Aharonov-Bohm Effect
Kevin Lucht
Outline
• Experimental Setup
• Aharonov-Bohm Effect
• Berry Phases
Goal
• The potential fields in quantum mechanics are physically relevant
What is it?
• For dynamics we consider fields (𝐸 or 𝐵) real
𝑚𝑎⃗ = 𝑞(𝐸 + 𝑣×𝐵)
⃗
⃗
• Potential fields are auxiliary (𝜙 or 𝐴)
𝐸 = −∇𝜙, 𝐵 = ∇×𝐴⃗
Aharonov and Bohm showed in quantum mechanics both are physical
• Without a field 𝐵, non-zero 𝐴⃗ affects
https://www.feynmanlectures.caltech.edu/II_15.html#Ch15-S5difference
Wave Exposition
Consider a double-slit with light
being emitted.
Two plane waves:
𝜓! (𝑟, 𝑡) = 𝐴! 𝑒 "($! % &'()
𝜓* (𝑟, 𝑡) = 𝐴* 𝑒 "($" % &'()
Superposition: 𝜓 𝑟 = 𝜓! 𝑟, 𝑡 + 𝜓* (𝑟, 𝑡)
Intensity at detector is
𝐼 = ∫ 𝜓 𝑟, 𝑡 𝜓 ∗ (𝑟, 𝑡)𝑑𝑡 ∝ 𝐴!* + 𝐴** + 2𝐴! 𝐴* cos(𝜙! − 𝜙* )
Interference is given by difference in phase 𝜙! − 𝜙*
https://www.feynmanlectures.caltech.edu/II_15.html#Ch15-S5difference
Solenoid Example
Consider single electron split at A.
With planewave solution of
𝜓! (𝑟) = 𝐴𝑒 "#! $
𝜓% 𝑟 = 𝐶𝑒 "#" $
, 𝜓 𝑟 = 𝜓! 𝑟 + 𝜓% (𝑟)
At the detector at F, interference given by 𝜙! − 𝜙%
Interference depends on the vector potential from the solenoid.
Aharonov and D Bohm. Significance of Electromagnetic Potentials in the Quantum Theory.
Solenoid Example
Hamiltonian with magnetism:
1 &
2=
𝐻 −𝑖ℏ∇ + 𝑒𝐴⃗ 𝑟⃗
2𝑚
𝐴⃗ 𝑟⃗ is non-zero outside solenoid.
Then the groundstate solution is:
"( # ⃗ % %
' ∫ , ⃗
$ -. ⃗
$
𝜓 𝑟 =𝑒 ℏ #$ 𝜓(𝑟/ )
Identify
( $
𝜙= − ℏ ∫$ 𝐴⃗ 𝑟′
⃗ ; 𝑑 𝑟′
⃗
$
Aharonov and D Bohm. Significance of Electromagnetic Potentials in the Quantum Theory. PRL, 1959.
Aharonov- Bohm Effect
Interference is then
𝑒 $& 𝑒 $&
𝜙! − 𝜙% = − = 𝐴⃗ 𝑟′ ⃗ + = 𝐴⃗ 𝑟′
⃗ ; 𝑑 𝑟′ ⃗ ; 𝑑 𝑟′
⃗
ℏ $$ ℏ $$ 𝐴→𝐵 𝐵→𝐹
𝑒
= > 𝐴⃗ 𝑟′
⃗ ; 𝑑 𝑟′
⃗
ℏ 𝐶→𝐴 𝐹→𝐶
By Stokes’ Theorem
𝑒 𝑒 The vector potential without a field produces a
> 𝐴⃗ 𝑟′
⃗ ; 𝑑 𝑟′
⃗ = Φ
ℏ ℏ phase difference which is measurable.
Experimental Results
• Experimental setup identical.
• Here, a’ signifies range of observation.
• Phase shift equal to enclosed flux.
R G Chambers. Shift of an Electron Interference Pattern by Enclosed Magnetic Flux. PRL, 1960.
Berry Phase
• Take a general Hamiltonian 𝐻(𝑥" ; 𝜆0 )
• 𝑥& are the degrees of freedom (e.g. (𝑥, 𝑦, 𝑧))
• 𝜆' parameters our system depends on (e.g. length 𝐿 of an infinite potential
well)
• Adiabatic theorem lets us slowly vary the parameter to take in the
same eigenstate.
• If we take the groundstate and adiabatically vary a parameter and
return back to the groundstate,
𝜓 → 𝑒 "1 |𝜓⟩
• An overall measurable phase difference appears.
Berry Phase
• For a closed contour, this phase is called the Berry phase and is
computed as
𝛾 = − > 𝒜" 𝜆 𝑑𝜆"
Where 𝒜" 𝜆 is the Berry connection
𝜕
𝒜" 𝜆 = −𝑖⟨𝜓| |𝜓⟩
𝜕𝜆"
Applying Stokes’ Theorem, we can define the Berry curvature
𝜕𝒜" 𝜕𝒜0
ℱ"0 𝜆 = −
𝜕𝜆0 𝜕𝜆"
Aharonov- Bohm Connection
• In essence, take 𝒜" 𝜆 = 𝐴⃗ (vector potential) then ℱ"0 𝜆 = ∇×𝐴⃗
• Region without 𝐵 we recover Aharonov- Bohm
• The Aharonov-Bohm is a Berry phase with real fields
Summary
• The phase difference is measurable, not just a phase
𝜓 𝑥 ~𝑒 "$ 𝜓 𝑥 , phase not measurable
𝜓 𝑥 → 𝑒 ",$ 𝜓 𝑥 , phase difference is
• Analogues classical effect called parallel transport.
• Taking a vector around a closed path on a curved surface rotates the vector
• The Aharonov-Bohm Effect shows the vector potential is real
• Produces measurable effects
• An application of Berry phases
• Useful notion for studying topological systems
