Today’s Outline - April 2, 2015

• Aharonov-Bohm Effect

• Problem 9.3

Homework Assignment #09:Chapter 10:5,6,7,8,9,10due Thursday, April 09, 2015

Midterm Exam #2:Thursday, April 16, 2015

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 1 / 10

Today’s Outline - April 2, 2015

• Aharonov-Bohm Effect

• Problem 9.3

Homework Assignment #09:Chapter 10:5,6,7,8,9,10due Thursday, April 09, 2015

Midterm Exam #2:Thursday, April 16, 2015

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 1 / 10

Today’s Outline - April 2, 2015

• Aharonov-Bohm Effect

• Problem 9.3

Homework Assignment #09:Chapter 10:5,6,7,8,9,10due Thursday, April 09, 2015

Midterm Exam #2:Thursday, April 16, 2015

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 1 / 10

Today’s Outline - April 2, 2015

• Aharonov-Bohm Effect

• Problem 9.3

Homework Assignment #09:Chapter 10:5,6,7,8,9,10due Thursday, April 09, 2015

Midterm Exam #2:Thursday, April 16, 2015

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 1 / 10

Today’s Outline - April 2, 2015

• Aharonov-Bohm Effect

• Problem 9.3

Homework Assignment #09:Chapter 10:5,6,7,8,9,10due Thursday, April 09, 2015

Midterm Exam #2:Thursday, April 16, 2015

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 1 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)

(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)

(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= q ~Ae igΨ′ +~ie ig (∇Ψ′)− q ~Ae igΨ′

(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′

(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Consider the more general case where a particle moves through a regionwhere ~B = ∇× ~A = 0 but ~A 6= 0

for a static ~A the Schrodingerequation becomes

this can be simplified by sub-stituting

Ψ = e igΨ′

but ∇g = (q/~)~A so

i~∂Ψ

∂t=

[1

2m

(~i∇− q ~A

)2+ V

]Ψ

g(~r) ≡ q

~

∫ ~r

O~A(~r ′) · d~r ′

∇Ψ = e ig (i∇g)Ψ′ + e ig (∇Ψ′)(~i∇− q ~A

)Ψ =

~ie ig (i∇g)Ψ′ +

~ie ig (∇Ψ′)− q ~Ae igΨ′

= ����q ~Ae igΨ′ +

~ie ig (∇Ψ′)−����

q ~Ae igΨ′(~i∇− q ~A

)2Ψ = −~2e ig∇2Ψ′

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 2 / 10

Aharonov-Bohm effect

Substituting into the Schro-dinger equation

Ψ′ satisfies the Schrodingerequation without ~A

i~e ig∂Ψ′

∂t= − 1

2m~2e ig∇2Ψ′ + Ve igΨ′

i~∂Ψ′

∂t= − ~2

2m∇2Ψ′ + VΨ′

thus the solution of a system where thereis a vector potential is trivial, just add ona phase factor e ig

Aharonov & Bohm proposed an experi-ment where an electron beam is split intwo and passed on either side of alongsolenoid before being recombined

the two beams should arrive with differentphases g± = ±(qΦ/2~)

I

B

A

Beamsplit

Beamrecombined

solenoid

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 3 / 10

Aharonov-Bohm effect

Substituting into the Schro-dinger equation

Ψ′ satisfies the Schrodingerequation without ~A

i~e ig∂Ψ′

∂t= − 1

2m~2e ig∇2Ψ′ + Ve igΨ′

i~∂Ψ′

∂t= − ~2

2m∇2Ψ′ + VΨ′

thus the solution of a system where thereis a vector potential is trivial, just add ona phase factor e ig

Aharonov & Bohm proposed an experi-ment where an electron beam is split intwo and passed on either side of alongsolenoid before being recombined

the two beams should arrive with differentphases g± = ±(qΦ/2~)

