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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

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

First & second quantizations

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”

Reminder, for boson

Anticommutation relation

Reminder, for boson

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.

Occupation number

◇Convenient to describe processes in which particles are created and annihilated; ◇Convenient to describe interactions.

Normal productOperators in second quantization Creation/Annihilation operations

April 27, 2016

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

−=−=

=−=+++

+

Occupation number

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.

Normal product (order)

Contraction

Already in normal form

Thus

We defined

Wick’s theorem

Wick’s theorem

One-body operator in second quantization

Two-body operator

The Hamiltonian becomes,

II Random Phase approximation and Tamm-Dancoff

approximation

Two-body operator

Hartree-Fock potential

Random Phase Approximation (RPA)

RPA equation

[ ] ++ = 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

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

Tamm-Dankoff Approximation (TDA)

TDA equation

For holes