1. Using electron creation and annihilation operators, define Cooper
pair creation and annihilation operators. Find their commutation relations.
Do they satisfy all the fundamental commutation relations of Bosons ?

We can define creation and annihilation operators
and
using the electron creation and annihilation operators
and
as follows

We note the following fundamental commutation relations between creation
and annihilation operators for Bosons and Fermions,

0

That is, Fermion creation operators anticommute. Similarly, the Fermion
annihilation operators anticommute,

0

The third commutation relation is between the creation and the annihilation
operators ,

Using the above anticommutation relations, we have for the Cooper
pair operators the following three relations,
Thus, the Cooper pair creation and annihilation operators are not
quite Bosons either. They are also not genuine Fermions!

2. For the Cooper pair wavefunction show that

We write the Cooper pair wavefunction as

Relabeling, we get
Using the anticommutation rule,
,
we have
Thus, we must have

3. Can the Cooper pair binding energy be expanded in a power series in
the strength of the interaction ? If not, then what does it imply ?

No. The result cannot be obtatained from the perturbation theory.

4. What is the essential difference between the bound states of two electrons
above the Fermi sea and another in the absence of a Fermi sea ?

In the absence of a Fermi sea the density of states varies as the
square root of energy, and a minimum strength of interaction is needed
before the two electrons can form a bound state.

5. In order to obtain an effective electron-electron interaction from
the electron-phonon interaction we have carried out a canonical transformation
where we require Why?

represents a unitary transformation. This implies that
So that,
One can check that our transformation satisfies the above constraint.

6. Show that for a coherent state

It can easily be checked by noting that

7. Is the BCS wavefunction an eignefunction of the superconducting Hamiltonian ?

No. It is a variational wavefunction which is used to obtain an estimate
of the ground state energy using the superconducting Hamiltonian.

8. Does the BCS wavefunction
contain a fixed number of
particles? If not then what does it mean to have
?

It does not contain a fixed number of particles.

9. At , what are the values of
and
for
and
?

The values are
and
for
and
and

10. We have shown that the phase and the number of particle operator are
dynamically conjugate variables like position and momentum. Do they
satisfy some uncertainty relation similar to that of position and
momentum ?

Yes. They satisfy the uncertainty relation given by

11. In the variational determination of the energy of the superconducting
state how do we include the constraint that the number of particles
is fixed.

We use the method of Lagrange's multipliers to ensure that the number
of particles is fixed in the calculation.

12. What is the minimum energy needed to create a single particle excitation
in the BCS ground state ?

13. For interpret the Bogoliubov-Valatin operators.

For , the Bogoliubov-Valatin operators become
and
when
the Bogoliubov-Valatin operators contain a mixture
of electron-like,
, and hole-like,
,
operators.