Decoding the Spin of Electrons: A Beginner’s Guide to Quantum Mechanics

 What Exactly is Spin?

Let's talk about spin. When we think of an electron, proton, or other elementary particles, we often imagine them somehow “spinning.” But what exactly is spinning, and how it works – we don't know for sure.

In classical physics, spinning objects have something called angular momentum. The faster they spin, the greater the angular momentum. Similarly, the heavier the object, the greater its angular momentum. Electrons and protons have something like "intrinsic angular momentum", which we call "spin". However, the value of this spin isn't just any number – it takes discrete values, multiples of half the Planck constant.

I consider it as quite possible that if we can one day fully understand what spin is, we might unlock the entire mystery of quantum mechanics. 

It is quite possible that if we can one day fully understand what spin is, we might unlock the entire mystery of quantum mechanics. 

For now, though, we have to be content with its mathematical description – which, unfortunately, is not quite the same as understanding the essence of the phenomenon.

Understanding Electron Spin

So, let’s dive into the mathematical description of electron spin, and I’ll try to make it a bit more accessible. An electron's spin is equal to half of a Planck constant. This is why we say the electron has a spin of ½.

In experiments, we can align the electron's spin axis in a specific direction, for example, upwards along the z-axis. This alignment defines the spin state, but it doesn’t fully describe the state vector. In quantum mechanics, we make a distinction between states and state vectors.

• State: What we observe.
• State Vector: Information that includes both what we see and what is invisible, yet still necessary.

A Model for Spin: Visible and Invisible Wheels

Can we visualize this? Maybe. But let’s remember, models can be misleading. What I propose is simply a mental tool – it might help, but it could also lead us astray.

Imagine an electron as a blue cog with a visible mark. The position of this mark in relation to an external coordinate system represents the electron's state – what we can directly control. However, alongside this visible cog, there’s an invisible gray cog, also marked. This cog is hidden from our view but plays a crucial role.

The Internal Phase

To fully describe the state vector of the electron, we not only need to know the position of the visible mark, but also the angle it makes with the mark on the invisible cog. Let’s call this angle the internal phase.

• Knowing the state (the position of the visible mark) is important.
• But we also need to know the internal phase, the relationship between the visible and invisible marks.

Now, let’s make this more interesting.

A 720-Degree Rotation

Imagine the gray, invisible cog is twice the size of the blue, visible one. If you rotate the blue cog 360 degrees, the gray cog only rotates by 180 degrees. To return both cogs to their original alignment, you would need to rotate the blue cog a full 720 degrees.

Think of it like this:

A More Detailed Model

For those of you following closely, I need to add a bit of complexity. My earlier analogy of two cogs is a bit too simple. Ideally, I should color the gray cog (instead of keeping it plain) to emphasize that the internal phase is relative and subjective. One person might define the "zero" phase when the marks align; another might choose red or green as the reference point. The key idea is that it takes a full 720-degree rotation of the visible cog for both to return to their original states.

Exploring Other Models

This model is just one of many. There are also examples in the literature involving cubes tied together with strings, or twisted strips resembling Möbius bands. However, I prefer my cog analogy – it’s simple and relatable. But again, it's a rough analogy. There’s something happening with the topology of space within the electron itself. It’s as if the electron “screws” itself into space when we rotate it.

Think of it like this: part of the electron exists in our space, while another part is in a sort of “anti-space,” where time flows in the opposite direction, some topologically twisted Einstein-Rosen bridge. When you rotate the electron by 360 degrees, part of it moves into anti-space, and vice versa. To return everything to the initial state, you need to rotate it another 360 degrees. It’s fascinating, but we’re far from fully understanding this yet.

The Spin State on a Sphere

Now that we've explored the analogy, let’s focus on the mathematical representation. We can describe the electron (the “visible” part) as a point on a unit radius sphere in three-dimensional space (x, y, z). This point indicates the spin direction. It can be described by using:

• Latitude and longitude, or
• Cartesian coordinates (nx, ny, nz) of a vector n of length 1.

Here’s the basic relationship:

nx²+ny²+nz² = 1

The conversion between spherical and Cartesian coordinates follows these formulas:

• $n\mathrm{x }= \sin (\varphi ) \cos (\theta )$
• $n\mathrm{y }= \sin (\varphi ) \sin (\theta )$
• $n\mathrm{z }= \cos (\varphi )$

In this model, ϕ (latitude) ranges from 0 to Pi, andθ (longitude) ranges from 0 to 2 Pi. At the poles (where phi = 0 or Pi), theta is undefined, but we often just assign it a value of zero for convenience.

Wrapping It Up

To summarize: the spin state (the visible part) is a point on the sphere that indicates the direction of the electron’s spin axis. Think of this spin axis as an arrow rather than a simple straight line.

In future posts, we’ll dive deeper into the concept of the state vector, its mathematical representation, and its relationship to the spin state. We’ll also explore how to project from four-dimensional space to three-dimensional space, allowing us to visualize the invisible internal phase, which, while not directly observable, plays a vital role in understanding the electron’s behavior.

What's Next?

