Abstract
This paper states that the induced charge should not be neglected in the electric Aharonov-Bohm (A-B) effect. If the induced charge is taken into account, the interference pattern of the moving charge will not change with the potential difference between the two metal tubes. It means that the scalar potential itself can not affect the phase of the moving charge, and the true factor affecting the phase of the moving charge is the energy of the system including the moving charge and the induced charge.
In classical physics, the concept of “force” is the most important, all the phenomena can be explained by the forces acting on the objects. In classical electrodynamics, the Lorentz force acting on a charge is determined by the electric field E and the magnetic field B at the position of the charge. So, the electric field E and the magnetic field B are considered as more fundamental quantities than the scalar potential φ and the vector potential A. But, in quantum mechanics, what appear in the Schrödinger equation are the scalar potential φ and the vector potential A instead of E and B. So, some physicists asserted that the potential functions φ and A are more fundamental than E and B1. Just for this reason, Y. Aharonov and D. Bohm predicted a new effect named by their names later2. This new effect asserts that the phase of a moving charge will be changed by the potential functions φ and A, even though the charge always move in a region where both E and B are zero, but the φ and A are not zero. This effect includes the electric A-B effect and the magnetic A-B effect. The magnetic A-B effect has been studied extensively in both theory and experiments3,4,5,6,7,8,9,10,11,12. The existence of the magnetic A-B effect has been supported by some experiments3,5. The theoretical and experimental studies on the magnetic A-B effect before 1989 have been well reviewed in the Ref. 8. But the electric A-B effect was much less studied13,14,15,16,17,18. Some experiments13,14,15,16 attempted to observe the electric A-B effect, but none of them could completely avoid the classical force acting on the moving charge due to the magnetic or electric fields in the experiment. The quantitative experimental result about the influence of scale potential has not been reported. Recently, a new experimental method18 has been advised, in which the moving electron is replaced by the moving hydrogen ion in order to lower the speed of the moving charge.
This paper will focus on the electric A-B effect in theory, especially, the possible experiment with two metal tubes proposed in the seminal paper by Aharonov and Bohm. It is found that a very important factor was neglected in their paper2, which is the induced charge on the inner surfaces of the metal tubes. If this induced charge is taken into account, we will find that the phase of the moving charge will not change with the electric potential difference between two metal tubes. So, it is more convincing to state that the real factor affecting the phase of a moving charge is the energy of the system including the moving charge and the induced charge, but not the scalar potential itself. The similar conclusions about the magnetic A-B effect6,10,11,12 and its theoretical proof10 have also been proposed before.
The electric A-B effect with induced charges neglected
First, let us repeat the possible experiment proposed by Aharonov and Bohm to demonstrate the electric A-B effect. As depictured in Fig. 1, a coherent charge beam is split into two parts and each part enters into a separate long cylindrical metal tube. After the beams pass through the metal tubes, they are combined to interfere coherently at the screen. By means of time-determining electrical “shutters” the charge beam is divided into wave packets, the length of each wave packet is long compared with its wavelength but short compared with the length of the metal tube. To analysis this experiment in more detail, we suppose the moving charge enter into the metal tubes at t0, comes out from the tubes at t3, in addition, t0 < t1 < t2 < t3. During the time interval from t1 to t2, while the moving electron is well inside the tubes, an electric potential difference V0 is applied between these two tubes. For example, the tube 1 is always connected to the zero potential point, and the tube 2 is connected to an external voltage generator, which makes the electric potential of the tube 2 to alternate in time as following:
Figure 1. Schematic experiment to demonstrate the electric A-B effect.
U(t) represents the external voltage generator. The charge enters into the tubes at t = t0, the external voltage generator is switched on at t = t1, the external voltage generator is switched off at t = t2, then, the charge leave the tubes at t = t3.
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To keep the potentials of the two tubes being zero in t < t1 and t > t2, the metal tube 2 should also be connected with the zero potential point in these two time intervals. Otherwise, the collision of the ions and the charges accumulation will make the potential of the tube 2 uncontrollable.
Let’s discuss this problem in the following situations:
The first situation
The external voltage generator is switched off, 
and 
represent the wave functions of the parts passing through the tubes 1 and 2, respectively, which are unperturbed by the external electric potential; 
is the coordinate of the moving charge. The total wave function 
is:
 |
and 
are determined by the following equations:
 |
where 
.
