Table of Contents 2.2 Ground State Energy and Elementary Theory of the Electron Gas.....................................41 2.2.1 Hamiltonian. …41 2.2.2 Ground State Energy 42 2.2.3 Modificatio of Elec ergy Levels due to the Coulom nteraction.................... 2.3 Hartree-Fock Equations for Atoms....................... 4 Problems.…52 Bo. n 55 3.1 Free Bosons. 3.1.1 Pair Distribution Function for Free Bosons.......... 55 *3.1.2 Two-Particle States of Bosons...................... 57 3.2 Weakly Interacting,Dilute Bose Gas. 60 3.2.1 Quantum Fluids and Bose-Einstein Condensation. 60 3.2.2 Bogoliuboy Theory of the Weakly Interacting Bose Gas 62 *393 Superftuidity..................................... problems ................................................. 72 4.Correlation Functions,Scattering,and Response 75 Scatt d Re spons 4.4.4。。。。”””””””0”4”4.44.4.4.4 75 4.2 Density Matrix,Correlation Functions.................... 82 4.3 Dynamical Susceptibility ............................... 85 4.4 Dispersion Relations ................................... 89 4.5 Spectral Representation. 90 4.6 Fluctuation-Dissipation Theorem... 91 4.7 Examples of Applications. 93 100 4.8.1 General Symmetry Relations 100 4.8.2 Symmetry Properties of the Response Function for Hermitian Operators...........................102 4.9 Sum Rules......... 。...。.。.。.。......,。,..107 4.9.1 General Structure of Sum Rules.. 4.9.2 Application to the Excitations in He II..............108 Problems.· ...109 Bibliography for Part I... 。。。 ..111XII Table of Contents 2.2 Ground State Energy and Elementary Theory of the Electron Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.2 Ground State Energy in the Hartree–Fock Approximation . . . . . . . . . . . . . . . . . 42 2.2.3 Modification of Electron Energy Levels due to the Coulomb Interaction . . . . . . . . . . . . . . . . . . . . 46 2.3 Hartree–Fock Equations for Atoms . . . . . . . . . . . . . . . . . . . . . . . 49 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3. Bosons ................................................... 55 3.1 Free Bosons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.1.1 Pair Distribution Function for Free Bosons . . . . . . . . . . 55 ∗3.1.2 Two-Particle States of Bosons . . . . . . . . . . . . . . . . . . . . . . 57 3.2 Weakly Interacting, Dilute Bose Gas . . . . . . . . . . . . . . . . . . . . . . 60 3.2.1 Quantum Fluids and Bose–Einstein Condensation . . . . 60 3.2.2 Bogoliubov Theory of the Weakly Interacting Bose Gas . . . . . . . . . . . . . . . . . 62 ∗3.2.3 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4. Correlation Functions, Scattering, and Response .......... 75 4.1 Scattering and Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Density Matrix, Correlation Functions . . . . . . . . . . . . . . . . . . . . 82 4.3 Dynamical Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4 Dispersion Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.5 Spectral Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.6 Fluctuation–Dissipation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.7 Examples of Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 ∗4.8 Symmetry Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.8.1 General Symmetry Relations . . . . . . . . . . . . . . . . . . . . . . . 100 4.8.2 Symmetry Properties of the Response Function for Hermitian Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.9 Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.9.1 General Structure of Sum Rules . . . . . . . . . . . . . . . . . . . . 107 4.9.2 Application to the Excitations in He II . . . . . . . . . . . . . . 108 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Bibliography for Part I ....................................... 111