I

B

A

Beamsplit

Beamrecombined

solenoid

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 3 / 10

Aharonov-Bohm effect

Substituting into the Schro-dinger equation

Ψ′ satisfies the Schrodingerequation without ~A

i~e ig∂Ψ′

∂t= − 1

2m~2e ig∇2Ψ′ + Ve igΨ′

i~∂Ψ′

∂t= − ~2

2m∇2Ψ′ + VΨ′

thus the solution of a system where thereis a vector potential is trivial, just add ona phase factor e ig

Aharonov & Bohm proposed an experi-ment where an electron beam is split intwo and passed on either side of alongsolenoid before being recombined

the two beams should arrive with differentphases g± = ±(qΦ/2~)

I

B

A

Beamsplit

Beamrecombined

solenoid

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 3 / 10

Aharonov-Bohm effect

Substituting into the Schro-dinger equation

Ψ′ satisfies the Schrodingerequation without ~A

i~e ig∂Ψ′

∂t= − 1

2m~2e ig∇2Ψ′ + Ve igΨ′

i~∂Ψ′

∂t= − ~2

2m∇2Ψ′ + VΨ′

thus the solution of a system where thereis a vector potential is trivial, just add ona phase factor e ig

Aharonov & Bohm proposed an experi-ment where an electron beam is split intwo and passed on either side of alongsolenoid before being recombined

the two beams should arrive with differentphases g± = ±(qΦ/2~)

I

B

A

Beamsplit

Beamrecombined

solenoid

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 3 / 10

Aharonov-Bohm effect

Substituting into the Schro-dinger equation

Ψ′ satisfies the Schrodingerequation without ~A

i~e ig∂Ψ′

∂t= − 1

2m~2e ig∇2Ψ′ + Ve igΨ′

i~∂Ψ′

∂t= − ~2

2m∇2Ψ′ + VΨ′

thus the solution of a system where thereis a vector potential is trivial, just add ona phase factor e ig

Aharonov & Bohm proposed an experi-ment where an electron beam is split intwo and passed on either side of alongsolenoid before being recombined

the two beams should arrive with differentphases g± = ±(qΦ/2~)

I

B

A

Beamsplit

Beamrecombined

solenoid

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 3 / 10

Aharonov-Bohm effect

Substituting into the Schro-dinger equation

Ψ′ satisfies the Schrodingerequation without ~A

i~e ig∂Ψ′

∂t= − 1

2m~2e ig∇2Ψ′ + Ve igΨ′

i~∂Ψ′

∂t= − ~2

2m∇2Ψ′ + VΨ′

thus the solution of a system where thereis a vector potential is trivial, just add ona phase factor e ig

Aharonov & Bohm proposed an experi-ment where an electron beam is split intwo and passed on either side of alongsolenoid before being recombined

the two beams should arrive with differentphases g± = ±(qΦ/2~)

I

B

A

Beamsplit

Beamrecombined

solenoid

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 3 / 10

Aharonov-Bohm effect

Substituting into the Schro-dinger equation

Ψ′ satisfies the Schrodingerequation without ~A

i~e ig∂Ψ′

∂t= − 1

2m~2e ig∇2Ψ′ + Ve igΨ′

i~∂Ψ′

∂t= − ~2

2m∇2Ψ′ + VΨ′

thus the solution of a system where thereis a vector potential is trivial, just add ona phase factor e ig

Aharonov & Bohm proposed an experi-ment where an electron beam is split intwo and passed on either side of alongsolenoid before being recombined

the two beams should arrive with differentphases g± = ±(qΦ/2~)

I

B

A

Beamsplit

Beamrecombined

solenoid

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 3 / 10

Aharonov-Bohm experiment

s: electron sourceo: observing planee,f: biprisma: confined field regiona’: extended field region

• modified electron microscope

• stray fields mitigated bybiprism made of analuminized quartz fiber (f)and two grounded metalplates (e)

• shift of n fringes forΦ = nhc/e

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 4 / 10

Aharonov-Bohm experiment

s: electron sourceo: observing planee,f: biprisma: confined field regiona’: extended field region

• modified electron microscope

• stray fields mitigated bybiprism made of analuminized quartz fiber (f)and two grounded metalplates (e)

