The Nature of the Cooper Pair - University of Liverpool

To show that a bound state exists, we need to solve the
Schrodinger’s equation. If V (r) is constant, the solution is
ψ(r) = sin(kr).
However, V (r) is not constant. We may still try a solution in
the form of a sum of sine functions:
Ψ(r) = �
ak sin(kr),
where a k are unknown constants that we must find.
k
When we study electrons in metal, we learn how k is quantised
by solving Schrodinger’s equation for a cube. Since we now
have spherical symmetry, we shall use a sphere of radius R
instead (not to be confused with the wavefunction R(r)).
Superconductivity 35