# Derivation of the gauge-independent relation between the phase and the electric field

The effective wavefunction and the charge current are given as:

--- (3.4)

---(3.13)

The wavefunction follows the Schrödinger equation:

--- (3.5)

Without losing the generality, we can take the Coulomb gauge:

--- (1)

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:

--- (2)

Also, we have:

By using (1) and (2), we also have:

(3.5) becomes:

--- (3)

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:

--- (3.16)

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:

we have:

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 .

By defining:

We have:

--- (4)

Hence, if we apply a constant voltage between two points, the phase difference grows as:

#### Note:

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.