Cooper Pair
Source: A.L.Fetter, J.D.Walecka, “Quantum Theory of Many-Particle Systems”, McGraw Hill (71)
Consider 2 fermions interacting with potential :
(1)
Using , we have
where C is a normalization constant so that . For box normalization, , where is the volume of the system.
For the ground state of H:
Multiply (1) with gives:
Writing (1) as, we have:
(2)
where we have used the completeness of .
Since , we have
According to the Pauli exclusion principle, is the filled fermi sphere with . Hence, the sum in (2) is restricted to .
Let:
Hence:
,
,
,
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:
The jacobian of the transformation can be obtained as follows:
1st, consider the 1-dim case:
In cartesian coordinates, the 3-dim J has the form:
where each JI has the form of the 1-dim case.
Hence:
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:
Writing , eq(2)
becomes:
(3)
where we have used to denote the restriction , ie..
Likewise, becomes:
(4)
To simplify the notation, we’ll set .
Eqs (3) is known as the Bethe-Goldstone eq.[1]
Model Solution:
Non-local generalization of V:
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: .
A non-local potential is separable if it can be written as: .
In the non-local generalization of a local potential, the natural way is to write:
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: 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.:
becomes:
where V(k) is the fourier transform of V defined by:
and we have defined
Eq(4) thus becomes:
(5)
which implies:
Similarly, eq(3) is now:
Using this to calculate gives:
Putting this into eq(5) gives:
Dividing both sides by gives:
(6)
Eq(6) is the eigenvalue equation which determines the energy shift:
Let
so that .
Consider 1stA:
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:
,
which is also the minimum value of t in the integration. Hence, if , the integrand explodes at and A is infinite. For , 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 negligibleexcept around .
The overall behavior of is shown in fig 36.2. The eigenvalues are given by the intersects of and the stright line .
For > 0, ie, repulsive e-e interaction,there’s only 1 solution at . 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 kCk, 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. ForK 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 neglectB, so (6) becomes:
where the angular part is integrated. Note that 0 and the integral is positive.
Setting , we have:
where
Now:
Assuming V(t)→0 sufficiently fast as t→∞, we have:
(the integral involving is dropped)
for
Hence:
See comments on pp.325-6.
1
[1] H.A.Bethe, J.Goldstone, Proc Roy Soc (London) A238, 551 (57)