In the upcoming posts, expect more formulas and visual aids as we continue unraveling the mysteries of quantum mechanics. 

Will we discover new insights? Only time will tell. But one thing is for sure – the journey is just beginning.

One Law I want to know

John Archibald Wheeler proposed The One law for the Universe; Boundary of the boundary is zero - BB Principle. 

In symbols:

∂∂ = 0.

From Warner Allen Miller ,"The Geometrodynamic Content of the Regge Equations as Illuminated by the Boundary of a Boundary Principle", (pp. 201-227 in Between Quantum and Cosmos: Studies and Essays in Honor of John Archibald Wheeler, Wojciech Hubert Zurek, Alwyn van der Merwe, Warner Allen Miller, Princeton University Press 1988):

All our laws of nature would be then consequences of this one. Looks nice and simple indeed. In topology and in algebraic topology it holds indeed, either as a theorem or as a postulated fundamental property.

In my vision there is also one law: it is the "Law of Free Fall". Everything moves along a geodesic of some kind of "geometry". Can these two visions, Wheeler's and mine, be merged into one vision? Is free fall another expression of BB? Is BB just another expression of Free Fall? And what about Free Will?

And then: why THIS law and not some other? And why one law and not two or three? Or even an infinite number of laws? Why?

While looking for answers, enlarging my knowledge (and my ignorance as well, but only linearly), I am asking myself: "Am I of this world?"

"And so we wander on our way asking ourselves—if truth be known, muttering to  ourselves really—“Am  I  of  this  world  or  the  other?”  and  answering  “I  am  of  both.”  And  we remind ourselves of this as we go along."

 

"Through this circular river symbol, the moat, the tale warns us that this water is not just any water but a certain kind. It is a boundary water, much like the circle the maiden drew around herself to keep the Devil away. When one crosses into or through a circle, one  is entering  into  or  passing  through  to  another  state  of  being,  another  state  of awareness, or lack of one. "

Clarissa Pinkola Estes, "Women Who Run With The Wolves", 

I want to know

 

P.S.1

Found by Laura for me to watch. A good one indeed. Recommend for everybody!

 P.S.2. Just chceked my researchGate mail. Discovered a message from September:

"Hallo Arkadiusz,

I looked to your your Preprint Time of arrival operator in the momentum space, it is interesting and please see my paper (blow) , where one can see the relation to the Fujiwara–Kobe time operator, sec 4 of your preprint.

https://www.mdpi.com/2624-960X/2/2/15
or
https://www.researchgate.net/publication/340491512_Time_Operator_Real_Tunneling_Time_in_Strong_Field_Interaction_and_the_Attoclock_Open_access_journal_Quantum_Report_httpswwwmdpicomjournalquantumrep

P.S.3 While Igor Bayak wrote me 

"Уважаемый Аркадиуш, здравствуйте!

Посмотрите, пож., на уравнение 3.16 из статьи Хаотическая динамика электрона. Вы ее легко найдете в моем профиле. Там речь о динамическом решении для нулей дзета функции Римана."

Riemann Zeta! Beatiful piece of art!

P.S.4 Another worth considering quotation from "Women Who Run With The Wolves" mentioned in the main text of this note:

"The young and the injured are uninitiated. Neither knows much about the dark predator and are, therefore, credulous. But, fortunately, when the predator is on the move, it leaves behind unmistakable tracks in dreams. These tracks eventually lead to its discovery, capture and containment."

"Wild Ways teaches people when not to act 'nice' about protecting their souls. The instinctive nature knows that being 'sweet' in these instances only makes the predator smile. When the soul is being threatened, it is not only acceptable to draw the line and mean it, it is required."

P.S.5. I am back to conformal infinity. It is a part of U(2). Perhaps I will write a note about it. 

P.S.6. From the battlefield Planet Earth;

P.S.7. I have on my table these two books. Oh God, how much I would like to know their content and master it!. Why it is not possible for us to bring these books close to our heads and download their content (even if it may take a while), and push the button "Understand" and another button "Now"? Why?

But I clearly realize that different people may have extremely different standards of "understanding"....

Different level of "depth". The next button would be "Integrate". Of course the first book would have nothing to ingrate with. Would be instantaneous. But with each new piece of added knowledge integration would take exponentially longer time.

By integration I mean making the whole more than just the sum of parts. For instance: when we learn about complex numbers, we understand much better real numbers about which we have learned before. We create a huge number of connections and interactions between the old and the new knowledge. Thus, approximately, when we have m pieces of old knowledge and add m pieces of new knowledge we can optimally-ideally gain mxn rather than just m+n. But there is also useless knowledge: a knowledge that cannot be integrated with our other knowledge, because it belongs to a different "species" of knowledge. For instance knowledge of blacksmithing or  knitting will hardly add anything useful  to your knowledge of algebraic topology.  We all have finite resources and we have to choose whether we want to know superficially possibly many subjects, or to know really deeply at least one. People with extremely high IQ seem to be naturally gravitating towards the first option.

And, as John Wheeler has noticed: the more we know about a given subject, the more we are painfully aware of how little we, in fact,  know.

P.S.8. You should not miss or neglect this one: