) +The vector x⊥ = (x, y) is the projection of x orthogonal to B.The trial frequencies Ω = (ΩB , Ω⊥ , Ωk ) are arbitrary functions of p0 , x0 , and B.Inserting the decomposition (5.140) into (5.138), we expand the exponential of theinteraction, exp {−Aint [p, x]/h̄}, and obtain a series of expectation values of powersp0 ,x0of the interaction h Anint [p, x] iΩ, defined in general by the path integralh O[p, x] ipΩ0 ,x0=1p0 ,x0ZΩID3p3(3)D x3 O[p, x] δ (x0 − x)(2πh̄) δ(p0 − p)(2πh̄)31 p0 ,x0× exp − AΩ[p, x] ,(5.142)h̄p0 ,x0where the local partition function in phase space ZΩis the normalization factorp0 ,x0which ensures that h 1 iΩ= 1. From Eq. (3.826) we know thatp0 ,x0ZΩ≡ e−βFp0 ,x0= le3 (h̄β)h̄βΩk /2h̄βΩ+ /2h̄βΩ− /2,sinh h̄βΩ+ /2 sinh h̄βΩ− /2 sinh h̄βΩk /2(5.143)4865 Variational Perturbation Theorywhere Ω± ≡ ΩB ± Ω⊥ .
In comparison to (3.826), the classical Boltzmann factore−βH(p0 ,x0 ) is absent due to the shift of the integration variables in the action (5.141).Note that the fluctuations p(τ ) − p0 decouple from p0 just as x(τ ) − x0 decoupledfrom x0 due to the absence of zero frequencies in the fluctuations.Rewriting the perturbation series as a cumulant expansion, evaluating theexpectation values, and integrating out the momenta on the right-hand side ofEq. (5.138) leads to a series representation for the effective classical potentialVeff (x0 ).
Since it is impossible to sum up the series, the perturbation expansion(N )must be truncated, leading to an Nth-order approximation WΩ (x0 ) for the ef(N )fective classical potential. Since the parameters Ω are arbitrary, WΩ (x0 ) shoulddepend minimally on Ω. This determines the optimal values of Ω to be equal(N )(N )(N )to Ω(N ) (x0 ) = (ΩB (x0 ), Ω⊥ (x0 ), Ωk (x0 )) of Nth order.
Reinserting these into(N )(N )WΩ (x0 ) yields the optimal approximation W (N ) (x0 ) ≡ WΩ(N) (x0 ).The first-order approximation to the effective classical potential is then, withωk = 0, ω⊥ = ωB ,MΩB (x0 )[ωB − ΩB (x0 )] b2⊥ (x0 ) a2⊥ (x0 )2* 2 +p0 ,x0hiM 21 2 2e2+ωB − Ω⊥ (x0 ) − Ωk ak (x0 ) −.22|x| Ω(1)WΩ (x0 ) = FΩp0 ,x0 −(5.144)The smearing of the Coulomb potential is performed as in Section 5.10. This yieldsthe result (5.117).
with the longitudinal widtha2k (x0 )=x0G(2)(τ, τ )zzh̄βΩk (x0 )h̄βΩk (x0 )1=coth−1 ,222βMΩk (x0 )"#(5.145)and an analog transverse width.The quantity b2⊥ (x0 ) is new in this discussion based on the canonical path integral. It denotes the expectation value associated with the z-component of theangular momentum1p0 ,x0b2⊥ (x0 ) ≡,(5.146)h lz iΩMΩBwhich can also be written asb2⊥ (x0 ) =Ep0 ,x02 Dx(τ )py (τ ).ΩMΩT 1(5.147)DAccording to Eq.
(3.355), the correlation function x(τ )py (τ )Dx(τ )py (τ 0 )Ep0 ,x0Ω(2)(2)Ep0 ,x0Ωis given by= iM∂τ 0 Gω2 ,B,xx (τ, τ 0 ) − MωB Gω2 ,B,xy (τ, τ 0 ),(5.148)where the expressions on the right-hand side are those of Eqs. (3.326) and (3.328),with ω replaced by Ω.H. Kleinert, PATH INTEGRALS4875.11 Hydrogen Atom in Strong Magnetic Field 3 21 0 −10510 1520Figure 5.12 Effective classical potential of atom in strongq magnetic field plotted alongtwo directions: once as a function of the coordinate ρ0 = x20 + y02 perpendicular to thefield lines at z0 = 0 (solid curves), and once parallel to the magnetic field as a functionof z0 at ρ0 = 0 (dashed curves). The inverse temperature is fixed at β = 100, and thestrengths of the magnetic field B are varied (all in natural units).The variational energy (5.144) is minimized at each x0 , and the resultingW (N ) (x0 ) is displayed for a low temperature and different magnetic fields in Fig.
5.12.The plots show the anisotropy with respect to the magnetic field direction. Theanisotropy grows when lowering the temperature and increasing the field strength.Far away from the proton at the origin, the potential becomes isotropic, due to thedecreasing influence of the Coulomb interaction. Analytically, this is seen by goingto the limits ρ0 → ∞ or z0 → ∞, where the expectation value of the Coulombpotential tends to zero, leaving an effective classical potential(1)2WΩ (x0 ) → FΩp0 ,x0 − MΩB (ω⊥ − ΩB ) b2⊥ + M ω⊥− Ω2⊥ a2⊥ −M 2 2Ωa.2 k k(1)(5.149)(1)This is x0 -independent, and optimization yields the constants ΩB = Ω⊥ = ωB and(1)Ωk = 0, with the asymptotic energy1βh̄ωBW (1) (x0 ) → − log.βsinh βh̄ωB(5.150)The B = 0 -curves agree, of course, with those obtained from the previous variationalperturbation theory of the hydrogen atom [26].For large temperatures, the anisotropy decreases since the violent thermal fluctuations have a smaller preference of the z-direction.4885.125 Variational Perturbation TheoryVariational Approach to Excitation EnergiesAs explained in Section 5.4, the success of the above variational treatment is rootedin the fact that for smooth potentials, the ground state energy can be approximatedquite well by the optimal expectation value of the Hamiltonian operators in a Gaussian wave packet.
The question arises as to whether the energies of excited statescan also be obtained by calculating an optimized expectation value between excitedoscillator wave functions. If the potential shape has only a rough similarity withthat of a harmonic oscillator, when there are no multiple minima, the answer ispositive. Consider again the anharmonic oscillator with the actionh̄βZA[x] =0dτ1M 2ẋ + ω 2 x2 + gx4 .24(5.151)As for the ground state, we replace the action A by AxΩ0 +Axint0 with the trial oscillatoraction centered around the arbitrary point x0AxΩ0=Ω21(x − x0 )2 ,dτ M ẋ2 +22#(5.152)ω 2 − Ω2gdτ M(x − x0 )2 + x424(5.153)Zh̄β0"and a remainderAxint0=Z0β"#(n)to be treated as an interaction.
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