Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets - ed.4 - 2006 (523104), страница 8
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. . . . . . . . . . . . .Illustration of eigenvalues of fluctuation matrix . . . . .Finite-lattice effects in internal energy E and specific heat........991231451743.13.2Pole in Fourier transform of Green functions Gp,aω (t) . . . . . . . . . 220pSubtracted periodic Green function Gω,e(τ ) − 1/ω and antiperiodicGreen function Gaω,e(τ ) for frequencies ω = (0, 5, 10)/h̄β . . . . . . .
221Two poles in Fourier transform of Green function Gp,aω 2 (t) . . . . . . 222pSubtracted periodic Green function Gω2 ,e (τ ) − 1/h̄βω 2 and antiperiodic Green function Gaω2 ,e (τ ) for frequencies ω = (0, 5, 10)/h̄β . . . . 243Poles in complex β-plane of Fourier integral . . . .
. . . . . . . . . 270Density of states for weak and strong damping in natural units . . . 271Perturbation expansion of free energy up to order g 3 . . . . . . . . . 283Diagrammatic solution of recursion relation for the generating functional W [j[ of all connected correlation functions . . . .
. . . . . . 290Diagrammatic representation of functional differential equation . . . 295Diagrammatic representation of recursion relation . . . . . . . . . . 297Vacuum diagrams up to five loops and their multiplicities . . . . . . 298Diagrammatic differentiations for deriving tree decomposition ofconnected correlation functions . . . . . . . . . . .
. . . . . . . . . . 303Effective potential for ω 2 > 0 and ω 2 < 0 in mean-field approximation309Local fluctuation width of harmonic oscillator . . . . . . . . . . . . 327Magnetization curves in double-well potential . . . . . . . . . . . . 337Plot of reduced Feynman integrals â2LV (x) . . .
. . . . . . . . . . . . 3653.33.43.53.63.73.83.93.103.113.123.133.143.153.164.14.24.34.4. .. .. .C........Left: Determination of energy eigenvalues E (n) in semiclassical expansion; Right: Comparison between exact and semiclassical energies 415Solution for screening function f (ξ) in Thomas-Fermi model . . . .
. 419Orbits in Coulomb potential . . . . . . . . . . . . . . . . . . . . . . 435Circular orbits in momentum space for E > 0 . . . . . . . . . . . . . 438xxxiiixxxiv4.54.64.7Geometry of scattering in momentum space . . . . . . . . . . . . . . 439Classical trajectories in Coulomb potential . . . . . . . . . . .
. . . 445Oscillations in differential Mott scattering cross section . . . . . . . 4465.15.25.35.45.5Illustration of convexity of exponential function e−x . . . . . . . .Approximate free energy F1 of anharmonic oscillator . . . . .
. .Effective classical potential of double well . . . . . . . . . . . . . .Free energy F1 in double-well potential . . . . . . . . . . . . . . .Comparison of approximate effective classical potentials W1 (x0 ) andW3 (x0 ) with exact V eff cl (x0 ) . . . . . . . .
. . . . . . . . . . . . . .Effective classical potential W1 (x0 ) for double-well potential and various numbers of time slices . . . . . . . . . . . . . . . . . . . . . .Approximate particle density of anharmonic oscillator . . . . . . .Particle density in double-well potential . . . .
. . . . . . . . . .Approximate effective classical potential W1 (r) of Coulomb systemat various temperatures . . . . . . . . . . . . . . . . . . . . . . . .Particle distribution in Coulomb potential at different T 6= 0 . . .First-order variational result for binding energy of atom in strongmagnetic field . . . . . . . .
. . . . . . . . . . . . . . . . . . . . .Effective classical potential of atom in strong magnetic field . . .One-particle reducible vacuum diagram . . . . . . . . . . . . . . .Typical Ω-dependence of approximations W1,2,3 at T = 0 . . . . . .Typical Ω-dependence of Nth approximations WN at T = 0 . . .New plateaus in WN developing for higher orders N ≥ 15 . . . . .Trial frequencies ΩN extremizing variational approximation WN atT = 0 for odd N ≤ 91 .
