Understanding Magnetic Potentials and Spin Precession: A Geometric Perspective

[sirchasm]

I'm trying to get a picture of magnetic potential, as in how to relate it to spin precession.
(So have I got the right picture so far?)

Classically, B is measured as the derivative of curl (which is a circulation integral) "around" a conducting current (that is, perpendicular to the direction of current I).
Gauss' Law says the field is divergence-free, which means it circulates linearly, and the integral is just BL, for any circular integral along a circumference L, of a potential around say, a 1-d current in a wire.

Einstein and de Haas tried to measure precessional forces to explain magnets, by envisaging electrons as being in orbits and precessing, as a gyroscope does in a gravitational potential. It didn't quite explain magnetism because it didn't allow for Pauli's exclusion principle that 'admits' two electrons, which share an orbital; magnetic forces are in fact related to electron's spin angular momentum, not their orbital momentum. Or electrons rotate "in place", rather than orbiting like gyroscopes.

Aharonov-Bohm phase precession is due to a magnetic potential in a curl-free region, which rotates the spin precession angles in a divided beam of electrons.
It looks like a bump for 1/2 the beam, and like a dip for the other 1/2. One spin phase shifts up or 'squeezes' over the dip, the other shifts down or 'expands' over the bump in potential. The phase-difference is preserved, as momentum carries the matter-wave to a detector.

So it's about how to relate the geometry of a magnetic field (potential) to the geometry of spin in electrons.
Is potential just equal to the precession angle, since a change in this angle, will change the local potential gradient?

It's a question (I think), of seeing the geometry and topology involved. The algebra is the really tricky bit with complex quantum spaces with a dimension of $$\mathbb C^{2n}$$. Anyways, there's this somewhat dated article in a '81 SciAm about the A-B geometry and neutron spin-precession. Quite an interesting geometric picture; they relate geodesics (over an inclined conic surface) to parallel transport, then what a "path-lifting" does, and so on.

You can relate the observed patterns of interference to the topology and geometry of spin precession. Spin is a 'component' of the Schrodinger wavefunction(?), and charge is too(?), whereas position and momentum are another component, or IOW the wavefunction is a "mixed wave" of spin, charge, and mass wave components? There's a major difference between the A-B experiment and neutron interferometry, wrt to the magnetic field applied, since the neutrons cross classical field-lines, or do not travel through a curl-free region, but see a field that rotates the precessional angle, over time, whereas the electrons see just a potential and preserve a single phase shift?

Have I left something out of this picture?

 

[eljose79]

I would like to clarify and add some information to your understanding of magnetic potential and its relation to spin precession.

Firstly, the classical definition of magnetic field (B) as the derivative of curl is correct. This is because the magnetic field is a vector field that describes the strength and direction of the magnetic force at any given point in space. The curl of this vector field gives us the circulation or rotational component of the magnetic field.

Secondly, Gauss' Law for magnetism states that the magnetic field is divergence-free, meaning that it does not have a source or sink. This is important because it tells us that the magnetic field lines form closed loops, and any change in the magnetic field must be due to a change in the current or a nearby magnetic material.

Now, let's talk about the relationship between magnetic potential and spin precession. As you correctly pointed out, classical theories of magnetism, such as the one proposed by Einstein and de Haas, did not fully explain the phenomenon of magnetism because they did not take into account the quantum nature of electrons. It was later discovered that the magnetic moment of an electron is not solely due to its orbital motion, but also its spin angular momentum.

The Aharonov-Bohm effect, as you mentioned, is a quantum phenomenon that shows the relationship between magnetic potential and spin precession. In this effect, a magnetic potential in a curl-free region can cause a phase shift in the spin of an electron passing through it. This is because the spin of an electron is affected by the magnetic field, and a change in the magnetic potential will result in a change in the spin precession angle.

In terms of the geometry and topology involved, it is important to note that the magnetic potential is not equal to the precession angle. The potential gradient, which is the change in potential over a given distance, can affect the precession angle of the spin. This is because the spin precession is dependent on the strength and direction of the magnetic field, which is described by the potential gradient.

In summary, the relationship between magnetic potential and spin precession is a complex one, as it involves both classical and quantum theories. It is important to understand the concept of magnetic potential and its role in describing the magnetic field, as well as its effect on the spin of particles. I hope this clarifies and adds to your understanding of this topic.

 

[denian]

I would say that your understanding of magnetic potential and spin precession is on the right track. The relationship between the two can be described through the concept of gauge invariance, which states that the physics of a system should not depend on the specific choice of gauge (or mathematical representation) used to describe it. In the case of magnetic potential and spin precession, this means that the magnetic potential can be seen as a gauge field that interacts with the spin of particles, causing them to precess.

In classical physics, the magnetic field is described as a derivative of the curl of a conducting current. This means that the field is related to the circulation of the current, and can be measured by integrating the potential around a circular path. However, in quantum mechanics, the concept of spin must be taken into account. As you mentioned, spin is related to the angular momentum of particles and plays a crucial role in their behavior in a magnetic field.

The Aharonov-Bohm effect, which you also mentioned, is a good example of this relationship between magnetic potential and spin precession. This effect shows that the magnetic potential can affect the phase of a particle's wavefunction, which in turn affects its behavior. In the case of the A-B effect, the potential in a curl-free region causes the spin precession angles to change, leading to a phase shift in the particle's wavefunction. This phase shift can then be observed in the interference patterns of the particle.

Overall, understanding the relationship between magnetic potential and spin precession is important in understanding the behavior of particles in a magnetic field. It involves both the geometry and topology of the field, as well as the concept of gauge invariance. Your understanding of this relationship is a good starting point, but further study and research can deepen your understanding of this complex topic.