Vector Potential

Potentials

Electric Field

From Gauss

(where is the electric field)

While from Faraday, in absence of magnetic field

which implies that the field is conservative, by Stokes

(since the curl is null, due to the arbitrariness of the circulation must be 0 given by every ; the converse holds due to the arbitrariness of , if the circulation is 0 then the curl must be identically null).

Given the famous vector identity for a scalar field

we can always express the field as the gradient of a scalar field .

Since we know that field is the potential, and physically it represents the work done by the field per unit charge against itself to bring the charge from infinity to a position , hence that work is negative, we conventionally put a negative sign in

is the scalar potential of .

In the non conservative case (presence of time varying magnetic field):

Magnetic Field

From Gauss

While from Ampere, in absence of electric field

Another famous vector identity for vector fields this time says

So we can always express as the curl of another vector field :

is called the vector potential of .

Time varying magnetic field

Now look at

This means the inner sum must be the gradient of a scalar field:

(where is for convention to make the original scalar potential)

so we have to account for the time evolution of the vector potential which creates a non conservative time varying electric field, nice.

Gauge Freedom

To understand gauge freedom we can make a comparison with integration and derivation: we know that the primitive of a function is not unique but it is defined up to a constant, since in derivation the constant disappears. The same is true for scalar potentials, we only care about potential differences:

(this is an example of how we could choose the constant, and we usually put )

So we can say that a scalar potential is defined up to a constant.

For vector potentials, they are defined up to a scalar field, since

Since the potentials are defined up to something, these fields should be the same

Now look at Faraday for the new fields

These are the relations between the vector and scalar potential when we shift the vector potential by a scalar field.

These choices are called gauges, and we are free to choose (gauge invariance).

Schrödinger equation

Expanding the Schrödinger equation for a free electron in an electric and magnetic field

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