Physics:Quantum Aharonov-Bohm effect - Revision history

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	== See also ==		== See also ==
	* [[Physics:Quantum Berry phase]]		{{#invoke:PhysicsQC\|tocHeadingAndList\|Physics:Quantum basics/See also}}
	* [[Physics:~~Quantum Aharonov-Casher effect]]~~
	* [[Physics:Quantum electromagnetic field]]
	* [[Physics:Quantum gauge field]]
	* [[Physics:Quantum ~~Wavefunction]]~~

	== References ==		== References ==

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	\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]		\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]
	\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged ~~particles are~~ affected by electromagnetic potentials even ~~in regions~~ where the classical electric and magnetic fields ~~vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]]~~.<ref>{{~~cite~~ web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect ~~\|title=Aharonov-Bohm effect~~ \|publisher=Wikipedia \|access-date=~~20 May~~ 2026}}</ref>		\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum-mechanical phenomenon in which an electrically charged particle can be affected by electromagnetic potentials even while moving through a region where the classical electric and magnetic fields are zero.<ref>{{Cite web \|title=Aharonov-Bohm effect \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|publisher=Wikipedia \|access-date=2026-05-23}}</ref>

	~~In the magnetic version, an~~ electron ~~wave is split into two paths~~ around a confined magnetic flux. The electrons ~~travel~~ through field~~-free regions~~, but the vector potential ~~changes their relative phase~~. When the paths recombine, the interference pattern shifts ~~according to the enclosed magnetic flux~~.		The effect is most often illustrated with electron interference around a confined magnetic flux. The electrons do not pass through the magnetic field, but their wavefunction phases depend on the vector potential associated with the enclosed flux. When the two paths recombine, the interference pattern shifts.
	}}		}}
	~~== Phase and potentials ==~~
	Classically, only electric and magnetic fields exert forces on charged particles. In quantum theory, the phase of the wavefunction can respond to the vector potential itself. This makes the effect an important example of the phase sensitivity of quantum amplitudes.

	The ~~effect also connects naturally with [[Physics~~:Quantum ~~Berry~~ phase~~\|geometric~~ phase]], gauge ~~fields~~, and ~~topological descriptions of~~ quantum ~~systems~~.		== Magnetic Aharonov-Bohm effect ==
			In the standard magnetic version, a coherent electron beam is split into two paths that pass on opposite sides of a long solenoid. The magnetic field is confined inside the solenoid, so the electrons travel in a region where the classical Lorentz force is zero. Nevertheless, the vector potential changes the relative phase of the electron wavefunctions.

			The phase shift around a closed path is proportional to the magnetic flux enclosed by the path:

			<math display="block">\Delta \phi = {q \over \hbar}\oint \mathbf{A}\cdot d\mathbf{l} = {q\Phi_B \over \hbar}.</math>

			Here <math>q</math> is the particle charge, <math>\mathbf{A}</math> is the vector potential, and <math>\Phi_B</math> is the enclosed magnetic flux. The observable consequence is a shift in the interference fringes.

			== Potentials and phase ==
			Classical electromagnetism is often described in terms of the electric and magnetic fields, because those fields determine the local force on a charged particle. Quantum mechanics is more phase-sensitive. The wavefunction phase couples directly to the electromagnetic potentials, and interference can reveal this coupling even when the fields vanish along the particle paths.

			This does not make the vector potential an arbitrary mathematical decoration. In quantum theory, gauge potentials help determine phase relations. Gauge transformations can change the local expression for the potential, but they do not change the measured phase shift around a closed loop.

			== Electric version ==
			There is also an electric Aharonov-Bohm effect, in which a scalar potential changes the phase of a charged particle even when the particle is not exposed to a classical electric field along its path. The electric and magnetic forms both show that quantum phases can encode information about potentials in field-free regions.

	== Significance ==		== Significance ==
	The Aharonov-Bohm effect is important in electron interferometry, mesoscopic physics, superconducting circuits, and discussions of gauge invariance. It shows that the quantum description cannot always be reduced to local classical forces acting along a path.		The Aharonov-Bohm effect is important for several reasons:

	~~It also~~ provides a ~~clean conceptual bridge between~~ [[Physics:Quantum ~~Wavefunction~~\|~~wavefunctions~~]], ~~electromagnetic potentials, and [[Physics:Quantum gauge field\|~~gauge fields]].		* it demonstrates the physical significance of electromagnetic potentials in quantum mechanics;
			* it connects interference, gauge invariance, and topology;
			* it provides a clear example of how global path information can affect a local measurement;
			* it underlies many ideas in mesoscopic physics, superconducting circuits, and topological quantum systems.

			The effect is closely related to [[Physics:Quantum Berry phase\|Berry phase]] and other geometric phases. In both cases, the phase acquired by a quantum state depends on a path in a space of parameters or gauge fields, not merely on local forces along the path.

