Download 6.1.5. Number Representation: Operators

6.1.5. Number Representation: Operators
Consider a 1-P operator A  p, x  . Given the complete orthonormal basis
   , we
can write
A  p, x      A  
  A  


where the matrix elements A   A 
where 
are numbers.
Using   â  ,
is the vacuum, we can write
   A aˆ 
 â

Note that the vacuum projector   serves to confine  to the 1-particle
 Â   0 if the number of particles in either  or  is not one.
subspace, i.e.,
By removing this restriction, we obtain the desired many body version
    A 
â â 

Next, we consider the 2-P potential
N
1 N
V   V  xi , x j    V  xi , x j 
2 i  j 1
i  j 1
   , a basis vector for the 2-P Hilbert
Given the complete orthonormal 1-P basis
space is
 1 
2

1

2
 1 2
where, to avoid ambiguity, we have used subscript to indicate the particle occupying
the state. Taking the hermitian conjugate, we obtain the adjoint basis vector


1
2



2
 1 
2 
1
  1 2
where the order of the factors is to be noted with care.
condition
I 
 ,
1

2 2

1

  1 2
 ,
1  2
we can write the 2-P potential (1st quantized) operator as
Using the completeness
V

1

2  
1

2 2

V 
1
1

2 2

1

1
 12 12 V  1 2  1 2
2  
Using
1 2  
 1 2  2 
1
1


2
 â â 

1

2


  â â 


  â â
we can write the 2nd quantized version of V as
1
Vˆ   â aˆ   V   aˆ â
2  
where
 V    d 3x1  d 3x 2 *  x1  *  x 2 V  x1 , x 2    x 2    x1 
As before, the vacuum projector   serves to confine Vˆ to the 2-particle
subspace. Removing this restriction then gives
1
Vˆ    V  â â â â
2 
(a)
Note that , and , refer to the particle at x1 and x2, respectively. Thus, in eq(a),
the first and last operators refer to particle 1.