The effective wavefunction and the charge current are given as:
The wavefunction follows the Schrödinger equation:
Without losing the generality, we can take the Coulomb gauge:
Now, we assume that the charge density is constant and uniform:
In this case, the currents should be divergence-free . Then, by applying on (3.13), we have:
Also, we have:
By using (1) and (2), we also have:
Since this is valid only on the Coulomb gauge, we will rewrite it as a gauge-independent expression.
The gauge-invariant phase is defined as:
As the RHS of (3.16) is gauge-invariant, we can take the Coulomb gauge and use (3) without breaking the gauge-invariance of LHS. By applying on (3.16), and assuming that the currents are uniform:
where we used the relationship between the electric field and potentials :
Because is gauge-invariant, the following relationship is gauge-independent:
Where is a voltage (electric potential difference) between the two points .
Hence, if we apply a constant voltage between two points, the phase difference grows as:
In the textbook, it says "We have to complete our derivation to regard more general conditions!" on p.27. In the derivation of (4), we used the following two assumptions:
- The charge density is constant and uniform (independent of ).
- The currents are uniform (independent of ).
These are the same assumptions used in the next section "3.7 Josephson Junctions". So we can use the relationship (4) in that section.