	== See also ==		== See also ==
	* [[Physics:Quantum Wavefunction]]
	* [[Physics:Quantum Berry phase]]		* [[Physics:Quantum Berry phase]]
			* [[Physics:Quantum Aharonov-Casher effect]]
			* [[Physics:Quantum electromagnetic field]]
	* [[Physics:Quantum gauge field]]		* [[Physics:Quantum gauge field]]
	* [[Physics:Quantum ~~electromagnetic field~~]]		* [[Physics:Quantum Wavefunction]]

	== References ==		== References ==

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	{{Short description\|Quantum phase effect caused by electromagnetic potentials}}		{{Short description\|Quantum phase effect caused by electromagnetic potentials}}
			{{Quantum book backlink\|Quantum dynamics and evolution}}
			{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum Aharonov-Casher effect\|next label=Aharonov-Casher effect}}
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	~~{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum Aharonov-Casher effect\|next label=Aharonov-Casher effect}}~~
	\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]		\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]
	\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>		\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>
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	In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.		In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.
	}}		}}

	== Phase and potentials ==		== Phase and potentials ==
	Classically, only electric and magnetic fields exert forces on charged particles. In quantum theory, the phase of the wavefunction can respond to the vector potential itself. This makes the effect an important example of the phase sensitivity of quantum amplitudes.		Classically, only electric and magnetic fields exert forces on charged particles. In quantum theory, the phase of the wavefunction can respond to the vector potential itself. This makes the effect an important example of the phase sensitivity of quantum amplitudes.

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	{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum ~~Scattering theory~~\|next label=~~Scattering theory~~}}		{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum Aharonov-Casher effect\|next label=Aharonov-Casher effect}}
	\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]		\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]
	\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>		\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

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	{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum Scattering theory\|next label=Scattering theory}}		{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum Scattering theory\|next label=Scattering theory}}
			\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]
	[[File:Quantum_aharonov_bohm_effect_yellow.png\|~~thumb\|right\|320px~~\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]		\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.		In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.
			}}

	== Phase and potentials ==		== Phase and potentials ==

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	~~<div style="display~~:~~flex; gap:24px; align~~-~~items:flex-start; max-width:1280px; clear:both;">~~		[[File:Quantum_aharonov_bohm_effect_yellow.png\|thumb\|right\|320px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]
	~~<div style="flex:1 1 680px; min-width:0; line-height:1.55;">~~
	The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>		The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.		In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.
	~~</div>~~
	~~<div style="flex:0 0 380px; max-width:38%;">~~
	~~[[File:Quantum_aharonov_bohm_effect_yellow.png\|thumb\|360px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]~~
	~~</div>~~
	~~</div>~~

	~~<div style="clear:both;"></div>~~

	== Phase and potentials ==		== Phase and potentials ==

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	~~[[File~~:~~Quantum_aharonov_bohm_effect_yellow.png\|thumb\|280px\|Aharonov~~-~~Bohm phase shift around a shielded magnetic flux~~.]]		<div style="display:flex; gap:24px; align-items:flex-start; max-width:1280px; clear:both;">
			<div style="flex:1 1 680px; min-width:0; line-height:1.55;">
	The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>		The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>

	In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.		In the magnetic version, an electron wave is split into two paths around a confined magnetic flux. The electrons travel through field-free regions, but the vector potential changes their relative phase. When the paths recombine, the interference pattern shifts according to the enclosed magnetic flux.
			</div>
			<div style="flex:0 0 380px; max-width:38%;">
			[[File:Quantum_aharonov_bohm_effect_yellow.png\|thumb\|360px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]
			</div>
			</div>

			<div style="clear:both;"></div>

	== Phase and potentials ==		== Phase and potentials ==

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	\|backlink={{Quantum book backlink\|Quantum dynamics and evolution}}		\|backlink={{Quantum book backlink\|Quantum dynamics and evolution}}
	{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum ~~Scattering theory~~\|next label=~~Scattering theory~~}}		{{Quantum article nav\|previous=Physics:Quantum Berry phase\|previous label=Berry phase\|next=Physics:Quantum Aharonov-Casher effect\|next label=Aharonov-Casher effect}}
	\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]		\|image=[[File:Quantum_aharonov_bohm_effect_yellow.png\|430px\|Aharonov-Bohm phase shift around a shielded magnetic flux.]]
	\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>		\|text=The '''Quantum Aharonov-Bohm effect''' is a quantum phenomenon in which charged particles are affected by electromagnetic potentials even in regions where the classical electric and magnetic fields vanish. It demonstrates that potentials can have direct physical significance in [[Physics:Quantum mechanics\|quantum mechanics]].<ref>{{cite web \|url=https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect \|title=Aharonov-Bohm effect \|publisher=Wikipedia \|access-date=20 May 2026}}</ref>