• shift of n fringes forΦ = nhc/e

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 4 / 10

Aharonov-Bohm experiment

s: electron sourceo: observing planee,f: biprisma: confined field regiona’: extended field region

• modified electron microscope

• stray fields mitigated bybiprism made of analuminized quartz fiber (f)and two grounded metalplates (e)

• shift of n fringes forΦ = nhc/e

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 4 / 10

Aharonov-Bohm experiment

s: electron sourceo: observing planee,f: biprisma: confined field regiona’: extended field region

• modified electron microscope

• stray fields mitigated bybiprism made of analuminized quartz fiber (f)and two grounded metalplates (e)

• shift of n fringes forΦ = nhc/e

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 4 / 10

Aharonov-Bohm experiment

s: electron sourceo: observing planee,f: biprisma: confined field regiona’: extended field region

• modified electron microscope

• stray fields mitigated bybiprism made of analuminized quartz fiber (f)and two grounded metalplates (e)

• shift of n fringes forΦ = nhc/e

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 4 / 10

Aharonov-Bohm experiment

(a) is with no additional field ap-plied in extended region

(b) 25mG applied and no fringeshift since both envelope and fringeshift by same amount, up to 300mG showed no shift

This calibration experiment shows that there is a quantum effect in theregion where there is both a flux AND a field

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 5 / 10

Aharonov-Bohm experiment

(a) is with no additional field ap-plied in extended region

(b) 25mG applied and no fringeshift since both envelope and fringeshift by same amount, up to 300mG showed no shift

This calibration experiment shows that there is a quantum effect in theregion where there is both a flux AND a field

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 5 / 10

Aharonov-Bohm experiment

(a) is with no additional field ap-plied in extended region

(b) 25mG applied and no fringeshift since both envelope and fringeshift by same amount, up to 300mG showed no shift

This calibration experiment shows that there is a quantum effect in theregion where there is both a flux AND a field

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 5 / 10

Aharonov-Bohm experiment

(a) is with no additional field ap-plied in extended region

(b) 25mG applied and no fringeshift since both envelope and fringeshift by same amount, up to 300mG showed no shift

This calibration experiment shows that there is a quantum effect in theregion where there is both a flux AND a field

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 5 / 10

Aharonov-Bohm experiment

(a) iron whisker produces con-fined field and flux with a gradientalong the z-axis manifested in tiltedfringes

(b) direct imaging, with the whiskeroutside the shadow of the biprismfiber, due to Fresnel diffraction inthe shadow of the fiber

(c) higher taper, again using Fres-nel diffraction in fiber shadow butwith no biprism

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 6 / 10

Aharonov-Bohm experiment

(a) iron whisker produces con-fined field and flux with a gradientalong the z-axis manifested in tiltedfringes

(b) direct imaging, with the whiskeroutside the shadow of the biprismfiber, due to Fresnel diffraction inthe shadow of the fiber

(c) higher taper, again using Fres-nel diffraction in fiber shadow butwith no biprism

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 6 / 10

Aharonov-Bohm experiment

(a) iron whisker produces con-fined field and flux with a gradientalong the z-axis manifested in tiltedfringes

(b) direct imaging, with the whiskeroutside the shadow of the biprismfiber, due to Fresnel diffraction inthe shadow of the fiber

(c) higher taper, again using Fres-nel diffraction in fiber shadow butwith no biprism

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 6 / 10

Aharonov-Bohm experiment

(a) iron whisker produces con-fined field and flux with a gradientalong the z-axis manifested in tiltedfringes

(b) direct imaging, with the whiskeroutside the shadow of the biprismfiber, due to Fresnel diffraction inthe shadow of the fiber

(c) higher taper, again using Fres-nel diffraction in fiber shadow butwith no biprism

“Shift of an electron interference pattern by enclosed magnetic flux,” R.G. Chambers,

Phys. Rev. Lett. 5, 3-5 (1960).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 6 / 10

