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Lecture IV 9
Lecture IV: Second Quantisation
We have seen how the elementary excitations of the quantum chain
can be presented
in terms of new elementary quasi-particles by the ladder
operator formalism. Can this
approach be generalised to accommodate other many-body systems?
The answer is provided
by the method of second quantisation an essential tool for the
development of interacting
many-body field theories. The first part of this section is
devoted largely to formalism
the second part to applications aimed at developing fluency.
Reference: see Feynmans book on Statistical Mechanics
Notations and Definitions
Consider a single-particle Schrodinger equation:
H| = |
How can one construct a many-body wavefunction?Particle
indistinguishability demands symmetrisation:
n
n
0
1
1
1
0
1
2
3
4
2
3
1
Fermions Bosons
0
e.g. two-particle wavefunction for fermions i.e. particle 1 in
state 1, particle 2...
F(x1, x2) 12
(
state 11 (
particle 1
x1 )2(x2) 2(x1)1(x2))
In Dirac notation:
|1, 2F 12
(|1 |2 |2 |1)
General normalised, symmetrised, N-particle wavefunctionof
bosons ( = +1) or fermions ( = 1)
|1, 2, . . . N 1
N!
=0n!
P
P|P1 |P2 . . . |PN
n no. of particles in state (for fermions, Pauli exclusion: n =
0, 1, i.e. |1, 2, . . . N is a Slater determinant)
Lecture Notes October 2005
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Lecture IV 10
P: Summation over N! permutations of
{1, . . . N
}required by particle indistinguishability Parity P no. of
transpositions of two elements which brings permutation
(P1, P2, P N) back to ordered sequence (1, 2, N)Evidently, first
quantised representation looks clumsy!
motivates alternative representation...
Second quantisation
Define vacuum state: |, and set of field operators a and
adjoints a no hats!
a| = 0, 1=0
n!
Ni=1
ai | = |1, 2, . . . N
cf. bosonic ladder operators for phonons N.B. ambiguity of
ordering?
Field operators fulfil commutation relations for bosons
(fermions)a, a
= ,
a, a
=
a, a
= 0
where [A, B] AB BA is the commutator (anti-commutator) Operator
a creates particle in state , and a annihilates it Commutation
relations imply Pauli exclusion for fermions: aa = 0 Any N-particle
wavefunction can be generated by application of set of
N operators to a unique vacuum state
e.g. |1, 2 = a2a1|
Symmetry of wavefunction under particle interchange maintained
by
commutation relations of field operators
e.g. |1, 2 = a2
a1| = a
1a2|
(So, providing one maintains a consistent ordering
convention,the nature of that convention doesnt matter)
Fock space: Defining FN to be linear span of all N-particle
states |1, 2, NFock space F is defined as direct sum N=0FN
General state | of the Fock space is linear combination of
stateswith any number of particles Note that the vacuum state |
(sometimes written as |0) is distinct from zero!
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Lecture IV 11
+ a+a
a
0
a
... F0
a
F1F2
Change of basis:
Using the resolution of identity 1 ||, we havea|
| =
a|| |
i.e. a
=
|
a, and a =
|
a
E.g. Fourier representation: a ak, a a(x)
a(x) =k
eikx/
L x|k ak, ak = 1
L
L0
dx eikxa(x)
Occupation number operator: n = aa measures no. of particles in
state
e.g. (bosons)
aa(a)
n| = a
1 + aaaa
(a
)
n1| = (a)n| + (a)2a(a)n1| = = n(a)n|Exercise: check for
fermions
Lecture Notes October 2005
