second quantization - royal institute of technology · pdf filein second quantization one...
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KTH ROYAL INSTITUTE OF TECHNOLOGY
Second Quantization
Creation and annihilation operators. Occupation number. Anticonmutation relations. Normal product. Wick’s theorem. One-body operator in second quantization. Hartree-Fock potential. Two-particle Random Phase Approximation (RPA). Two- particle Tamm-Damkoff Approximation (TDA).
April 21, 2017
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SI2380, Advanced Quantum Mechanics Edwin Langmann
D. Cohen: Lecture notes in quantum mechanics http://arxiv.org/abs/quant-ph/0605180v3
K. Heyde, The nuclear shell model, Springer-Verlag 2004
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First & second quantizations
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In Second Quantization one introduces the creation operator such that a state can be written as
Vacuum
There is no particle to annihilation in “vacuum”
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Reminder, for boson
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Anticommutation relation
Reminder, for boson
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Hole stateBelow the Fermi level (FL) all states are occupied and one can not place a particle there. In other words, the A-particle state |0⟩, with all levels hi occupied, is the ground state of the inert (frozen) double magic core.
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Occupation number
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◇Convenient to describe processes in which particles are created and annihilated; ◇Convenient to describe interactions.
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Normal productOperators in second quantization Creation/Annihilation operations
April 27, 2016
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Occupation Number FormalismFermion Creation and Annihilation Operators
Boson Creation and Annihilation Operators
Fermion examples
€
ci n1n2...ni ... = −1( ) i∑ ni n1n2...ni-1...
ci+ n1n2...ni... = −1( ) i∑ 1- ni( )n1n2...ni+1...
i∑ = n1 + n2 + ...+ ni-1 i.e. number of particles to left of ith
ai n1n2...ni ... = ni1/2 n1n2...ni ...
ai+ n1n2...ni... = ni +1( )1/ 2
n1n2...ni ...
( )( ) 11111101111110111110110c
11011001101100111111100c111
4
113
−=−=
=−=+++
+
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Occupation number
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Normal product (order)
Normal ordering for fermions is defined by rearranging all creation operators to the left of annihilation operators, but keeping track of the anti-commutations relations at each operator exchange.
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Normal product (order)
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Contraction
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Already in normal form
Thus
We defined
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Wick’s theorem
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Wick’s theorem
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One-body operator in second quantization
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Two-body operator
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The Hamiltonian becomes,
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II Random Phase approximation and Tamm-Dancoff
approximation
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Two-body operator
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Hartree-Fock potential
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Random Phase Approximation (RPA)
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RPA equation
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[ ] ++ = QQH , ω!
miim
miimim
mi aaYaaXQ +++ ∑∑ −=,,
0=RPAQ
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛
YX
YX
ABBA
1001
** ω!
p-h phonon operator
RPA equation
mε
Fermi Energy
iε
mnijnjmi
nijmijmnimnjmi
vBvA
~~)(
=
+−= δδεε
Closed shell: 1p-1h correlation
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Tamm-Damkoff Approximation (TDA)
The difference between TDA and RPA is that we use ➢The simple particle-hole vacuum |HF> in TDA ➢The correlated ground state in the RPA
mε
iε
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Tamm-Dankoff Approximation (TDA)
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TDA equation
For holes
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