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Chalmers University of Technology Many Body Solid State Physics 2007 Mattuck chapter 5 - 7 Chalmers University of Technology Contents • Quantum Vacuum: how to solve the equations of nothing • Birds eye of Diagrams: start of the elementary part of the book (page 118->) • Learning how to count: occupation number formalism • Any questions ?!? Chalmers University of Technology Quantum vacuum • Meaning of the vacuum of amplitude Chalmers University of Technology Vacuum amplitude E0 W0 Φ0|H 1|Φ0 ... Fermi vacuum 0 | 11..111000... R(t) = probability (amplitude) that if the system at t=0 is in the Fermi vacuum, then at t = t the system is in the Fermi vacuum = “no particle propagator” R (t ) | 0 | 0 Chalmers University of Technology Vacuum amplitude R(t ) 0 | U (t ) | 0 eiW0t U(t) = time development operator d E0 W0 lim i ln R(t ) t (1i ) dt Chalmers University of Technology Pinball vacuum amplitude Chalmers University of Technology Pinball vacuum amplitude G + P= O + O O + O +… G L G + L O +… O Chalmers University of Technology Quantum one-particle vacuum amplitude 2 p H 0 U (r ) 2m “Vacuum polarisation” or “vacuum fluctuation” t Zeroth First Second Third Chalmers University of Technology Quantum one-particle vacuum amplitude =- “Nevertheless it is important to retain such diagrams which violates conservation of particle number to prove the linked cluster theorem.” Chalmers University of Technology Quantum one-particle vacuum amplitude Topological equivalence t3 t3 t3 t3 t2 t2 t2 t1 t1 t1 t2 t1 t Chalmers University of Technology Quantum one-particle vacuum amplitude R 1 + + + + +… + + +… Chalmers University of Technology Quantum one-particle vacuum amplitude Linked cluster theorem ln R(t ) All linked diagrams Which can be shown via entities like + + = x These gives us the possibility to get the ground state energy even when the perturbation in strong. Chalmers University of Technology The many body case R=1+ + + … = all diagrams starting and beginning in the ground state Again E0 is only sum over linked diagrams We can get E0 in some approximation, eg. Hartree-Fock: E0 = W0 + + Chalmers University of Technology Bird’s eye view of MBP Field theoretic ingredient Significance in MB theory Occupation number formalism Express arbitrary state of MB system Primitive operators from which all MB operators can be built Quasi particle energies, momentum distribution and more Ground state energy Collective excitations, non equilibrium properties Equilibrium thermodynamic properties Temperature dependent properties Creation and destruction operators Single particle propagator Vacuum amplitude Two-particle propagator Finite temperature vacuum amplitude Finite temperature propagator Chalmers University of Technology Second quantization (again) • A way to write the wave function in a compact way (no Slater determinant crap) • A way to treat the particle type automatically (fermions and bosons) • Can refer to any basis (momentum, real…) • A way to vary particle number Chalmers University of Technology Second quantization (again) Extended Hilbert | 000... space = | 001... | 010... | 100... No particle One particle | 011... | 101... ... | 110... Two particles … Chalmers University of Technology Chalmers University of Technology Chalmers University of Technology Chalmers University of Technology Chalmers University of Technology Chalmers University of Technology