8.6 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aharonov–Bohm effect | 3/5 | https://en.wikipedia.org/wiki/Aharonov–Bohm_effect | reference | science, encyclopedia | 2026-05-05T10:54:30.845762+00:00 | kb-cron |
In quantum mechanics the same particle can travel between two points by a variety of paths. Therefore, this phase difference can be observed by placing a solenoid between the slits of a double-slit experiment (or equivalent). An ideal solenoid (i.e. infinitely long and with a perfectly uniform current distribution) encloses a magnetic field
B
{\displaystyle \mathbf {B} }
, but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an electron) passing outside experiences no magnetic field
B
{\displaystyle \mathbf {B} }
. (This idealization simplifies the analysis but it's important to realize that the Aharonov–Bohm effect does not rely on it, provided the magnetic flux returns outside the electron paths, for example if one path goes through a toroidal solenoid and the other around it, and the solenoid is shielded so that it produces no external magnetic field.) However, there is a (curl-free) vector potential
A
{\displaystyle \mathbf {A} }
outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the solenoid current is turned on or off. This corresponds to an observable shift of the interference fringes on the observation plane. The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization occurs because the superconducting wave function must be single valued: its phase difference
Δ
φ
{\displaystyle \Delta \varphi }
around a closed loop must be an integer multiple of
2
π
{\displaystyle 2\pi }
(with the charge
q
=
2
e
{\displaystyle q=2e}
for the electron Cooper pairs), and thus the flux must be a multiple of
h
/
2
e
{\displaystyle h/2e}
. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by F. London in 1948 using a phenomenological model. The first claimed experimental confirmation was by Robert G. Chambers in 1960, in an electron interferometer with a magnetic field produced by a thin iron whisker, and other early work is summarized in Olariu and Popèscu (1984). However, subsequent authors questioned the validity of several of these early results because the electrons may not have been completely shielded from the magnetic fields. An early experiment in which an unambiguous Aharonov–Bohm effect was observed by completely excluding the magnetic field from the electron path (with the help of a superconducting film) was performed by Tonomura et al. in 1986. The effect's scope and application continues to expand. Webb et al. (1985) demonstrated Aharonov–Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986) and Imry & Webb (1989). Bachtold et al. (1999) detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004).
=== Monopoles and Dirac strings === The magnetic Aharonov–Bohm effect is also closely related to Dirac's argument that the existence of a magnetic monopole can be accommodated by the existing magnetic source-free Maxwell's equations if both electric and magnetic charges are quantized. A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as a Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole "charge" g. The Dirac string starts from, and terminates on, a magnetic monopole. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates charge-quantization. That is,
2
q
e
q
m
ℏ
c
{\displaystyle 2{\frac {q_{\text{e}}q_{\text{m}}}{\hbar c}}}
must be an integer (in cgs units) for any electric charge qe and magnetic charge qm. Like the electromagnetic potential A the Dirac string is not gauge invariant (it moves around with fixed endpoints under a gauge transformation) and so is also not directly measurable. An equivalent conclusion can be reached by considering a static magnetic field
B
=
q
M
r
2
ϕ
^
{\displaystyle \mathbf {B} ={\frac {q_{M}}{r^{2}}}\,{\hat {\boldsymbol {\phi }}}}
. It is impossible to construct a single vector potential A without a singularity whose curl leads to this B field. However, two vector potentials in different regions can be used to describe such a B field:
A
±
=
±
q
M
(
1
+
−
cos
θ
)
r
sin
θ
ϕ
^
{\displaystyle \mathbf {A} _{\pm }=\pm {\frac {q_{M}\!\left(1{\overset {-}{+}}\cos \theta \right)}{r\sin \theta }}\,{\hat {\boldsymbol {\phi }}}}
for the regions (+)
θ
<
π
−
ϵ
{\displaystyle \theta <\pi -\epsilon }
and (-)
θ
>
ϵ
{\displaystyle \theta >\epsilon }
. The gauge transformation
−
2
q
M
ϕ
{\displaystyle -2q_{M}\phi }
can be used to relate the two potentials, leading to the relation between wavefunctions:
ψ
(
I
I
)
=
exp
(
−
2
i
q
q
M
ϕ
ℏ
c
)
ψ
(
I
)
{\displaystyle \psi ^{(\mathrm {II} )}=\exp \!\left({\frac {-2iqq_{M}\phi }{\hbar c}}\right)\psi ^{(\mathrm {I} )}}
. Thus, we can conclude that each wavefunction must be single-valued as the gauge requires that the state ket expansion is unique with respect to the expansion of the position eigenkets. Thus,
2
q
q
M
ℏ
c
∈
Z
{\displaystyle {\frac {2qq_{M}}{\hbar c}}\in \mathbb {Z} }
and the magnetic charge is quantized.