Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 97
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In this way the effectiveclassical potential explains the stability of matter in quantum physics, i.e., the factthat atomic electrons do not fall into the origin. The effective classical potential ofthe Coulomb system is then by the isotropic version of (5.97)|x | 3e23 sinh[h̄βΩ(x0 )/2]−erf q 0− MΩ2 (x0 )a2 (x0 ). (5.115)W1Ω (x0 ) = ln2βh̄βΩ(x0 )/2|x0 |22a (x0 )Minimizing W1 (r0 ) with respect to a2 (r0 ) gives an equation analogous to Eq. (5.37)determining the frequency Ω2 (r0 ) to be2 1122Ω2 (r0 ) = e2 √e−r0 /2a .23/23 2π (a )(5.116)We have gone to atomic units in which e = M= h̄= kB = 1, so that energies andtemperatures are measured in units of E0 = Me4 /Mh̄2 ≈ 4.36 × 10−11 erg ≈ 27.21eVand T0 = E0 /kB ≈ 31575K, respectively. Solving (5.116) together with (5.24),we find a2 (r0 ) and the approximate effective classical potential (5.32).
The resultH. Kleinert, PATH INTEGRALS4795.10 Application to Coulomb and Yukawa Potentialsis shown in Fig. 5.9 as a dashed curve. The above approximation may now beimproved by treating the fluctuations anisotropically, as described in the previoussection, with different Ω2 (x0 ) for radial and tangential fluctuations δxL and δxT ,and the effective potential following Eqs. (5.97) and (5.98). For the anisotropicallysmeared Coulomb potential we calculate from (5.108):s1 Z1e2r02 u2hi .du qVa2L ,a2T (r0 ) = −exp − 2 a2 u2 + a2 (1 − u2 ) 2π −1a2 u2 + a2 (1 − u2 )LTLT(5.117)qqIntroducing the variable λ = a2L u/ a2L λ2 + a2T (1 − λ2 ), which runs through thesame interval [−1, 1] as u, we rewrite this as2Va2L ,a2T (r0 ) = −esa2L2πZ1dλ−1exp[−(r02 /2a2L )λ2 ].a2L(1 − λ2 ) + a2T λ2(5.118)Extremization of W1 (r0 ) in Eq.
(5.97) yields the equations for the trial frequenciess∂λ2 exp[−(r02 /2a2L )λ2 ]a2L 12Ω2T (r0 ) ≡dλV=e,222π −1 [a2L (1 − λ2 ) + a2T λ2 ]2∂a2T aT ,aL"#∂1 ∂1 222+ 2 2 Va2 ,a2[Ω + 2ΩT ] =Ω (r0 ) ≡T L3 L3 ∂a2L∂aT2 11= e2 √ qexp(−r02 /2a2L ).3 2π a2 a4L TZ(5.119)(5.120)These equations have to be solved together witha2L,T (r0 )"#βΩL,T (r0 )βΩL,T (r0 )=coth−1 .222βΩL,T (r0 )1(5.121)Upon inserting the solutions into (5.97), we find the approximate effective classicalpotential plotted in Fig. 5.9 as a solid curve. Let us calculate the approximate particle distribution functions using a three-dimensional anisotropic version of Eq. (5.94).With the potential W1 (r0 ), we arrive at the integral [6]ρ1 (r) =Z3d x0= 2πZ0∞2 /2a2L222e−(x0 +y0 )/2aT e−βW1 (r0 )23/22πa2L (r0 ) 2πaT (r0 ) (2πβ)e−(z0 −r)qdr02 /2a2L222e−r0 (1−λ )/2aT e−βW1 (r0 ).dλ q−12πa2T (r0 ) (2πβ)3/22πa2L (r0 )Z1e−(λr0 −r)(5.122)The resulting curves to different temperatures are plotted in Fig.
5.10 and comparedwith the exact distribution as given by Storer. The distribution obtained from theearlier isotropic approximation (5.116) to the trial frequency Ω2 (r0 ) is also shown.4805 Variational Perturbation Theory√3Figure 5.10 Particle distribution g(r) ≡ 2πβ ρ(r) in Coulomb potential at different temperatures T (the same as in Fig. 5.9), calculated once in the isotropic andonce in the anisotropic approximation. The dotted curves show the classical distribution. For low and intermediate temperatures the exact distributions of R.G.
