8.8 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Aharonov–Bohm effect | 5/5 | https://en.wikipedia.org/wiki/Aharonov–Bohm_effect | reference | science, encyclopedia | 2026-05-05T10:54:30.845762+00:00 | kb-cron |
The Aharonov–Bohm effect can be understood from the fact that one can only measure absolute values of the wave function. While this allows for measurement of phase differences through quantum interference experiments, there is no way to specify a wavefunction with constant absolute phase. In the absence of an electromagnetic field one can come close by declaring the eigenfunction of the momentum operator with zero momentum to be the function "1" (ignoring normalization problems) and specifying wave functions relative to this eigenfunction "1". In this representation the i-momentum operator is (up to a factor
ℏ
/
i
{\displaystyle \hbar /i}
) the differential operator
∂
i
=
∂
∂
x
i
{\displaystyle \partial _{i}={\frac {\partial }{\partial x^{i}}}}
. However, by gauge invariance, it is equally valid to declare the zero momentum eigenfunction to be
e
−
i
ϕ
(
x
)
{\displaystyle e^{-i\phi (x)}}
at the cost of representing the i-momentum operator (up to a factor) as
∇
i
=
∂
i
+
i
(
∂
i
ϕ
)
{\displaystyle \nabla _{i}=\partial _{i}+i(\partial _{i}\phi )}
i.e. with a pure gauge vector potential
A
=
d
ϕ
{\displaystyle A=d\phi }
. There is no real asymmetry because representing the former in terms of the latter is just as messy as representing the latter in terms of the former. This means that it is physically more natural to describe wave "functions", in the language of differential geometry, as sections in a complex line bundle with a hermitian metric and a U(1)-connection
∇
{\displaystyle \nabla }
. The curvature form of the connection,
i
F
=
∇
∧
∇
{\displaystyle iF=\nabla \wedge \nabla }
, is, up to the factor i, the Faraday tensor of the electromagnetic field strength. The Aharonov–Bohm effect is then a manifestation of the fact that a connection with zero curvature (i.e. flat), need not be trivial since it can have monodromy along a topologically nontrivial path fully contained in the zero curvature (i.e. field-free) region. By definition this means that sections that are parallelly translated along a topologically non trivial path pick up a phase, so that covariant constant sections cannot be defined over the whole field-free region. Given a trivialization of the line-bundle, a non-vanishing section, the U(1)-connection is given by the 1-form corresponding to the electromagnetic four-potential A as
∇
=
d
+
i
A
{\displaystyle \nabla =d+iA\,}
where d means exterior derivation on the Minkowski space. The monodromy is the holonomy of the flat connection. The holonomy of a connection, flat or non flat, around a closed loop
γ
{\displaystyle \gamma }
is
e
i
∫
γ
A
{\displaystyle e^{i\int _{\gamma }A}}
(one can show this does not depend on the trivialization but only on the connection). For a flat connection one can find a gauge transformation in any simply connected field free region(acting on wave functions and connections) that gauges away the vector potential. However, if the monodromy is nontrivial, there is no such gauge transformation for the whole outside region. In fact as a consequence of Stokes' theorem, the holonomy is determined by the magnetic flux through a surface
σ
{\displaystyle \sigma }
bounding the loop
γ
{\displaystyle \gamma }
, but such a surface may exist only if
σ
{\displaystyle \sigma }
passes through a region of non trivial field:
e
i
∫
∂
σ
A
=
e
i
∫
σ
d
A
=
e
i
∫
σ
F
{\displaystyle e^{i\int _{\partial \sigma }A}=e^{i\int _{\sigma }dA}=e^{i\int _{\sigma }F}}
The monodromy of the flat connection only depends on the topological type of the loop in the field free region (in fact on the loops homology class). The holonomy description is general, however, and works inside as well as outside the superconductor. Outside of the conducting tube containing the magnetic field, the field strength
F
=
0
{\displaystyle F=0}
. In other words, outside the tube the connection is flat, and the monodromy of the loop contained in the field-free region depends only on the winding number around the tube. The monodromy of the connection for a loop going round once (winding number 1) is the phase difference of a particle interfering by propagating left and right of the superconducting tube containing the magnetic field. If one wants to ignore the physics inside the superconductor and only describe the physics in the outside region, it becomes natural and mathematically convenient to describe the quantum electron by a section in a complex line bundle with an "external" flat connection
∇
{\displaystyle \nabla }
with monodromy
α
=
{\displaystyle \alpha =}
magnetic flux through the tube /
(
ℏ
/
e
)
{\displaystyle (\hbar /e)}
rather than an external EM field
F
{\displaystyle F}
. The Schrödinger equation readily generalizes to this situation by using the Laplacian of the connection for the (free) Hamiltonian
H
=
1
2
m
∇
∗
∇
{\displaystyle H={\frac {1}{2m}}\nabla ^{*}\nabla }
. Equivalently, one can work in two simply connected regions with cuts that pass from the tube towards or away from the detection screen. In each of these regions the ordinary free Schrödinger equations would have to be solved, but in passing from one region to the other, in only one of the two connected components of the intersection (effectively in only one of the slits) a monodromy factor
e
i
α
{\displaystyle e^{i\alpha }}
is picked up, which results in the shift in the interference pattern as one changes the flux. Effects with similar mathematical interpretation can be found in other fields. For example, in classical statistical physics, quantization of a molecular motor motion in a stochastic environment can be interpreted as an Aharonov–Bohm effect induced by a gauge field acting in the space of control parameters.
== See also ==
== References ==
== Further reading == D. J. Thouless (1998). "§2.2 Gauge invariance and the Aharonov–Bohm effect". Topological quantum numbers in nonrelativistic physics. World Scientific. pp. 18ff. ISBN 978-981-02-3025-8.
== External links == The David Bohm Society page about the Aharonov–Bohm effect.