The second situation
The external voltage generator is switched on. 
and 
represent the wave functions perturbed by the external electric potential. The total wave function 
is:
 |
and 
are determined by the following equations:
 |
Then, comparing the eq. (3) and (5), the wave functions 
and 
for the charge in the two beams are given by:
 |
When t > t2, the total wave function become:
 |
Comparing the equations (2) and (7), after the two beams come out from the tubes, an additional phase difference between these two beams appears in the second situation, which is
 |
So, when these two beams meet at the screen, the interference pattern will change with the value of 
.
The discussion above is quite similar to the original paper by Aharonov and Bohm2, which also appears in some modern quantum mechanics textbooks19. But, in the discussion above, no induced charge was taken into account. No induced charge appearing on the surfaces of the metal tubes means that the electric field of the moving charge exists in all the space. So, in these two situations above, the electric field of the moving charge is not zero in the interior of the metal, furthermore, this electric field can penetrate the metal tube and exists in the region outside the metal tubes. Obviously, the discussion above is not reliable in principle. The induced charges should be taken into account.
In classical electrodynamics, if a charge is placed outside a metal surface, the induced charges will appear on the metal surface, where the metal surface is idealized as a mathematical surface of zero thickness. In quantum mechanics, Ref. 20 showed that the induced charges appear in a 0.2 nm thick layer near the metal surface in reality, but, the potential energy between the external charge and the induced charges can be described to a good approximation by the results of classical electrodynamics.
As depicted in the Fig. 2 a charge q is placed in a cavity that is totally within the metal, and the outer surface of the metal is connected to the zero potential point. Suppose the total induced charges on the inner surface is 
. As long as there are free electrons in the metal, the electric field E at every point within the metal should be zero. As long as the Coulomb’s law holds, or in other words, the potential of a point charge q is 
, the Gauss’s law should hold too. So, if we draw a Gaussian surface G surrounding the cavity, for E = 0 everywhere on the Gaussian surface:
Figure 2. A charge is placed in a metal cavity, the induced charges will appear on the inner surface of metal cavity ensuring that the electric field E at every point within the metal is zero.
 |
Then, we have
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The electric A-B effect with influence of the induced charges
So, in experiments, once the moving charge q (suppose q = ±e) get into the metal tubes, an induced charge 
will appear on the inner surfaces of the metal tubes. (If the moving charge q is a proton, the induced charge 
will be an electron; If the moving charge q is an electron, then the induced charge 
will be a hole in the Fermi gas of the metal tube. The induced charge 
is located on the inner surfaces of the tubes.) At the same time, another induced charge 
will appear on the outer surfaces of the metal tubes. During the time interval from t0 to t1, both the tubes are connected to the zero potential point, so, the induced charge 
on the outer surfaces of the metal tubes will flow into the zero potential point. Therefore, only the induced charge 
on the inner surfaces of the tubes needs to be taken into account.
When the charge q moves in the metal tubes, for the induced charge 
appears on the inner surfaces of the tubes, the electric field of the moving charge q will be shielded by the induced charge 
, and the resultant electric field produced by the charges q and 
only exists in the region enclosed by the inner surface of the metal tubes. For the moving charge q and the induced charge 
attract each other, the coordinate y(y = y1, y2, y3) of the induced charge 
is dependent on the coordinate x(x = x1, x2, x3) of the moving charge q. Therefore, the wave function of the induced charge 
changes with the position of the moving charge q, at the same time, the wave function of the moving charge q is also perturbed by the induced charge 
21,22. So, both the moving charge q and the induced charge 
are not free particles, we should take the moving charge q and the induced charge 
as a system. Let 
and 
represent the wave functions of this system with the external voltage generator being switched off or on, respectively.
In the region outside the metal tube (i.e. t < t0 or t > t3), the moving charge is a free particle and has no relationship with the induced charges and the potential difference U(t) between the two tubes. So, we need only to discuss the revolution of the wave function of the charges q and 
in the region enclosed by the metal tubes, i.e. the time dependence of the wave function from t0 to t3.
So, the third situation: the external voltage generator is switched off, but, the induced charge 
is included. Then the total wave function is:
 |
where 
and 
represent the wave functions of the parts passing through the tubes 1 and 2, respectively, which are unperturbed by the external electric potential U(t). The wave functions 
and 
satisfy the following Schrödinger equation:
 |
where, t0 < t < t3; 
is the mass of the induced charge 
; ϕ(y) is the potential function experienced by the induced charge 
in the metal tubes, which ensures that the induced charge 
can only move in the interior of the metal and cannot leave out from the surfaces of the metal tubes, (noticing: ϕ(y) is independent of the potential difference U(t) between the two metal tubes.); 
; and 
is the interaction energy between the moving charge q and the induced charge 
.