Storer,J. Math. Phys. 9 , 964 (1968) are well represented by the two lowest energy levels for which ρ(r)=π −1 e−2r eβ/2 +(1/8π)(1 − r + r 2 /2)e−r eβ/8 +(1/38 π)(243 − 324r +216r 2 −48r 3 +4r 4 )e−2r/3 eβ/18 .5.11Hydrogen Atom in Strong Magnetic FieldThe recent discovery of magnetars [7] has renewed interest in the behavior of chargedparticle systems in the presence of extremely strong external magnetic fields. In thisnew type of neutron stars, electrons and protons from decaying neutrons producemagnetic fields B reaching up to 1015 G, much larger than those in neutron starsand white dwarfs, where B is of order 1010 − 1012 G and 106 − 108 G, respectively.Analytic treatments of the strong-field properties of an atomic system are difficult, even in the zero-temperature limit.
The reason is a logarithmic asymptoticbehavior of the ground state energy to be derived in Eq. (5.132). In the weak-fieldlimit, on the other hand, perturbative approaches yield well-known series expansionsin powers of B 2 up to B 60 [8]. These are useful, however, only for B B0 , where B0is the atomic magnetic field strength B0 = e3 M 2 /h̄3 ≈ 2.35 × 105 T = 2.35 × 109 G.So far, the most reliable values for strong uniform fields were obtained by numerical calculations [9].The variational approach can be used to derive a single analytic expression forthe effective classical potential applicable to all field strengths and temperatures [10].The Hamiltonian of the electron in a hydrogen atom in a uniform external magneticH. Kleinert, PATH INTEGRALS4815.11 Hydrogen Atom in Strong Magnetic Fieldfield pointing along the positive z-axis is the obvious extension of the expression inEq.
(2.622) by a Coulomb potential:1 2 M 2 2e2H(p, x) =p + ωB x − ωB lz (p, x) −,2M2|x|(5.123)where ωB denotes the B-dependent magnetic frequency ωL /2 = eB/2Mc ofEq. (2.627), i.e., half the Landau or cyclotron frequency. The magnetic vectorpotential has been chosen in the symmetric gauge (2.616). Recall that lz is thez-component of the orbital angular momentum lz (p, x) = (x × p)z [see (2.623)].At first, we restrict ourselves here to zero temperature. From the imaginary-timeversion of the classical action (2.616) we see that the particle distribution functionin the orthogonal direction of the magnetic field is, for xb = 0, yb = 0, proportionaltoMexp − ωB x2a .2(5.124)This is the same distribution as for a transverse harmonic oscillator with frequencyωB .
Being at zero temperature, the first-order variational energy requires knowingthe smeared potential at the origin. Allowing for a different smearing width a2k anda2⊥ along an orthogonal to the magnetic field, we may use Eq. (5.118) to writeVa2 ,a2 (0) =k⊥vu2u−e t1 Z11dλ.222πak −1 (1 − λ ) + λ2 a2⊥ /a2k(5.125)Performing the integral yieldsVa2 ,a2 (0) =k⊥vuu−e2 tak1.2 2 arccosha⊥− a⊥ )2π(a2k(5.126)Since the ground state energies of the parallel and orthogonal oscillators are Ωk /2and 2 × Ω⊥ /2, we obtain immediately the first-order variational energyW1 (0) = Ω⊥ +ΩkM+ Va2 ,a2 (0) + (ωk2 − Ω2k )a2k + M(ωB2 − Ω2⊥ )a2⊥ ,k ⊥22(5.127)with ωk = 0 and a2k,⊥ = 1/2Ωk,⊥ . In this expression we have ignored the second termin the Hamiltonian (5.123), since the angular momentum lz of the ground state musthave a zero expectation value.For very strong magnetic fields, the transverse variational frequency ΩT willbecome equal to ωB , such that in this limitsΩkΩk8ωW (0) ≈ ωB +− e2log B .4πΩk(5.128)4825 Variational Perturbation TheoryExtremizing this in Ωk yieldsΩk ≈4e42πωlog2 4 B ,πe(5.129)and thus an approximate ground state energy(0)E1 ≈ ωB −2πωe4log2 4 B .πe(5.130)The approach to very strong fields can be found by extremizing the energy (5.127)also in Ω⊥ .
