By Takafumi Kita
This booklet offers a theoretical, step by step finished clarification of superconductivity for undergraduate and graduate scholars who've accomplished common classes on thermodynamics and quantum mechanics. To this finish, it adopts the original technique of beginning with the statistical mechanics of quantum perfect gases and successively including and clarifying parts and strategies indispensible for knowing it. They comprise the spin-statistics theorem, moment quantization, density matrices, the Bloch–De Dominicis theorem, the variational precept in statistical mechanics, beautiful interplay and sure states. plentiful examples in their utilization also are supplied by way of subject matters from complicated statistical mechanics comparable to two-particle correlations of quantum excellent gases, derivation of the Hartree–Fock equations, and Landau’s Fermi-liquid conception, between others. With those preliminaries, the basic mean-field equations of superconductivity are derived with greatest mathematical readability in line with a coherent kingdom by way of the Cooper-pair construction operator, a quasiparticle box for describing the excitation and the variational precept in statistical mechanics. they've got the virtue that the part coherence end result of the Cooper-pair condensation will be basically visible making the superfluidity understandable clearly. consequently, they're utilized to homogeneous instances to explain the BCS conception for traditional s-wave superconductors and its extension to the p-wave superfluidity of 3He. Later, the mean-field equations are simplified to the Eilenberger and Ginzburg–Landau equations for you to describe inhomogeneous superconductivity comparable to Abrikosov’s flux-line lattice concisely and transparently. Chapters give you the most up-to-date experiences at the quasiclassical idea of superconductivity and a discovery of p-wave superfluidity in liquid 3He. The booklet serves as a customary reference for complicated classes of statistical mechanics with workouts in addition to particular answers.
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Additional info for Statistical Mechanics of Superconductivity
D Á 0 eˇ. / ˇ. e / d D ˇ. e / d ˇ1 Z 1 ˇ ˇ C A ˇ ˇÁ 0 0 ˇ. ˇe Á 1 1 U; Á e ˇ. 13) in the last equality. 18). The above general derivation also implies that the energy dependence of the density of states manifests itself in the proportionality constant between PV and U . 2=d /U for d dimensions. 3 Introducing Dimensionless Variables Using a set of appropriate dimensionless variables often enables us to make physical arguments clearer and find similarities among apparently different systems. 20) The order of lQ is roughly the mean interparticle spacing of particles with spin component ˛.
10 / O . 1 / O . 2 / O . N/ D ı. 10 ; 1 / O . 2 / C 1 O . N/ ı. 10 ; 2 / O . 1 / O . 3 / O . 4 Bra-Kets for Many-Body Wave Functions 31 C C N 1 C N ı. 10 ; N/ O . 1/ O . 1/ O . 2/ O . O . N 1/ N/ O . 23) to move the annihilation operator O . 22) to the left of j0i and use O . j0 /j0i D 0 repeatedly for j D 1; ; N 0 . 22) is shown to hold as 1 h0j O . 20 / O . 10 / O . 1 / O . 2 /j0i 2Š 1 D h0j O . 20 / ı. 10 ; 1 / O . 2 / C ı. 10 ; 2 / O . 1 / 2Š C 2 O . 1 / O . 2 / O .
1 / O . 2 / O . 10 / j0i h 10 ; 20 j 1 ; 2 i D 1 ı. 10 ; 1 /h0j O . 20 / O . 2 /j0i C ı. 10 ; 2 /h0j O . 20 / O . 1 /j0i 2Š 1 ı. 10 ; 1 /ı. 20 ; 2 / C ı. 10 ; 2 /ı. 20 ; 1 / : D 2Š D Thus, we have constructed the eigenspace of PO . 5) by Z Z Z jˆ i Á d d 1 d 2 j 1; 2; ; N iˆ . 1; 2; ; N /: h 1; 2; ; N jˆ . 25) The ket has the following properties: O . 1 /jˆ i D p Z N d O . 2 / O . 1 /jˆ i D O. N 1/ d 0 N 0 3 j 20 ; Z d ; 0 N iˆ 0 N j 30 ; . 1 ; 20 ; ; p O . 2 / O . 1 /jˆ i D N Š j0iˆ . 1 ; 2 ; 0 N iˆ ; 0 N /; .
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