The fourth situation: the external voltage generator is switched on, and the perturbed wave function 
of the system will become
 |
where 
and 
represent the wave functions perturbed by the external electric potential U(t). 
and 
satisfy the following Schrödinger equations:
 |
In the equations (14), qU(t) and 
are the potential energies of the moving charge q and the induced charge 
in the electric field U(t), respectively. For 
, the sum of 
and 
is zero i.e. the interaction energy between q and U(t) is completely counteracted by the interaction energy between 
and U(t). Therefore, the equations (14) will become:
 |
Now, the potential difference U(t) has disappeared from the eq. (15), i.e. the Schrödinger equations (15) are independent of the potential difference U(t), so, the wave functions 
and 
should also be independent of U(t). Comparing the equations (12) and (15), which are exactly same to each other, so, the following equations are obvious:
 |
In this situation, the total wave function is:
 |
For 
is the wave function of the charges q and 
with the external potential difference being U(t), but, 
is the wave function with the external potential difference being zero, 
means that the wave function will not change with the potential difference U(t). So, when the two beams meet at the screen, the interference pattern will not change with U(t), i.e. the phase shift as eq. (8) predicted by the electric A-B effect will not appear.
Discussions and Conclusions
Why does not the phase shift predicted by the electric A-B effect appear if the induced charge is included? Because, if there were no induced charge on the surfaces of the metal tubes, the electric field of the moving charge would exist in all the space, this field could penetrate the metal tubes and overlap with the electric field between the two tubes applied by the external voltage generator. While the moving charge q is in the tube 2, its potential energy is qU(t); while the moving charge q is in the tube 1, its potential energy is 0. According to quantum mechanics, the wave function of the moving charge is:
 |
where E1 and E2 are the energies of the parts passing through the tube 1 and 2, respectively. They are given by:
 |
where EK is the kinetic energy of the moving charge. Obviously 
, so, the time dependence of the 
is different with that of 
. Therefore, when these two beams come out from the tubes, a phase shift due to the potential difference U(t) between the tubes will appear, this is the electric A-B effect predicted by Aharonov and Bohm. This analysis is consistent with the eq. (8). The eq. (8) shows the phase shift Δφ is proportion to qV0, which represents the potential energy of the moving charge q. So, the eq. (8) strongly implies the phase shift Δφ arise from the interaction energy between the moving charge q and the electric field U(t)6.
But in a real experiment, when the moving charge q moves in the metal tubes, there must be an induced charge 
appearing on the inner surfaces of the metal tubes. Just for the appearance of the induced charge 
, the resultant electric field produced by the moving charge q and the induced charge 
can only exist in the region enclosed by the inner surfaces of the metal tubes. So, this resultant electric field cannot overlap with the electric field between the tubes. The potential energy (related to U(t)) of the system including the moving charge q and the induced charge 
is zero no matter the moving charge q is in the tube 1 or tube 2. The wave function of the system:
 |
where, E1 is the energy of the system with the moving charge passing through the tube 1, E2 is the energy of the system with the moving charge passing through the tube 2. E1 and E2 are given by:
 |
where, EK and 
are the kinetic energies of the moving charge q and the induced charge 
; the other quantities are defined as above.
For 
, so:
 |
Obviously,
 |
So, the time dependence of 
is same to that of 
. Therefore, when these two beams come out from the tubes, no phase shift due to the potential difference U(t) will appear. When these two parts meet at the screen, the interference pattern will not change with the potential difference U(t) between the two tubes.
In this situation, the potential difference U(t) between the two metal tubes still exists, but the phase shift due to U(t) does not appear. So, it is difficult to state the scalar potential can affect the phase of a moving charge; it is more reasonable to state that the real factor affecting the phase of a moving charge should be the potential energy of the system including the moving charge q and the induced charge q′.
Additional Information
How to cite this article: Wang, R.-F. Absence of the Electric Aharonov-Bohm Effect due to Induced Charges. Sci. Rep. 5, 14279; doi: 10.1038/srep14279 (2015).
Acknowledgments
This work is supported by “the Fundamental Research Funds for the Central Universities, No: 2009JBM102”.
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