Problems
225
so that s ˆ
2
|0› = −|0›. It follows that
k
θ
2
θ
3
exp(θ k s ˆ )
k |0› = (1 + θ k s ˆ
k
k −
k s ˆ . . .) 0
k
2
− 3
| ›
= (cosθ k + sin θ k s ˆ ) 0
k | ›
= (cosθ k + sin θ k c ˆ
† c ˆ
† ) 0
(17.139)
k −k
| ›
and hence
∏
|ground› BCS =
(cos θ k + sin θ k c ˆ
† c ˆ
† )
k −k
|0›.
(17.140)
k
As for the superfluid, (17.140) represents a coherent superposition of correlated pairs, with no restraint on the particle number.
We should emphasize that the above is only the barest outline of a simple
version of BCS theory, with no electromagnetic interactions, from which many
subtleties have been omitted. Consider, for example, the binding energy E b
of a pair, to calculate which one needs to evaluate the constant γ in (17.114).
To a good approximation one finds (see for example Enz 1992) E b ≈ 3Δ
2 /E F .
One can also calculate the approximate spatial extension of a pair, which is
denoted by the coherence length ξ and is of order v F /πΔ where k F = mv F
is the Fermi momentum. If we compare E b to the Coulomb repulsion at a
distance ξ we find
E b /(α/ξ) ∼ a 0 /ξ
(17.141)
where a 0 is the Bohr radius. Numerical values show that the right-hand side
of (17.141), in conventional superconductors, is of order 10
−3 . Hence the pairs
are not really bound, only correlated, and as many as 10
6 pairs may have their
centres of mass within one coherence length of each other. Nevertheless, the
simple theory presented here contains the essential features which underlie all
attempts to understand the dynamical occurrence of spontaneous symmetry
breaking in fermionic systems.
We now proceed to an important application in particle physics.
Problems
17.1 Verify (17.29).
17.2 Verify (17.35).
17.3 Derive (17.43) and (17.44).
17.4 Let
1
ˆ
U = exp[ λθ(ˆ a
2
− a ˆ
†2
λ
)]
2
where [a, ˆ a ˆ
† ] = 1 and λ, θ are real parameters.
225
so that s ˆ
2
|0› = −|0›. It follows that
k
θ
2
θ
3
exp(θ k s ˆ )
k |0› = (1 + θ k s ˆ
k
k −
k s ˆ . . .) 0
k
2
− 3
| ›
= (cosθ k + sin θ k s ˆ ) 0
k | ›
= (cosθ k + sin θ k c ˆ
† c ˆ
† ) 0
(17.139)
k −k
| ›
and hence
∏
|ground› BCS =
(cos θ k + sin θ k c ˆ
† c ˆ
† )
k −k
|0›.
(17.140)
k
As for the superfluid, (17.140) represents a coherent superposition of correlated pairs, with no restraint on the particle number.
We should emphasize that the above is only the barest outline of a simple
version of BCS theory, with no electromagnetic interactions, from which many
subtleties have been omitted. Consider, for example, the binding energy E b
of a pair, to calculate which one needs to evaluate the constant γ in (17.114).
To a good approximation one finds (see for example Enz 1992) E b ≈ 3Δ
2 /E F .
One can also calculate the approximate spatial extension of a pair, which is
denoted by the coherence length ξ and is of order v F /πΔ where k F = mv F
is the Fermi momentum. If we compare E b to the Coulomb repulsion at a
distance ξ we find
E b /(α/ξ) ∼ a 0 /ξ
(17.141)
where a 0 is the Bohr radius. Numerical values show that the right-hand side
of (17.141), in conventional superconductors, is of order 10
−3 . Hence the pairs
are not really bound, only correlated, and as many as 10
6 pairs may have their
centres of mass within one coherence length of each other. Nevertheless, the
simple theory presented here contains the essential features which underlie all
attempts to understand the dynamical occurrence of spontaneous symmetry
breaking in fermionic systems.
We now proceed to an important application in particle physics.
Problems
17.1 Verify (17.29).
17.2 Verify (17.35).
17.3 Derive (17.43) and (17.44).
17.4 Let
1
ˆ
U = exp[ λθ(ˆ a
2
− a ˆ
†2
λ
)]
2
where [a, ˆ a ˆ
† ] = 1 and λ, θ are real parameters.