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):

{
   "Param3D" :
   {
      "Component" : [
      "Torus1",
      "Torus2",
      "Torus3"
      ],
      "Fx" : [
      "sin(pi/8)*cos(v)/(1-cos(pi/8)*sin(u))",
      "sin(pi/4)*cos(v)/(1-cos(pi/4)*sin(u+v))",
      "sin(3*pi/8)*cos(v)/(1-cos(3*pi/8)*sin(u+v))"
      ],
      "Fy" : [
      "sin(pi/8)*sin(v)/(1-cos(pi/8)*sin(u))",
      "sin(pi/4)*sin(v)/(1-cos(pi/4)*sin(u+v))",
      "sin(3*pi/8)*sin(v)/(1-cos(3*pi/8)*sin(u+v))"
      ],
      "Fz" : [
      "cos(pi/8)*cos(u)/(1-cos(pi/8)*sin(u))",
      "cos(pi/4)*cos(u+v)/(1-cos(pi/4)*sin(u+v))",
      "cos(3*pi/8)*cos(u+v)/(1-cos(3*pi/8)*sin(u+v))"
      ],
      "Name" : [
      "Torus"
      ],
      "Umax" : [
      "3.52*pi/8",
      "2*pi",
      "2*pi"
      ],
      "Umin" : [
      "-3*pi/2",
      "0",
      "0"
      ],
      "Vmax" : [
      "2*pi",
      "3*pi/2",
      "2*pi"
      ],
      "Vmin" : [
      "0",
      "0",
      "0"
      ]
   }

}

And here are the results:

You will notice that in the script I messed up, and exchanged cos(φ/2) with sin(φ/2), so you may want to fix it, changing also the arguments. I will fix it next time.  

What’s Next? More Tori, More Fun!

Okay, this post is getting long, and I don’t want to scare anyone away before we really get into the quantum weeds. So, let’s pause here, and in the next post, we’ll dive deeper into the toroidal beauty of spin and perhaps even solve the mystery of why the universe insists on being so complicated.

A quantum spin, like autumn’s bloom,
A fleeting dance in nature’s room,
With every turn, a petal falls,
Yet beauty lingers through it all.

Until then, happy math-ing!

P.S. 11-10-24 9:44 Reading Mary Rose Barrington, Ian Stevenson and Zofia Weaver  - "A World In A Grain Of Sand. The Clairvoyance Of Stefan Ossowiecki",  McFarland & Company (2005).

There, p. 32, Experiment by Gustave Geley, a physician who had turned his considerable talents to the field of psychical research and become both the Director of the Institut Metapsychique International (IMI) in Paris and the editor of its journal, the Revue Metapsychique (RM):

Experiment 3: 21st September Chez Prince Lubomirski, in the same conditions as in experiment no. 2

I recognize the envelope as containing one of the two letters prepared by me. I did not know which of the two it was. I opened it up only after M. Ossowiecki had finished speaking.

TARGET MATERIAL
1. An oriental landscape; some camels.
2. [drawing of a fish, facing right]
3. The ringing of bells.
4. The scent of Mimosa.
5. Long live Poland! 

 

WORDS SPOKEN BY M. OSSOWIECKI

"This thing, it's long.
"This was written by a man. It's chaotic! It's so chaotic I can't grasp it! It's four or five ideas, a broth of ideas!
"Something of a certain size. Something that swims. I see some work on a fish that reminds me of a corassin [a very large Polish fish]. It's not writing, but there is a fish. What's the connection between this fish and Poland? I can't understand it. It's an exclamation: Long live Poland!
"Why Poland?
"I smell something, perfume, delicious perfume." (M. Ossowiecki seems to breathe in these scents.)
"There is also something to do with nature.
"There is one more thing, three things in this broth. I see the fish, I am going to draw it." (He makes the drawing shown below.) "Why these scents?
"There is a numbering system, 1st, 2nd, 3rd, 4th, 5th. After the No. 2 there is nothing written.
"Something to do with nature ... I don't see any more." 

 

To sum up, out of the five different ideas comprising this target, M. Ossowiecki fully grasped those behind Nos. 2, 4 and 5. He perceived nothing for No. 3, and had a very incomplete idea of No. 1.
He had a clear, dominant and persistent notion of fish; but, curiously enough, his drawing is not at all like mine. His fish is wide across and looks to the left. Mine was long in the body and looked to the right. For target No. 4, he was aware of delicious perfumes, without specifying that it was mimosa that was in question. The idea behind No. 5 was given in its entirety. 

During the whole of the evening that we spent together M. Ossowiecki remained obsessed by the drawing of the fish .

How shall I explain with physics and math these experiments? There must be some holographic principle at work. Some kind of associative navigation in the space of information, beyond space and time. What kind of space is this space of information? Is it just, in the first approximation,  some space of probability distributions (signed measures)? Or, in a toy model, a subspace of the space of bilinear forms on some vector space? How to "read the wave" stored holographically in this space?   

 P.S. 12-10-24 12:25 Here is Bjab's beauty:

{

   "MathModels" : [

   {

      "Param3D" :

      {

         "Component" : [

         "Torus1",

         "Torus2",

         "Torus3"

         ],

         "Fx" : [

         "cos(pi/8)*cos(v-u)/(1-sin(pi/8)*sin(u+v))",

         "cos(pi/4)*cos(v-u)/(1-sin(pi/4)*sin(u+v))",

         "cos(3*pi/8)*cos(v-u)/(1-sin(3*pi/8)*sin(u+v))"

         ],

         "Fy" : [

         "cos(pi/8)*sin(v-u)/(1-sin(pi/8)*sin(u+v))",

         "cos(pi/4)*sin(v-u)/(1-sin(pi/4)*sin(u+v))",

         "cos(3*pi/8)*sin(v-u)/(1-sin(3*pi/8)*sin(u+v))"

         ],

         "Fz" : [

         "sin(pi/8)*cos(u+v)/(1-sin(pi/8)*sin(u+v))",

         "sin(pi/4)*cos(u+v)/(1-sin(pi/4)*sin(u+v))",

         "sin(3*pi/8)*cos(u+v)/(1-sin(3*pi/8)*sin(u+v))"

         ],

         "Name" : [

         "Torus"

         ],

         "Umax" : [

         "pi/2",

         "pi/2",

         "pi/2"

         ],

         "Umin" : [

         "-pi/2",

         "-pi/2",

         "-pi/2"

         ],

         "Vmax" : [

         "pi/2",

         "pi/2",

         "pi/2"

         ],

         "Vmin" : [

         "-pi/2",

         "-pi/2",

         "-pi/2"

         ]

      }

   }

   ]

}