Going over to atomic units with e = 1 and m = 1, where energies aremeasured in units of 0 = Me4 /h̄2 ≡ 2 Ryd ≈ 27.21 eV, temperatures in 0 /kB ≈3.16 × 105 K, distances in Bohr radii aB = h̄2 /Me2 ≈ 0.53 × 10−8 cm, and magneticfield strengths in B0 = e3 M 2 /h̄3 ≈ 2.35 × 105 T = 2.35 × 109 G, the extremizationyields2aa22ln B − 2lnln B ++ 2 + b + O(ln−3 B),Ω⊥ = √πln B ln B(5.131)with abbreviations a = 2 − ln 2 ≈ 1.307 and b = ln(π/2) − 2 ≈ −1.548.
Theassociated optimized ground state energy is, up to terms of order ln−2 B,BΩk ≈ ,2E(0)!qB1(B) =−2πln2 B − 4 ln B lnln B + 4 ln2 ln B − 4b lnln B + 2(b + 2) ln B + b2−i1 h 28 ln ln B − 8b lnln B + 2b2+ O(ln−2 B).ln B(5.132)The prefactor 1/π of the leading ln2 B-term using a variational ansatz of the type(5.64), (5.65) for the transverse degree of freedom is in contrast to the value 1/2calculated in the textbook by Landau and Lifshitz [11]. The calculation of higherorders in variational perturbation theory would drive our value towards 1/2.The convergence of the expansion (5.132) is quite slow. At a magnetic fieldstrength B = 105 B0 , which corresponds to 2.35 × 1010 T = 2.35 × 1014 G, the contribution from the first six terms is 22.87 [2 Ryd].
The next three terms suppressedby a factor ln−1 B contribute −2.29 [2 Ryd], while an estimate for the ln−2 B-termsyields nearly −0.3 [2 Ryd]. Thus we find ε(1) (105 ) = 20.58 ± 0.3 [2 Ryd].Table 5.2 Example for competing leading six terms in large-B expansion (5.132) atB = 105 B0 ≈ 2.35 × 1014 G.(1/π)ln2 B42.1912−(4/π)ln B lnln B−35.8181(4/π) ln2 ln B7.6019−(4b/π) lnln B4.8173[2(b + 2)/π] ln Bb2 /π3.30980.7632Table 5.2 lists the values of the first six terms of Eq.
(5.132). This shows inparticular the significance of the second term in (5.132), which is of the same orderof the leading first term, but with an opposite sign.H. Kleinert, PATH INTEGRALS4835.11 Hydrogen Atom in Strong Magnetic Field30Landau: − log2 B 2 /2(B)/2Ryd102086 10420100005002000100015002000B/B0Figure 5.11 First-order variational result for binding energy (5.133) as a function ofthe strength of the magnetic field. The dots indicate the values derived in the referencegiven in Ref.
[12]. The long-dashed curve on the left-hand side shows the simple estimate0.5 ln2 B of the textbook by Landau and Lifshitz [11]. The right-hand side shows thesuccessive approximations from the strong-field expansion (5.134) for N = 0, 1, 2, 3, 4,with decreasing dash length. Fat curve is our variational approximation.The field dependence of the binding energyε(B) ≡B− E (0)2(5.133)is plotted in Fig.
5.11, where it is compared with the results of other authors whoused completely different methods, with satisfactory agreement [12]. On the strongcoupling side we have plotted successive orders of a strong-field expansion [24]. Thecurves result fromqan iterative solution of the sequence of implicit equations for thequantity w(B) = (4B)/2 for N = 1, 2, 3, 4:w=NX1Ban (B, w),log 2 +2wn=1(5.134)wheres1B √2π2a1 ≡ − (γ +log 2), a2 ≡log, a3 ≡ −− πw , a4 =212wB2w 2R∞qs√2 πD,B 2(5.135)and D denotes the integral D ≡ γ −2 0 dy (y/ y 2 + 1−1) log y ≈ −0.03648, whereγ ≈ 0.5773 is the Euler-Mascheroni constant (2.461).Our results are of similar accuracy as those of other first-order calculations basedon an operator optimization method [25].
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