In this presentation I present Aharonov-Bohm effect, a quantum phenomenon in which a particle is effected by electromagnetic fields even when traveling through a region of space in which both electric and magnetic field are zero. I will describe theoretical background of the effect, present some experimental verifications and show how this phenomenon can be practically used in modern devices for precise measurement of magnetic field.

1. AHARONAOV-BOHM EFFECT
Presented By
Anzar Ali
PH14006
Instructor Dr. K.P. Yogendran
Indian Institute of Science Education and Research
Mohali Punjab
18/04/2015

2. 1) Introduction
2) Maxwell’s equation
3) Gauge transformation
4) Charge particle in electromagnetic field
5) Aharonov-Bohm effect experiment
6) Practical use of Aharonov-Bohm effect
7) Conclusion
8) References
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Outline of the talk

3. Introduction
 Description of electromagnetic phenomena can be
simplified by introduction of electromagnetic potentials:
scalar potential φ and vector potential A.
 Until the beginning of the twenty century it was
widely believed that potentials are only a mathematical
construct to simplify calculations and that they contain
no physical significance.
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4.  With the development of quantum mechanics in the
early twenty century, this view was put under question,
because Schrödinger equation, basic equation of
quantum mechanics, doesn’t contain fields but
potentials.
 Schrodinger Equation:-
Introduction (Continue)
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−
ℏ2
2𝑚
𝛻2
ψ + V x ψ = 𝐸ψ

5. The heart of the
experiment is the effect in
which wave function
acquire some additional
phase when travelling
through space with no
electromagnetic fields,
only potentials. This is
called Aharonov-Bohm
effect.
Introduction (Continue)
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6. Maxwell Equations
𝜵. 𝑬 =
ρ
𝜀0
𝜵 × 𝑬 = −
𝜕𝑩
𝜕𝑡
𝜵. 𝑩 = 0
𝜵 × 𝑩 = μ0 𝑱 + μ0ε0
𝜕𝑬
𝜕𝑡
𝑩 = 𝜵 × 𝑨
𝑬 = −𝜵ɸ −
𝜕𝑨
𝜕𝑡
1- 2-
3- 4-
From equation (2)
From equation (3)
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(5)
(6)
In 1861, Scottish physicist and mathematician James Clerk
Maxwell wrote four differential equations[1].

7. Gauge Transformation
𝑨′ = 𝑨 + 𝜵λ
ɸ′ = ɸ −
𝜕λ
𝜕𝑡
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Electromagnetic Potential have another important
property.
Where function λ is called “gauge function.”
Because we can satisfy Maxwell’s equation with different
potentials, we can say that the equations are gauge
invariant.

8. Charged particle in electromagnetic field
[
1
2𝑚
−𝑖ℏ𝛁 − 𝑒𝑨 𝒓 )2
+ 𝑒ɸ 𝒓 + 𝑉 𝒓 𝜓 = 𝑖ℏ
𝜕𝜓
𝜕𝑡
For a charged particle in electromagnetic field, Hamiltonian
is of the form[2].
𝐻 =
1
2𝑚
(𝒑 − 𝑒𝑨(𝒓) )2 + 𝑒ɸ(r)
If we write 𝒑 = −𝑖ℏ𝜵 and put Hamiltonian in Schrodinger
equation, we get
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9. Vector potential of solenoid magnet
𝑬 = −𝛁ɸ = 0 ɸ = 0
𝐵 = 𝛻 × 𝑨 = 0
𝑨 =
φ 𝑚
2𝜋𝑟
φ
To solve Schrodinger equation, we must determine
A = ? and ɸ = ?
Because solenoid is uncharged
Vector potential outside the solenoid :-
𝑨. 𝑑𝒓 = 𝛁 × 𝑨 . 𝑑𝑺 = 𝑩. 𝑑𝑺 = φ 𝒎
C S S
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10. Wavefunction in vector potential
ψ 𝐫, t = eig(𝐫)ψ’(r, t) Where 𝑔 𝒓 =
𝑒
ℏ
0
𝑟
𝑨 𝒓 . 𝑑𝒓
In term of ψ’, the gradient of ψ is
𝜵𝜓 = 𝑒 𝑖𝑔 𝒓
𝑖𝜵𝑔 𝒓 𝜓′
+ 𝑒 𝑖𝑔 𝒓
(𝜵ψ′
)
Because
𝜵𝑔 𝑟 =
𝑒
ℏ
𝑨
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To describe wavefunction of charged particle, we have to
solve Schrodinger equation. In our case it can be simplified
by writing the wavefunction in the form

11. −𝑖ℏ𝜵 − 𝑒𝑨 ψ = −𝑖ℏ𝑒 𝑖𝑔 𝒓
𝜵ψ′
(−𝑖ℏ𝜵 − 𝑒𝑨)2
ψ = −ℏ2
𝑒 𝑖𝑔 𝒓
𝜵2
ψ′
−
ℏ2
2𝑚
𝜵2
ψ′
− 𝑉ψ′
= 𝑖ℏ
𝜕ψ′
𝜕𝑡
The solution in presence of vector field is the same
wavefunction, multiplied by phase factor 𝑒 𝑖𝑔(𝑟).
Wavefunction in vector potential (Continue)
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12. Aharonov-Bohm effect Experiment
ψ1 = 𝐴𝑒 𝑖𝑘𝑥1 ψ2 = 𝐴𝑒 𝑖𝑘𝑥2
∆𝚽0 = 𝑘(𝑥1 − 𝑥2)
∆𝚽 = 𝑔1 − 𝑔2
∆𝚽 = 𝑒/ℏ[ 𝑨 𝒓 . 𝑑𝒓 − 𝑨 𝒓 . 𝑑𝒓]
∆𝚽 = 𝑒/ℏ 𝑨 𝒓 . 𝑑𝒓
∆𝚽 = 𝑒φ 𝑚/ℏ
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𝐶1 𝐶2
Phys. Rev., 115, 485, (1959)

13. Practical use of Aharonov-Bohm effect
 Phase shift between electron beams strongly depends on
enclosed magnetic flux.
 Interference pattern shifts one fringe for every ∆ɸ 𝑚 =
ℎ
𝑒0
= 4.1 × 10−15
𝑇𝑚2
, which is very small value[6].
 In principle, the effect enables us measurement of
extremely small differences in magnetic flux.
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14. Conclusion
The main objective of this talk was to show that in
quantum mechanics, electromagnetic potentials appear
to be more fundamental physical entities than fields.
Aharonov-Bohm effect is phenomenon which can’t be
describe in terms of classical mechanics and is of purely
quantum origin. The effect was confirmed by many
different experiments and today it’s existence is widely
accepted.
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15. References
1) Griffits, D.J., Introduction to electrodynamics
2) Griffits, D.J., Introduction to Quantum Mechanics
3) Sakurai, J.J., Modern Quantum Mechanics
4) Y. Aharonov and D. Bohm, Significance of electromagnetic potential
in the quantum theory Phys Rev., 115, 485, (1959)
5) R.G. Chambers, Shift of an electron interference pattern by enclosed
magnetic flux, Phys. Rev. Lett. 5, 3, (1960)
6) N. Osakabe et. al., Experimental confirmation of Aharonov-Bohm
effect using a toroid magnetic field confined by a superconductor,
Phys. Rev. A 34, 815, (1986)
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