5.2 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aharonov–Bohm effect | 2/5 | https://en.wikipedia.org/wiki/Aharonov–Bohm_effect | reference | science, encyclopedia | 2026-05-05T10:54:30.845762+00:00 | kb-cron |
=== Global action vs. local forces === Similarly, the Aharonov–Bohm effect illustrates that the Lagrangian approach to dynamics, based on energies, is not just a computational aid to the Newtonian approach, based on forces. Thus the Aharonov–Bohm effect validates the view that forces are an incomplete way to formulate physics, and potential energies must be used instead. In fact Richard Feynman complained that he had been taught electromagnetism from the perspective of electromagnetic fields, and he wished later in life he had been taught to think in terms of the electromagnetic potential instead, as this would be more fundamental. In Feynman's path-integral view of dynamics, the potential field directly changes the phase of an electron wave function, and it is these changes in phase that lead to measurable quantities.
=== Locality of electromagnetic effects === The Aharonov–Bohm effect shows that the local E and B fields do not contain full information about the electromagnetic field, and the electromagnetic four-potential, (Φ, A), must be used instead. By Stokes' theorem, the magnitude of the Aharonov–Bohm effect can be calculated using the electromagnetic fields alone, or using the four-potential alone. But when using just the electromagnetic fields, the effect depends on the field values in a region from which the test particle is excluded. In contrast, when using just the four-potential, the effect only depends on the potential in the region where the test particle is allowed. Therefore, one must either abandon the principle of locality, which most physicists are reluctant to do, or accept that the electromagnetic four-potential offers a more complete description of electromagnetism than the electric and magnetic fields can. On the other hand, the Aharonov–Bohm effect is crucially quantum mechanical; quantum mechanics is well known to feature non-local effects (albeit still disallowing superluminal communication), and Vaidman has argued that this is just a non-local quantum effect in a different form. In classical electromagnetism the two descriptions were equivalent. With the addition of quantum theory, though, the electromagnetic potentials Φ and A are seen as being more fundamental. Despite this, all observable effects end up being expressible in terms of the electromagnetic fields, E and B. This is interesting because, while you can calculate the electromagnetic field from the four-potential, due to gauge freedom the reverse is not true.
== Magnetic solenoid effect == The magnetic Aharonov–Bohm effect can be seen as a result of the requirement that quantum physics must be invariant with respect to the gauge choice for the electromagnetic potential, of which the magnetic vector potential
A
{\displaystyle \mathbf {A} }
forms part. Electromagnetic theory implies that a particle with electric charge
q
{\displaystyle q}
traveling along some path
P
{\displaystyle P}
in a region with zero magnetic field
B
{\displaystyle \mathbf {B} }
, but non-zero
A
{\displaystyle \mathbf {A} }
(by
B
=
0
=
∇
×
A
{\displaystyle \mathbf {B} =\mathbf {0} =\nabla \times \mathbf {A} }
), acquires a phase shift
φ
{\displaystyle \varphi }
, given in SI units by
φ
=
q
ℏ
∫
P
A
⋅
d
x
,
{\displaystyle \varphi ={\frac {q}{\hbar }}\int _{P}\mathbf {A} \cdot d\mathbf {x} ,}
Therefore, particles, with the same start and end points, but traveling along two different routes will acquire a phase difference
Δ
φ
{\displaystyle \Delta \varphi }
determined by the magnetic flux
Φ
B
{\displaystyle \Phi _{B}}
through the area between the paths (via Stokes' theorem and
∇
×
A
=
B
{\displaystyle \nabla \times \mathbf {A} =\mathbf {B} }
), and given by:
Δ
φ
=
q
Φ
B
ℏ
.
{\displaystyle \Delta \varphi ={\frac {q\,\Phi _{B}}{\hbar }}.}