Infinite solenoid with magnetic flux

[yxgao]

Hi,
I have a question as follows.

For an infinite solenoid with magnetic flux [tex]\Phi[/tex], for what values of [tex]\Phi[/tex] "is the quantum mechanical motion of an electron constrained to stray far from the solenoid exactly the same as it would be if [tex]\Phi[/tex] were zero"?

Solution:
[tex]\Phi[/tex] = [tex]n \Phi_{o}[/tex] with n = integer and [tex]\Phi_{o} = \frac{h c}{e}[/tex]

Can someone please explain the solution? I don't understand the concepts involved here. Thanks.

 

[dextercioby]

Hi,
I have a question as follows.


For an infinite solenoid with magnetic flux [tex]\Phi[/tex], for what values of [tex]\Phi[/tex] "is the quantum mechanical motion of an electron constrained to stray far from the solenoid exactly the same as it would be if [tex]\Phi[/tex] were zero"?

Solution:
[tex]\Phi[/tex] = [tex]n \Phi_{o}[/tex] with n = integer and [tex]\Phi_{o} = \frac{h c}{e}[/tex]

Can someone please explain the solution? I don't understand the concepts involved here. Thanks.

It deals with the famous Bohm-Aharonov effect and magnetic fkux quantization in superconductors.The trick is to exploit the fact that the magetic field is gauge invariant and in the Hamiltonian it enters not through the field,but through the magnetic vector potential.

A nice discussion is made here:
http://hep.ucsd.edu/~branson/130/130b/130b_notes_prod/node50.html
This is node 50,read node 51 as well.

Daniel.

 

[wais]

The solution given is related to the quantum mechanical concept of quantization, which states that certain physical quantities can only take on discrete values. In this case, the magnetic flux \Phi is quantized, meaning it can only have certain values determined by the integer n and the fundamental magnetic flux unit \Phi_{o}. This is known as the flux quantization condition.

Now, for an electron moving in the presence of an infinite solenoid, its motion is affected by the magnetic field created by the solenoid. When the magnetic flux \Phi is equal to zero, there is no magnetic field and thus the electron's motion is not affected by it. This is why the quantum mechanical motion is the same as if \Phi were zero.

However, when \Phi is not equal to zero, the electron's motion is constrained by the magnetic field and it can only take on certain discrete values determined by the quantization condition. So, for the electron's motion to be exactly the same as if \Phi were zero, the magnetic flux must be a multiple of the fundamental unit \Phi_{o}.

In summary, the solution is saying that for the quantum mechanical motion of an electron to be unaffected by the magnetic field of an infinite solenoid, the magnetic flux must be quantized, with values determined by the quantization condition.