Aharonov-Bohm effect

The Aharonov-Bohm effect (according to David Bohm and Yakir Aharonov ) is a quantum mechanical phenomenon in which a magnetic field influences the interference of electron beams , although they are not in the classic range of influence that can be expected from . The main cause of the effect is that it is influenced by the magnetic vector potential and not by the magnetic field itself. ${\displaystyle {\vec {B}}}$ ${\displaystyle {\vec {B}}}$

The Aharonov-Bohm Effect was selected as one of the Seven Wonders in the Quantum World by New Scientist magazine .

Aharonov and Bohm published their work in 1959. Werner Ehrenberg and Raymond E. Siday were able to predict the effect as early as 1949. Apparently, however, Walter Franz presented the effect in 1939 - 20 years before Aharonov and Bohm - in a seminar of the Physical Society , Gauverein Ostland in Danzig.

experiment

Schematic representation of the experiment. The electrons pass two gaps and form an interference pattern on the observation screen. This interference pattern changes depending on whether the magnetic field is switched on or off.

In the experiment, charged particles ( electrons ) run past a cylinder on different sides in which there is a magnetic field . The cylinder is surrounded by a wall that cannot be penetrated by the particles; outside the magnetic field is zero. Nevertheless, the outcome of the experiment depends on whether the magnetic field is switched on or off, because in the first case the vector potential is also present outside the cylinder. Imagine a radial vector potential. Its rotation and thus the magnetic field is zero outside of the cylinder, but the vector potential itself is nowhere zero. ${\displaystyle B}$ ${\displaystyle {\vec {A}}}$ ${\displaystyle \mathrm {rot} \,{\vec {A}}}$

The superposition of the wave functions behind the cylinder results in an interference pattern that is influenced by the vector potential, since the wave functions receive a different phase shift on paths to the right and left of the cylinder .

Experiments of this kind were made in the early 1960s a. a. carried out by Möllenstedt and Robert G. Chambers .

theory

Classically, a charged particle in a magnetic field is influenced by the Lorentz force of the magnetic field, according to the equation of motion :

${\displaystyle m\,{\vec {a}}=q\cdot \left({\vec {v}}\times {\vec {B}}+{\vec {E}}\right)}$

With

• the mass of the particle ${\displaystyle m}$
• its acceleration ${\displaystyle {\vec {a}}}$
• its electrical charge ${\displaystyle q}$
• its speed ${\displaystyle {\vec {v}}}$
• the magnetic flux density ${\displaystyle {\vec {B}}}$
• the vector product . ${\displaystyle \times }$

Classically, an effect can only be expected where the magnetic field is different from zero (apart from the electric field , which is insignificant here). ${\displaystyle {\vec {B}}}$ ${\displaystyle {\vec {E}}}$

In quantum mechanics, on the other hand, the behavior of the particle is described by the Hamilton operator :

${\displaystyle H={\frac {1}{2m}}\left({\vec {p}}-q{\vec {A}}({\vec {r}},t)\right)^{2}+q\,\Phi ({\vec {r}},t).}$

With

• the canonical momentum operator ${\displaystyle {\vec {p}}={\tfrac {\hbar }{i}}{\vec {\nabla }}}$
• the kinetic momentum operator ${\displaystyle {\vec {\Pi }}={\vec {p}}-q{\vec {A}}({\vec {r}},t)}$
• the vector potential ${\displaystyle {\vec {A}}}$
• the place ${\displaystyle {\vec {r}}}$
• the time  t
• the scalar electrical potential , which is insignificant here. ${\displaystyle \Phi }$

Vector potential and magnetic field are related by the rotation operator: ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {B}}}$

${\displaystyle {\vec {B}}=\mathrm {rot} \,{\vec {A}}:={\vec {\nabla }}\times {\vec {A}}}$

As a result, the vector potential is generally only determined up to the gradient of any scalar function , since the rotation of a gradient field disappears for scalar fields that are continuously differentiable twice (see calibration transformation ). ${\displaystyle {\vec {A}}}$ ${\displaystyle {\vec {\nabla }}f}$ ${\displaystyle f}$

interpretation

Sometimes the conclusion is drawn from the effect that the vector potential has a more fundamental meaning in quantum mechanics than the associated force field . However, this does not apply to the essentials: Ultimately, the magnetic flux is decisive, which can be expressed by a curve integral : ${\displaystyle \Phi _{B}}$

