Cooper Pair
Source: A.L.Fetter, J.D.Walecka, “Quantum Theory of Many-Particle Systems”,
McGraw Hill (71)
Consider 2 fermions interacting with potential V ( x1 , x2 )  V (1,2) :
H (1,2) (1,2)  E (1,2)
(1)
H (1,2)  T1  T2  V (1,2)  H 0  V (1,2)
H 0 (1,2) n (1,2)   n n (1,2)
2
Using
Ti 
pi
,
2m
we have
 n (1,2)   k ,k (1,2)  C exp ik1  x1  ik 2  x 2 
1
 n   k ,k 
1
2
2

2 2
k1  k 22
2m

where C is a normalization constant so that  n  n  1 . For box normalization,
C
1
, where  is the volume of the system.

For the ground state of H:
Multiply (1) with  0
gives:
 0 H 0  V   E  0 
  0   0 V   E
Writing (1) as E  H 0   V , we have:
 

n
1
1
V  
E  H0
E  H0

n
 n V 
n
1
 n  n V 
E n
1
1

 0  0 V   
 n  n V 
E  0
n0 E   n
1
 n  n V 
n0 E   n
 0  
where we have used the completeness of  n .
0   0 0  1
Since  m  n   mn , we have
1
(2)
According to the Pauli exclusion principle,  0 is the filled fermi sphere with
k1 , k 2  k F . Hence, the sum in (2) is restricted to n  k1 , k 2   k F .
Let:
K  k1  k 2
R
1
x1  x2 
2
1
k1  k 2 
2
x  x1  x 2
k
Hence:
x
x
x2  R 
,
2
2
R 
x  1
1 

 R   ,
x1 R x1 x 2
x1  R 
2 
1
R  
2
1 2
2 2 2 2
2
     R  2  H 0  
R  
2
4m
m
K
K
k1   k ,
k2   k
2
2
2
1
2
2
1 2
2 K 2 2k 2
2
k  k  K  2k   n   k1 ,k2 

2
4m
m
k1  x1  k 2  x2  K  R  k  x
2
1
2
2
 n   k ,k  C exp iK  R  ik  x 
1
2
2 2 2 2
 R    V ( x)
4m
m
 (1,2)   (R, x)  C exp(iK  R) K ( x)
H 
2K 2
E
 E K ,k
4m
 2 2

   V ( x) K  E K ,k K
 m

where
The last equation is simply that written in center of mass coordinates with
reduced mass m/2. In substituting these into eq (2), we identity:

1
exp( iK  R) K ,k

1
 n  exp( iK  R) exp( ik  x)


n  2 3  d 3 k
The jacobian of the transformation ( x1 , x2 )  ( R, x) can be obtained as
follows:
2
1st, consider the 1-dim case:
x1
J  abs R
x2
R
x1
1
1
x  abs
2 1
x2
1
1 
2
x
In cartesian coordinates, the 3-dim J has the form:
J1
J 0
0
Hence:
0
J2
0
d x d
3
1
0
0 1
J3
3
where each JI has the form of the 1-dim case.
x 2   d 3 R  d 3 xJ   d 3 R  d 3 x
The integration limits of R and x are sample shape dependent. However, in the
limit of    and short range interactions, we can neglect the surface
corrections & let each runs through a volume Thus:
1
 n V   3  d 3 R  d 3 x exp( iK  R) exp( ik  x)V ( x) exp( iK  R) K ,k ( x)
 2
1
 1  d 3 x exp( ik  x)V ( x) K ,k ( x)  k V  K ,k
 2
 2 2
Writing E K ,k 
, eq(2)
m
1
  0  
 n  n V  becomes:
n0 E   n
 K ,k ( x) 
1

exp( ik  x)  

dt exp( it  x)
2 

3

m
  2 t2
2
 t V 
K ,k
(3)
where we have used  to denote the restriction k1 , k 2  k F , ie.
Likewise,
k2 
 0  0 V   E
becomes:
m
k V    2
2
To simplify the notation, we’ll set
(4)
v
m
V.
2
Eqs (3) is known as the Bethe-Goldstone eq.1
1
H.A.Bethe, J.Goldstone, Proc Roy Soc (London) A238, 551 (57)
3
1
K  t  kF .
2
Model Solution:
Non-local generalization of V:
V ( x)  AV ( x, x )
 V   A dx  dx  * ( x)V ( x, x ) ( x )
where A is a constant to maintain the correct dimensionality.
Note that V(x,x’) is not a 2-particle potential since we’re using 1 particle wave
functions to calculate its matrix elements. What V means is that the potential
experienced by the particle at one place depends on the potential & state of the
particle at some place else.
A local potential can be represented in non-local form: V ( x, x )  V ( x) ( x  x ) .
A non-local potential is separable if it can be written as: V ( x, x )  U ( x)W ( x ) .
In the non-local generalization of a local potential, the natural way is to write:
V ( x, x )  AV ( x)V ( x ) *
where the complex conjugate is to make V(x,x’) hermitian in case V(x) is
complex, and A has the dimension of energy density.
The only local potential that can be represented in separable hermitian non-local
form is a delta function: V ( x, x )  C ( x) where C is a constant. This comes
about because the separable condition requires the delta function to depend on
only 1 variable.
Non-local generalization of the Bethe-Goldstone eq.:
k V  K ,k 
1

