chalmers university of technology many body solid state physics 2007 mattuck chapter 5 - 7
Post on 22-Dec-2015
213 views
TRANSCRIPT
Chalmers University of Technology
Many Body Solid State Physics2007
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
...|Φ|HΦWE 01000
...111000..11|0
R(t) = probability (amplitude) that ifthe system at t=0 is in theFermi vacuum, then at t = t thesystem is in the Fermi vacuum
= “no particle propagator”
Fermi vacuum
0
)(
0 ||tR
Chalmers University of Technology
Vacuum amplitude
tiWetUtR 000 |)(|)(
U(t) = time development operator
)(lnlim)1(
00 tRdtdiWE
it
Chalmers University of Technology
Pinball vacuum amplitude
Chalmers University of Technology
Pinball vacuum amplitude
O O O
L
L
O
G
G
G
P= + + + +…
O
O
+ +…
Chalmers University of Technology
Quantum one-particle vacuum amplitude
)(2
2
0 rUmp
H
Zeroth First Second Third
“Vacuum polarisation” or “vacuum fluctuation”t
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
t1
t2
t3
t2
t2
t1 t1
t3 t3t
t1
t2
t3
Chalmers University of Technology
Quantum one-particle vacuum amplitude
1R + + + + +…
+ +…+
Chalmers University of Technology
Quantum one-particle vacuum amplitude
Linked cluster theorem
)(ln tR 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 andbeginning in the ground state
Again E0 is only sum over linked diagramsWe can get E0 in some approximation, eg. Hartree-Fock:
E0 = W0 + +
Chalmers University of Technology
Bird’s eye view of MBPField theoretic ingredient Significance in MB theoryOccupation number formalism Express arbitrary state of MB
systemCreation and destruction operators Primitive operators from which all
MB operators can be builtSingle particle propagator Quasi particle energies, momentum
distribution and moreVacuum amplitude Ground state energyTwo-particle propagator Collective excitations, non
equilibrium propertiesFinite temperature vacuum amplitude Equilibrium thermodynamic
propertiesFinite temperature propagator Temperature dependent properties
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)
...000|
...001|
...100|
Extended Hilbert space =
No particle One particle Two particles …
...010|
...011|
...101|
...110|
...
Chalmers University of Technology
Chalmers University of Technology
Chalmers University of Technology
Chalmers University of Technology
Chalmers University of Technology
Chalmers University of Technology