Advanced Solid State Physics

16.4.3 CooperPairs

We regard a two electron system in a cube of volume 1 and periodic boundary conditions. We write
the wave function in the basis of plane wave product state:

ψ(k 1 ,k 2 ;r 1 ,r 2 ) = exp [i(k 1 ·r 1 +k 2 ·r 2 )]. (311)

Furthermore, we assume that the two electrons are of opposite spin. We perform a coordinate trans-
formation according to

K=k 1 +k 2 , k=

1

2

(k 1 −k 2 ) (312)

and

R=

1

2

(r 1 +r 2 ), r=r 1 −r 2. (313)

Consequently,

ψ(K,k;R,r) = exp (iK·R) exp (ik·r), (314)

where the corresponding kinetic energy of the system is given by

εK+Ek=

~^2

m

(

1

4

K^2 +k^2

)

. (315)

We now investigate the special case ofK= 0 , i.e.k 1 =−k 2 and expand the wave function in terms
of the resulting basis functions:

χ(r) =

∑

k

gkexp (ik·r), (316)

Let us introduce an interaction HamiltonianHI. The correspondingSchrödingerequation reads

(H 0 +HI−ε)χ(r) =

∑

k′

[(Ek′−ε)gk′+HIgk′] exp

(

ik′·r

)

= 0. (317)

Taking the scalar product withexp (ik·r)yields

(Ek−ε)gk+

∑

k′

gk′

〈

k

∣∣

HI

∣∣

k′

〉

= 0. (318)

Transforming the sum overk′into an integral over the energyE′gives

(E−ε)g(E) +

∫

dE′g(E′)HI(E,E′)N(E′) = 0, (319)

whereN(E)is the number of states withK= 0 within the kinetic energy window of widthdE′
aroundE′. Detailed studies of the matrix elementsHI(E,E′)revealed that the contribution is most
important in an energy range slightly above the (one electron)FermienergyεF with thickness~ωD,
whereωDis theDebyephonon cutoff frequency. We approximateH(E,E′)within this energy region
by a constant, i.e.

H(E,E′) =−V=const. forE,E′∈[2εF, 2 εm] withεm=εF+~ωD. (320)