${\displaystyle \Phi _{B}(F)=\oint _{\!\!\!\Gamma }{\vec {A}}\cdot \mathrm {d} {\vec {r}}}$

The integration path must be closed , which is indicated by the circle in the integration symbol, but may also be outside the area . ${\displaystyle \Gamma }$ ${\displaystyle {\vec {B}}\neq 0}$

According to Stokes' theorem

${\displaystyle \oint _{\!\!\!\Gamma }{\vec {A}}\cdot \mathrm {d} {\vec {r}}=\iint _{F}(\operatorname {rot} {\vec {A}})\cdot {\vec {n}}\,\mathrm {d} ^{2}a}$

With

• the normal vector on the surface ${\displaystyle {\vec {n}}}$
• the two-dimensional surface element . ${\displaystyle \mathrm {d} ^{2}a}$

the line integral over the closed curve is identical to the flow of the magnetic flux density through the enclosed area : ${\displaystyle \Gamma }$ ${\displaystyle B}$ ${\displaystyle F}$

${\displaystyle \oint _{\!\!\!\Gamma }{\vec {A}}\cdot \mathrm {d} {\vec {r}}=\iint _{F}{\vec {B}}\cdot {\vec {n}}\,\mathrm {d} ^{2}a}$

In particular, Stokes' theorem shows why the selected calibration of the vector potential is irrelevant, since the curve integral can be written over as an area integral over and the rotation of the gradient field used for calibration disappears. ${\displaystyle {\vec {A}}}$ ${\displaystyle \operatorname {rot} {\vec {A}}}$

The effect can be interpreted as a consequence of the nontrivial topology of the calibration field: Because the space is not simply connected (the cylinder interior is "a hole in space"), the path integrals also do not disappear over closed curves .

literature

• Franz Schwabl : Quantum Mechanics (QM I) , Springer 2004, ISBN 3-540-43106-3 (chapter 7.5)
• Yakir Aharonov, David Bohm: Significance of Electromagnetic Potentials in the Quantum Theory. In: The Physical Review. 115, No. 3, 1959, pp. 485-491.
• Yakir Aharonov, David Bohm: Further Considerations on Electromagnetic Potentials in the Quantum Theory. In: The Physical Review. 123, No. 4, 1961, pp. 1511-1524.
• G. Möllenstedt, W. Bayh: Measurement of the continuous phase shift of electron waves in a space free of force fields by the magnetic vector potential of an air core coil . In: The natural sciences . 49, 1962, p. 81.
• Yoseph Imry, Richard A. Webb: Quantum Interference and the Aharonov-Bohm Effect. In: Scientific American. 260, No. 4, 1989, p. 56.
• M. Peshkin, A. Tonomura The Aharonov-Bohm effect , Springer Verlag 1989

Individual evidence

1. ^ Seven wonders of the quantum world , newscientist.com
2. ^ Ehrenberg, Siday The Refractive Index in Electron Optics and the Principles of Dynamics , Proceedings of the Physical Society B, Volume 62, 1949, pp. 8-21. Aharonov and Bohm did not find out about this work until after its publication and pointed out in their 1961 essay.
3. ↑ Hiley, BJ (2013): The Early History of the Aharonov-Bohm Effect . arxiv : 1304.4736 .
4. ↑ Typical experimental setup
5. ↑ Typical shift of the interference pattern ( Memento from April 26, 2016 in the Internet Archive ) (pdf; 26 kB)
6. ↑ Chambers Shift of an Electron Interference Pattern by Enclosed Magnetic Flux , Physical Review Letters, Volume 5 1960, pp. 3-5
7. ↑ The SI system is used here; in the CGS system would continuously and through and be replaced. ${\displaystyle q{\vec {A}}}$ ${\displaystyle q{\vec {B}}}$ ${\displaystyle q{\vec {A}}/c}$ ${\displaystyle q{\vec {B}}/c}$
8. ^ C. Nash, Bohm-Aharonov Effect, Encyclopedia of Mathematics, Springer