Self Assessment Quiz

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 ?

2. For the Cooper pair wavefunction show that
    $\displaystyle g_{\sigma}(\vec{k})=-g_{-\sigma}(-\vec{k})

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

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 ?

5. In order to obtain an effective electron-electron interaction from the electron-phonon interaction we have carried out a canonical transformation
    $\displaystyle \tilde{H}=e^{-S}He^{S}
    where we require $ S^{+}=-S.$ Why?

6. Show that for a coherent state
    $\displaystyle <\alpha\vert(a^{+})^{p}a^{q}\vert\alpha>=(\alpha*)^{p}\alpha^{q}

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

8. Does the BCS wavefunction $ \vert\psi_{BCS}>$ contain a fixed number of particles? If not then what does it mean to have $ <\psi_{BCS}\vert\psi_{BCS}>=1$ ?

9. At $ T=0$, what are the values of $ u_{\vec{k}}^{0}$ and $ v_{\vec{k}}^{0}$ for $ \vec{k}<\vec{k}_{F}$ and $ \vec{k}>\vec{k}_{F}$ ?

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 ?

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.

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

13. For $ \Delta=0,$ interpret the Bogoliubov-Valatin operators.