Preliminaries: N-particle ket with definite momentum appears in various notations as,
N PN p
P1
p
... p
P2
P N
p
p p ...p
p P ... p P
P1
2
P1
N
P2
P N
F ( N )
(1.1)
Definition of the wedge, 1
(1) N !
ˆ p p ...p p1 ... p N N !A 1 2 N
P
. ..p P N ; p P1 ..
P
p1 ... p N N !Sˆ p1 ...p N
1 N !
Also, slater determinant,
p
P1
...p P N ;
P
[1.42] | p1...p N | p1...pN det pi | p j
(1.2)
( 1) P p1 | pP1 ... p N | pP N ;
P
(1.3)
Since the creation operator maps an N-particle state into an (N+1)-particle state, the annihilation operator, being the adjoint, will map an N- particle particle state state into an (N−1) (N−1)-particle -particle state. To understand its properties, compute the sub-matrix elements of the operators (ap† )p1 ...p N p2 ...pN and (ap )p2 ...p N p1 ...pN . Using the definition of the wedge, (1.2), and the slater-determinant, (1.3), and eventually expanding the determinant in terms of its first column, *
| ap | p1 ...p N (ap† )p1 ...p N p2 ...pN p2 ...p N
p1 ... p N | ap† | p 2 ..... p N p 1 ... p N |p p 2 ... p N 1
det pi | pj P (1) p1 | pP ... p N | p P P
1
N
N
(1)
n 1
p n | p det p i | p j
(n)
(1.4)
n 1
The matrix elements ( ap )p2 ...p N p1... pN , meanwhile, come from taking the complex-conjugate of (1.4), (ap )p2 ...p N p1 ...pN
N
(1)n 1 p | p n det pj | pi
(n)
; det p j | p i
( n)
p2 ... pN | p1 ...(no pn )... p N ; (1.5)
n 1
From (1.4) and (1.5), we respectively get, ap p1 ... p N
N
(1)
n 1
p | p n p1 ...(no p n )... p N p1 ...(no p n )... p N ;
(1.6)
n 1
Let it be true, by construction, that {ap , ap } 0 0† {ap† , ap† } . Demonstrate the fundamental {ap , ap†} p | p . Compute apap† p1 ... p N and ap†ap p1 ... p N , which appears as, ap a p1 ... p N ap p p1 ... p N p | p p 1 ... p N † p
N
( 1)
n
p | p n p p 1 ...( no p n )... p N (1.7)
n 1
ap† ap p1 ... p N ap†
N
(1)n 1 p | p n
p1 ...( no p n )... p N
n 1
N
( 1)
n 1
p | p n p p 1 ...( no p n )... p N (1.8)
n 1
By adding (1.7) and (1.8), we realize the relation, {ap , ap† } p | p
(1.9)
Summary of results
Anticommutation relations,
{ap , ap†} pp
V 3
(p p); {ap , ap} 0 0 † {ap†, ap†};
(1.10)
property of ladder operator is to create/destroy particle,
ap† P N 0 1 pPN pPN ; ap pPN 1 PN PN ;
(1.11)
Field operators and their dual in position-space: they create destroy a particle at point x , 1 1 i p x / D.C . 3 e a d p e ipx/ ap† d 3 p † (x ) † ; ( x) x p 3/ 2 3/ 2 (2 ) (2 )
(1.12)
Field operators and their dual in momentum-space: they create destroy a particle with momentum p , 1 1 i p x / D.C . 3 a a (p) ap e ( ) d x e ipx/ † (x )d 3 x ap† a † (p ) a † ; (1.13) x 3/ 2 3/ 2 (2 ) (2 )
In a box,
(x) x
1
e V
ip x /
ap
p
a a(p) ap V
e
ip x /
x
(x)(2 )3
1
V
ip x/
e
ap
V d3 p (2 )
3
V e ip x / (x)(2 )3
V (2 ) d3x
V (2 )
3
3
e
ip x /
ap d 3 p; (1.14)
1
e V
[1.101] { (x), † (x)} (x x ); [1.102] { † (x ), † (x )} 0† 0 { (x ), (x )};
i p x /
(x) d 3 x
