The Spin Chronicles: Painting Quantum Tori (Part 1)

 Introduction: Spin, Tori, and Confusion Ahead!

Welcome, dear Reader, to another journey through the wild and wonderful world of quantum mechanics! Today, we dive headfirst into the realm of spin state vectors, grappling with a particularly beautiful concept: the spin tori. I promise we'll have some fun along the way, though I must issue a warning—things may get... twisty. 

Things may get... twisty

Especially when I casually decide to switch conventions that everyone else uses. Just a heads up!

A Quick Note on My Notation Shenanigans

Before we proceed, a quick confession. When I mention spherical coordinates, by φ, I mean latitude (measured from the North Pole), and by θ, I mean longitude. Now, the rest of the world? Well, they decided to do it the other way around! So if you're comparing my formulas to the ones in textbooks or online, remember that my φ is their θ and vice versa. Confused yet? Good. That's part of the charm.

Real Numbers vs. Complex—The Programmer's Dilemma

To keep things programmer-friendly, I’m sticking with real numbers, sines, and cosines. However, if you've cracked open a textbook or searched the internet, you've probably encountered those pesky complex numbers, accompanied by exp(iφ). While it's easy to translate between the two systems, the process can feel like translating Shakespeare into a meme—you're bound to lose some of the elegance along the way. So, I'll stick to the real stuff here and leave the complex translations as homework for the brave souls among you.

Spin Vectors: Real Numbers Edition

Now, let's get down to business. For us, the state vector is a column of four real numbers: X, Y, Z, and W, where the sum of their squares equals one. Textbooks, however, use two complex numbers a and b:

• a = X + iY
• b = Z + iW

It’s all the same thing, really—just more exciting when you throw in some complex numbers.

The Iconic State Vector |1)

Enter the state vector |1). Textbooks love this notation, but I’ll stick with using ) instead of the sharp "ket" symbol to avoid HTML shenanigans. It’s just a column with a 1 at the top and 0 at the bottom. For us, it looks like X=1, Y=Z=W=0. In quantum mechanics, this corresponds to a spin pointing along the z-axis. Pretty straightforward, right? Just don’t ask me to visualize it before my morning coffee.

Where the Math Gets Funky: Probabilities and Scalar Products

Now for the juicy part: if |u) is any state vector, the square of the scalar product |(1|u)|² gives us the probability that, when we measure the z-axis spin, we’ll get the result h/2 (the Planck constant divided by two). If u is our vector (X, Y, Z, W), and we write it as (X + iY, Z + iW), then the dot product of (1, 0) with this vector is X + iY. The square of the magnitude of this number is just X² + Y².

Angles, Angles, and More Angles

Let’s bring in some angles! Remember, I’m using φ, θ, and ψ. Here’s how they fit in:

• X = cos(φ/2)cos(ψ)
• Y = cos(φ/2)sin(ψ)
• Z = sin(φ/2)cos(ψ + θ)
• W = sin(φ/2)sin(ψ + θ)

Thus, X² + Y² = cos²(φ/2). And voilà! The probability that the spin is pointing up along the z-axis (with value h/2) is cos²(φ/2). And there you have it, folks—our φ angle now has physical meaning! We can even say, in true quantum mechanic fashion, that the "probability of transition" from state |u) to state |1) is cos²(φ/2). 

A 3D Quest: Visualizing the State Vectors

Now let’s level up. Suppose we want to visualize, in 3D, the set of state vectors where the probability of transition to the |1) state is 1/2. Easy enough: just take φ = π/2, since cos(π/4) = 1/√2, and cos²(π/4) = 1/2.

Let’s also recall our stereographic projection formulas:

• x = X/(1 - W)
• y = Y/(1 - W)
• z = Z/(1 - W)

Plug these into the expressions for X, Y, Z, and W, using a fixed φ, and we get some fancy formulas that lead to a very special surface. And guess what? That surface happens to be a torus! (Actually, it’s three nested tori, because one is never enough.)

Let’s Draw Some Tori (With MathMod)

Now, if you’re like me, staring at all these formulas will eventually lead to some existential questions, like “Why am I doing this?” But don't worry—I’ve got your back. To visualize these tori, I suggest you download and install MathMod. It’s free, it’s multiplatform, and, well, it works once you wrestle it into submission. I even wrote a script for you! Just save it as a .js file,copy and paste into Script Edit window,  run it, and let MathMod do the heavy lifting. With the mouse you can move the tori around.

Here is my script (you can also download it from here):

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.