1
2
d
2
 d x d
3
x exp( ik  x)V ( x) K ,k ( x)
becomes:
k V  K ,k 


A
1
 2
A
1
A

1
3
3
x  exp( ik  x)V ( x)V ( x ) * K ,k ( x )
V (k )  d 3 x V ( x ) * K ,k ( x )
V (k )W

where V(k) is the fourier transform of V defined by:
2
V (k )   d 3 x exp( ik  x)V ( x)
and we have defined
W   d 3 xV ( x) * K ,k ( x)
Eq(4) thus becomes:
4
 2 k2 
m
m 
k V   2 1 AV (k )W
2

  2
(5)
which implies:
1
 2 2
W 
 2 k2
mV (k ) A


Similarly, eq(3) is now:
 K ,k ( x) 

1

1

exp( ik  x)  
d 3t exp( it  x)
 2 
exp( ik  x)  

d 3t exp( it  x)
2 

2
2

m
  t

3

m
  2 t2

3
2
2

1
 t V 
K ,k
AV (t )W
2
Using this to calculate W gives:
W   d 3 xV ( x) * K ,k ( x)
 1

d 3 t exp( it  x)
m

  d 3 xV ( x) * 
exp( ik  x)  
AV (t )W 
3
2
2
2
1
  t  2

 2 
 



1

V (k ) *  
d 3t
 2 
3
V (t ) *

m

  t
2
2
2
  AV (t )W

1

 2 2
 2  k 2 

V (k ) *  
V
(
t
)
*
V
(
t
)
3
2
2
2

m

V
(
k
)





t

 2 
1

d 3t

m

1 
d 3t
m
2
V
(
k
)
*

V
(
t
)
*
V
(
t
)
 2 k2 

3
2
2
2

mV (k )
  t

 2 





Putting this into eq(5) gives:
m 
 2  k 2  2 1 V (k )W
  2



m 
d 3t
m
2
V
(
k
)
V
(
k
)
*

V
(
t
)
*
V
(
t
)
 2 k2 

2
3
2
2
2

mV (k )
 
  t
 2 



m
d 3t
m
2
V
(
k
)


V (t ) * 2 2 2 V (t )( 2  k 2 )
2
3

 
 (  t )
 2 






Dividing both sides by  2  k 2  gives:
1


m
2


1
d 3t
1
2
V
(
k
)

V (t ) * 2
V (t )  f ( 2 )
 2

2
3
2
 t
 2 
 (  k )

5
(6)
Eq(6) is the eigenvalue equation which determines the energy shift:

2 2
E 
 k2
m

Let
A

B
d 3t
2 
3
V (t ) *
1
V (t )
 t2
2
1
2
V (k )
2
(  k )
2
1
 A  B  f ( 2 ) .

Consider 1st A:
 is the region outside the union of the spheres shown in fig 36.1. The wave
vectors at the intersection of the spheres have a square magnetude:
so that
2
K
t  k   ,
2
2
S
2
F
which is also the minimum value of t in the integration. Hence, if   t S ,
the integrand explodes at t   and A is infinite. For   t S , A is finite,
negative, and decreases as increases. (see fig 36.2). As will be shown later, the
singularity near tS is logarithmic.
For B:
Owing to the presence of  in the denominator, as    , its contribution
is negligible except around   k .
The overall behavior of f ( 2 ) is shown in fig 36.2. The eigenvalues are given
by the intersects of f ( 2 ) and the stright line
1

.
For  > 0, ie, repulsive e-e interaction, there’s only 1 solution at   k . The
energy shift is therefore small.
For  < 0, ie, attractive e-e interaction, there’re 2 solutions. 1 near k and the other
at kC. If kC < k, we have E < 0. Hence, the 2 e’s will form a bound state ( cooper
pair ).
Evaluation:
Eq(6) can be solved as long as K < 2kF. For K > 2kF, the spheres in fig 36.1
become disjoined & A is singular for all . The most interesting case is when K = 0
for which the spheres collapse into 1 so that A and hence E are maximum. To
calculate kC , we can neglect B, so (6) becomes:
6
2

dt  t 2 V (t )

 k F 2 2 t 2   2
1
where the angular part is integrated. Note that 0 and the integral is positive.
Setting
x t
kF
, we have:
2

dx  x 2 V (k F x)
 kF 
2

x 2  ~ 2
1 2
1
where ~  
kF
Now:
x2
c2
c 1
1 

1

 1 

2
2
2
2
2  x  c x  c 
x c
x c
Assuming V(t)→0 sufficiently fast as t→∞, we have:
   1
1 
2

dx 

V ( k F x)
2 
~

 4 1  x   x   
1


2
 x  ~ 

V
(
k
x
)
ln
F
 x  ~ 
4 2
1


2
1  ~ 
V
(
k
)
ln
F
1  ~ 
4 2



 kF   

2
V
(
k
)
ln
F


4 2
 kF   

 4k F2 

2
V
(
k
)
ln
 2
F
2 
4 2
 kF   
(the integral involving
for k F  
Hence:



2
   V (k F ) 
 2  k F2  4k F2 exp 
4 2
See comments on pp.325-6.
7
d | V |2
is dropped)
dx