Effective wavefunction describing the flow of superconducting elections (Cooper pairs):
Schrödinger equation for :
I will show that this model is invariant under the gauge transformation:
(3.5) is equivalent to
A simple calculation shows the following commutation relations.
Hence we have:
By using (2), we have:
Similarly, by using (3), we have:
Using (3) again:
From (4) and (5), (1) is equivalent to:
Now we can show that the phase defined by the following equation is gauge-invariant:
First, in terms of the phase , the gauge transformation of can be written as:
So, by combining with the gauge transformation of , the combination becomes gauge-invariant. By integrating it through a path from to , the following combination gives a gauge-invariant phase up to a constant .
By setting , we get the expression (3.16)