A-B effect in a normal metal

There is no reason why theAharonov-Bohm effect couldnot be observed in a conductingloop

The key would be to have theelectrons traveling on either sideof the “solenoid” maintain co-herence until they reunite

Make a small loop out of gold,cool to very low temperatures toincrease the mean free path ofthe electrons, and vary the field

“Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” R.A. Webb, et

al., Phys. Rev. Lett. 54, 2696-2699 (1985).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 7 / 10

A-B effect in a normal metal

There is no reason why theAharonov-Bohm effect couldnot be observed in a conductingloop

The key would be to have theelectrons traveling on either sideof the “solenoid” maintain co-herence until they reunite

Make a small loop out of gold,cool to very low temperatures toincrease the mean free path ofthe electrons, and vary the field

“Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” R.A. Webb, et

al., Phys. Rev. Lett. 54, 2696-2699 (1985).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 7 / 10

A-B effect in a normal metal

There is no reason why theAharonov-Bohm effect couldnot be observed in a conductingloop

The key would be to have theelectrons traveling on either sideof the “solenoid” maintain co-herence until they reunite

Make a small loop out of gold,cool to very low temperatures toincrease the mean free path ofthe electrons, and vary the field

“Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” R.A. Webb, et

al., Phys. Rev. Lett. 54, 2696-2699 (1985).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 7 / 10

A-B effect in a normal metal

There is no reason why theAharonov-Bohm effect couldnot be observed in a conductingloop

The key would be to have theelectrons traveling on either sideof the “solenoid” maintain co-herence until they reunite

Make a small loop out of gold,cool to very low temperatures toincrease the mean free path ofthe electrons, and vary the field

“Observation of h/e Aharonov-Bohm oscillations in normal-metal rings,” R.A. Webb, et

al., Phys. Rev. Lett. 54, 2696-2699 (1985).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 7 / 10

Electrostatic A-B effect

“Magneto-electric Aharonov-Bohm effect in metal rings,” A. van Oudenaarden, et al.,

Nature 391, 768-770 (1998).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 8 / 10

Electrostatic A-B effect

“Magneto-electric Aharonov-Bohm effect in metal rings,” A. van Oudenaarden, et al.,

Nature 391, 768-770 (1998).

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 8 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]

= −i q~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

This can be solved using

ψn = e igψ′n

g ≡ q

~

∫ ~r

~R

~A(~r ′) · d~r ′

Suppose the charged particle is confined to abox centered at ~R by a potential V (~r − ~R)

Enψn =

{1

2m

[~i∇− q ~A(~r)

]2+ V (~r − ~R)

}ψn

Enψ′n =

[− ~2

2m∇2 + V (~r − ~R)

]ψ′n

The box is now carried around the solenoid(varying ~R and Berry’s phase is calculated

∇Rψn = ∇R

[e igψ′n(~r − ~R)

]= −i q

~~A(~R)e igψ′n(~r − ~R) + e ig∇Rψ

′n(~r − ~R)

now take the inner product 〈ψn|∇Rψn〉

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 9 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a

=qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10

General features of Aharonov-Bohm effect

〈ψn|∇Rψn〉 =

∫e−ig [ψ′n(~r − ~R)]∗e ig

[− iq

~~A(~R)ψ′n(~r− ~R) +∇Rψ

′n(~r− ~R)

]d3~r

= − iq

~~A(~R)−

∫[ψ′n(~r − ~R)]∗∇ψ′n(~r− ~R)d3~r

noting that 〈p〉 ∝ ∇ψ′ = 0for stationary states, the sec-ond term vanishes

and applying Berry’s formulafor the geometric phase

this is simply the Aharonov-Bohm effect!

〈ψn|∇Rψn〉 = −i q~~A(~R)

γn(T ) = i

∫ Rf

Ri

〈ψn|∇Rψn〉 · d ~R

=q

~

∮~A(~R) · d ~R

=q

~

∫(∇× ~A) · d~a =

qΦ

~

C. Segre (IIT) PHYS 406 - Spring 2015 April 02, 2015 10 / 10