# Aharonov - Bohm effect exercise

• A
• dude2
In summary, current through the solenoid creates a magnetic field which affects the particles in the screen.
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.

What have you already done towards a solution?

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.

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
dude2 said:
a) Calculate the vector potential in the space outside the solenoid.

Last edited:
Cryo said:
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 referring to?

dude2 said:
Thanks but for what subquestion of the three, your answer is referring to?

dude2 said:
) Calculate the vector potential in the space outside the solenoid

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

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

Have a look

Will do sir, thanks!

## 1. What is the Aharonov-Bohm effect exercise?

The Aharonov-Bohm effect exercise is a thought experiment that demonstrates the influence of electromagnetic fields on charged particles, even when the particles are not directly in contact with the field.

## 2. How does the Aharonov-Bohm effect work?

In the exercise, a charged particle is split into two paths, one passing through a region with a magnetic field and the other passing through a region with no magnetic field. Even though the particle does not directly interact with the magnetic field, its path is still affected by the field, resulting in an interference pattern when the paths are recombined.

## 3. What is the significance of the Aharonov-Bohm effect?

The Aharonov-Bohm effect challenges the traditional understanding of the relationship between particles and fields in classical physics. It also has implications for quantum mechanics, as it suggests that the classical concept of a particle's trajectory may not be applicable in the quantum realm.

## 4. Can the Aharonov-Bohm effect be observed in real life?

Yes, the Aharonov-Bohm effect has been experimentally observed in various systems, including electron diffraction and interferometry experiments. It has also been observed in superconducting circuits and in Bose-Einstein condensates.

## 5. Are there any practical applications of the Aharonov-Bohm effect?

While the Aharonov-Bohm effect is primarily a theoretical concept, it has been used in the development of sensitive magnetometers and in the study of topological phases of matter. It also has potential applications in quantum computing and information processing.

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