I am stuck on a question in Chapter 11 of Advanced Solid State Physics by Philip Phillips, which asks to do the Cooper instability calculation for triplet pairing.
I attempt to solve the Schroedinger equation
$ [-\frac{\hbar^2}{2 m} (\nabla^2_1 + \nabla^2_2)+V(r_1 - r_2)] \psi(r_1,r_2) = E \psi(r_1,r_2)$
with a antisymmetric wavefunction $\psi$ following the steps in the textbook.
First express the wavefunction in center of mass coordinate
$\psi(r_1,r_2) = \phi(r) e^{i Q \cdot R},$
where $r = r_1 - r_2, Q = k_1 + k_2$ and $R = (r_1 + r_2)/2$.
Expand $\phi(r)$ in a Fourier series
$\phi(r) = \displaystyle\sum_{k} \frac{e^{i k \cdot r}}{\sqrt{V}} \alpha_k$.
Spatial antisymmetry then requires that $\alpha_k = - \alpha_{-k}$, but I don't know how this requirement changes the subsequent calculations.
Any explanation and hint are appreciated.