# A Aharonov - Bohm effect exercise

#### dude2

Does anyone know the answers to this, or can hopefully guide me to a text that will help me solve this aharonov-bohm problem?
Here is the given:
Particles (of mass m, and charge q), are driven through two slits that have distance d between them, in a screen that is far away (L>>d) from the obstacle. Behind the obstacle is a solenoid, tha constant current is flowing through it (I).
a) Calculate the vector potential in the space outside the solenoid. b) Assume that without current, the solution to the Schroedinger equation is of this form:

Confirm that this:

will constitute a solution to the equation, in the presence of the EM field

where Ψο, corresponds to the solution without current.
c) Prove that the interference pattern of the particles in the screen, will move, with the presence of current.

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#### vanhees71

Gold Member
What have you already done towards a solution?

#### dude2

Nothing, because i don't know where to start...
I didn't ask for a specific solution.
I just posted this here, in case someonw recognized it and pointed me towards a solution on a book, or rather in some references that would help me reach the solution.

#### Cryo

Gold Member
Why not begin by finding the vector potential? You have a solenoid, with complex winding current, but its main purpose is to create the magnetic field in a localized region in the XY-plane, and directed out of the page (along Z). You can describe this aptly with a point-like magnetization M=M^zM=Mz^, where current density is then J=∇×MJ=∇×M.

And the vector potential (AA) is:

∇×∇×A=∇×B=μ0J=∇×μ0M∇×∇×A=∇×B=μ0J=∇×μ0M

So a suitable equation is:

∇×A=μ0M∇×A=μ0M

I would suggest using Stockes theorem and cyllindrical symmetry of the solenoid to find the solution
a) Calculate the vector potential in the space outside the solenoid.

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#### dude2

Why not begin by finsing the vector potential? You have a solenoid, with complex winding current, but its main purpose is to create the magnetic field in a localized region in the XY-plane, and directed out of the page (along Z). You can describe this aptly with a point-like magnetization $\mathbf{M}=M\mathbf{\hat{z}}$, where current density is then $\mathbf{J}=\boldsymbol{\nabla}\times\mathbf{M}$.

And the vector potential ($\mathbf{A}$) is:

$\boldsymbol{\nabla}\times\boldsymbol{\nabla}\times\mathbf{A}=\boldsymbol{\nabla}\times\mathbf{B}=\mu_0\mathbf{J}=\boldsymbol{\nabla}\times\mu_0\mathbf{M}$

So a suitable equation is:

$\boldsymbol{\nabla}\times\mathbf{A}=\mu_0 \mathbf{M}$

I would suggest using Stockes theorem and cyllindrical symmetry of the solenoid to find the solution
Thanks but for what subquestion of the three, your answer is refering to?

#### Cryo

Gold Member
Thanks but for what subquestion of the three, your answer is refering to?
) Calculate the vector potential in the space outside the solenoid

#### Cryo

Gold Member
As for references. Have a look in S. Weinberg's Lectures on Quantum Mechanics, Ch 10.4 (NB! Not the field theory book). I think Sakurai's Modern Quantum Mechanics also had a bit on this

#### dude2

Thanks! I am mostly interested in the questions b and c though. Does Weinberg contain something about them?

Gold Member
Have a look

#### dude2

Will do sir, thanks!

"Aharonov - Bohm effect exercise"

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