. . . . . . . . . . . . . . . . . . . . . . .Extremal and turning point frequencies ΩN in variational approximation WN at T = 0 for even and odd N ≤ 30 . . . . . . . . . . .Difference between approximate ground state energies E = WN andexact energies Eex . . . . . . . . . . . . . . . . . . . . . . . . . . .Logarithmic plot of kth terms in re-expanded perturbation series .Logarithmic plot of N-behavior of strong-coupling expansion coefficients . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Oscillations of approximate strong-coupling expansion coefficient b0as a function of N . . . . . . . . . . . . . . . . . . . . . . . . . .Ratio of approximate and exact ground state energy of anharmonicoscillator from lowest-order variational interpolation . .
. . . . . .Lowest two energies in double-well potential as function of couplingstrength g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Isotropic approximation to effective classical potential of Coulombsystem in first and second order . . . . . . . . . . . . . . . . . . .Isotropic and anisotropic approximations to effective classical potential of Coulomb system in first and second order . .
. . . . . . . .5.65.75.85.95.105.115.125.135.145.155.165.175.185.195.205.215.225.235.245.255.26....456467469470. 471. 472. 473. 474. 478. 480......483487494496502503. 504. 504. 505. 507. 509. 509. 515. 518. 524. 526H. Kleinert, PATH INTEGRALSxxxv5.27 Approach of the variational approximations of first, second, andthird order to the correct ground state energy . . . . . . . . . . .
.5.28 Variational interpolation of polaron energy . . . . . . . . . . . . . .5.29 Variational interpolation of polaron effective mass . . . . . . . . . .5.30 Temperature dependence of fluctuation widths of any point x(τ ) onthe path in a harmonic oscillator . . . .
. . . . . . . . . . . . . . . .(n)5.31 Temperature-dependence of first 9 functions Cβ , where β = 1/kB T .5.32 Plots of first-order approximation W̃1Ω,xm (xa ) to the effective classicalpotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.33 First-order approximation to effective classical potential W̃1 (xa ) . . .5.34 Trial frequency Ω(xa ) and minimum of trial oscillator xm (xa ) atdifferent temperatures and coupling strength g = 0.1 . . . . . . . .5.35 Trial frequency Ω(xa ) and minimum of trial oscillator xm (xa ) atdifferent temperatures and coupling strength g = 10 . .
. . . . . . .5.36 First-order approximation to particle density . . . . . . . . . . . . .5.37 First-order approximation to particle densities of the double-well forg = 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.38 Second-order approximation to particle density (dashed) comparedto exact results . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .5.39 Radial distribution function for an electron-proton pair . . . . . . .5.40 Plot of reduced Feynman integrals â2LV (x) . . . . . . . . . . . . . . .5285425435465515585595605605615625635645676.16.26.36.46.56.6Path with jumps in cyclic variable redrawn in extended zone scheme 580Illustration of path counting near reflecting wall . .
. . . . . . . . . 583Illustration of path counting in a box . . . . . . . . . . . . . . . . . 586Equivalence of paths in a box and paths on a circle with infinite wall 586Variational functions fN (c) for particle between walls up to N = 16 591Exponentially fast convergence of strong-coupling approximations .
. 5927.17.27.3599599Paths summed in partition function (7.9) . . . . . . . . . . . . . . .Periodic representation of paths summed in partition function (7.9)Among the w! permutations of the different windings around thecylinder, (w − 1)! are connected . . . . . . .
. . . . . . . . . . . . .7.4 Plot of the specific heat of free Bose gas . . . . . . . . . . . . . . . .7.5 Plot of functions ζν (z) appearing in Bose-Einstein thermodynamics .7.6 Specific heat of ideal Bose gas with phase transition at Tc . . . . . .7.7 Reentrant transition in phase diagram of Bose-Einstein condensationfor different interaction strengths . . . . . . . . . . . . . . . . .
. .7.8 Energies of elementary excitations of superfluid 4 He . . . . . . . . .7.9 Condensate fraction Ncond /N ≡ 1 − Nn /N as function of temperature7.10 Peak of specific heat in harmonic trap . . . . . . . . . . . . . . . . .7.11 Temperature behavior of specific heat of free Fermi gas . . . . . . .601602608614618619623630638xxxvi10.1 Edge dislocation in crystal associated with missing semi-infiniteplane of atoms as source of torsion . . . . . .
. . . . . . . . . . . . . 77910.2 Edge disclination in crystal associated with missing semi-infinite section of atoms as source of curvature . . . . . . . . . . . . . . . . . . 78010.3 Images under holonomic and nonholonomic mapping of δ-functionvariation . . . . . . . . . . . . . . . . . .
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