## Thursday, September 16, 2021

### The Electric Potential - Is It Really Unphysical?

Illustration of magnetic and electrical Ahranov-Bohm Effects.
(From Physics Today,  August, p. 63,  cf.  Vavagiakis et al)

In a previous (June 16) post we saw that the electrical potential is defined as the potential at a point A when work is done per unit positive charge in bringing a charge from infinity  (¥ ) to A, expressed:

V = W/ q =  Q / 4p eo

Although Maxwell’s equations e.g.

i)  Ñ X (E) J    + D / t

ii)             Ñ X E  - B / t

iii)         Ñ ·0

iv)          Ñ ·r

Contain only electric  (E ) and magnetic (B, H)  fields, they can also be expressed in terms of electric and magnetic potentials.  In this post we focus on these two potentials and attempt to discern which is real and physical – and which isn’t.  This was originally examined by Vavagiakis et al in the Quick Study section of Physics Today (Physics Today August, p. 62,  by Vavagiakis et al

Technically, as the authors note these potentials have no real significance.  The basic physics is contained in the forces on charged particles e.g. for the E-field

F = Q q/ 4p eo r2

And the spatial and temporal variations of the potential determine their field strength. For example the electric field strength:

E = F/q

Or: E = 1/q { Q q/ 4p eo r2

The main point is the forces are directly proportional to the field strengths and vanish where the field strengths vanish. Physicists are at liberty to add terms to the potentials that leave the fields invariant.

Few undergrads know that Maxwell’s classical field equations regarded the potentials as non physical.  This view was revised with the development of quantum mechanics. For example, as previously noted (see my post of Sept. 14th) the Heisenberg uncertainty principle is incompatible with the notion of a point particle. It is impossible to precisely fix the position of an electron in a hydrogen atom, for example, without losing all information about its momentum.

For this reason, QM replaced point particles with wave functions, e.g.

y(x)=  | y(x) |  exp i  qx

Defined by an amplitude | y(x) |   and a phase qx.(Note the phase is irrelevant in classical mechanics, demonstrated when one works out the Heisenberg principle for a classical object, say bullet).   The squared amplitude | y(x) | 2   meanwhile is the probability density of finding the particle at position  x, more completely written:

P ab  =     ò b a   | y(x) | 2  dx

Where the location must be between points a and b.

Why the need to include phases for charged particles?  Well, because in the modern format we include quantum mechanics when examining electric and magnetic potentials and in QM the phase of a particle is fundamentally connected to these  - specifically the magnetic potential A and the electrical potential  f.

The phase of a particle of charge q  over traversing a spatial trajectory  g  over a time interval t  changes by (op. cit.):

D q = q/ h  ( ò g ds · A  - ò  t    dt’ f)

Relative to the phase of a particle that traverses the same trajectory with vanishing potentials.  Because differences in phases can be observed in quantum physics experiments the equation has consequences for the physical significance of the potentials,  A  and f . In other words they are supposed to be real, physical quantities.  But are they?   The Quick Study essay in  Physics Today  (August, p. 62)  by Vavagiakis et al questions that assumption.

Along with the preceding they ask us to consider the top left graphic shown, which depicts a copper solenoid which generates a magnetic field B inside it.   That field is related to the magnetic potential through its curl, i.e.

B  =   Ñ X A

At the same time it prevents the field from existing outside it. And yet, when an electron beam (blue)  splits and passes around the solenoid, the respective electrons in each path interfere and exhibit a phase difference.   This is evidence of the magnetic potential's influence, i.e. that it is a physical existent.  When the wavefunctions interfere on the far (right) side of the solenoid they still have a total phase difference of:

q = q f(B)/ h

Where f(B) is the magnetic flux in the solenoid.  This result is called the Ahronov -Bohm effect.

But is there an electric Ahranov-Bohm effect? In this case consider two identically-charged particle passing through charged metallic tubes (Faraday "cages") as depicted in the right top graphic.   The object here is to induce an electric potential difference between the two particles but without encountering any E-fields. To obtain this effect a voltage  D is applied between the tubes/cages for a time Dwith the particles inside the tubes.  What do we find? Well, the E-field ≠   0  only outside the tubes. Also, limited to the time the particles are inside them. One finds the following phase- voltage relationship whereby the particles acquire a relative phase shift (ibid.):

D q = q/ c   x    DV/ m V   x  DT/ h s

This equation describes the electric version of the Ahranov -Bohm effect, i.e. in which the electric potential determines the evolution of the phase - say as a charged particle traverses a time DT.  (The units of microvolts, m V and nanosconds are included to give an appreciation of the scale in changing the electron's phase.)

Here's the kicker:  an observation of a phase shift would be a direct confirmation that the electric potential is physical.  However, up to now no such observation has ever been made. The further ramifications of this and especially the experimental impediments will be explored in a future post.

The E-field is non-zero only outside the cages and only during the time the particles are inside them. According to the phase- voltage relationship the particles acquire a relative phase shift.