The Spin Chronicles: Painting Quantum Tori (Part 2)

 The Mysterious Phase of Quantum Mechanics: A Spin on Reality

In quantum mechanics, the phase of a wave function is treated like that distant cousin you only hear about but never meet—it's assumed to be unobservable. The star of the show is the amplitude, or more precisely, the square of its modulus. That’s the part we can see and measure. Textbooks say: "Only this, and nothing else, is observable." And sure, that’s one way of looking at it.

But let's face it—opinions in quantum mechanics are as varied as the stars in the sky. Physicists are like travelers in a big, wide world, each with their own unique map of how quantum mechanics works. In fact, no two physicists share exactly the same view—unless, of course, one of them is just crunching numbers from a cookbook of equations. And even then, not all cookbooks are created equal!

The Wave Function and the Art of Spin

We often talk about the "wave function" when we’re trying to pinpoint the location of an object. You know, that quantum object that seems to be everywhere and nowhere at the same time? But today, let's zoom in on something a little more grounded: spin.

Now, spin is a different beast. The object in question stays put, but its spin axis can change direction. This is where we enter the world of the state vector. For spin-½ particles, this vector is made up of four real numbers: $X$, $Y$, $Z$, and $W$. Their squares add up to one, which is pretty neat. If you’re a fan of complex numbers (and let’s be real, who isn’t?), it’s just two complex numbers whose modulus squares also sum to one. The conversion between the two is surprisingly simple:

$$z_1=X+iY\text{and}{z}_2=Z+iW$$

This lets us break the real part and imaginary part down like so:

$$X=\text{Re}(z_1),Y=\text{Im}(z_1),Z=\text{Re}(z_2),W=\text{Im}(z_2)$$

Easy, right? Well, it gets more interesting.

From Numbers to Angles: Theta, Phi, and Psi

Instead of juggling numbers like $X$, $Y$, $Z$, and $W$, it’s often more intuitive to use angles—

θ, ϕ, and ψ. 

These angles help specify the position of our vector on a three-dimensional sphere that lives in four-dimensional space. So, the coordinates transform into:

$$X = \sin\left(\frac{\phi}{2}\right)\cos(\psi), \quad Y = \sin\left(\frac{\phi}{2}\right)\sin(\psi)$$
$$Z = \cos\left(\frac{\phi}{2}\right)\cos(\psi + \theta), \quad W = \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)$$

The Projection into 3D Space

At this point, we need to bring things back into the three-dimensional world we're used to. To do that, we use something called stereographic projection, which I’ve covered in detail a couple of blog posts ago. This allows us to map the points from four-dimensional space into our three-dimensional world. The result gives us familiar coordinates:

$$x(\theta, \phi, \psi) = \frac{\sin\left(\frac{\phi}{2}\right)\cos(\psi)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$ $$y(\theta, \phi, \psi) = \frac{\sin\left(\frac{\phi}{2}\right)\sin(\psi)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$ $$z(\theta, \phi, \psi) = \frac{\cos\left(\frac{\phi}{2}\right)\cos(\psi + \theta)}{1 - \cos\left(\frac{\phi}{2}\right)\sin(\psi + \theta)}$$

What's the Deal with Theta, Phi, and Psi?

So what’s the physical meaning behind these angles? In our lab, the $z$-axis points upwards (think of it as "North" on a globe), and the ϕ angle measures the tilt of the spin axis—essentially, it's latitude. θ is the longitude, with the x-axis as 0 degrees and the y-axis at 90 degrees (or π/2 if you’re feeling mathematically fancy).

But the ψ angle? That’s where things get interesting. Psi is like the "invisible phase," a ghostly presence that Richard Feynman himself couldn't ignore when he wrote about spin. And today, we’re going to shine a light on this elusive angle.

Painting the Invisible: Great Circles and Wheels

Our mission here is to bring these invisible aspects of the state vector to life. As we vary psi from 0 to $2\pi$, something magical happens—a closed curve appears. This curve, on a three-dimensional sphere in four-dimensional space, is known as a "great circle."

You’ve seen great circles before—they’re the big loops you get when slicing through a globe. Meridians and the equator are great circles. The same idea applies here, except now we’re in four dimensions. These invisible great circles can be projected back into our familiar 3D space as... wait for it... circles! That’s why this blog post is titled Invisible Wheels.

A Torus of Invisible Wheels

Now, imagine a large torus made of these invisible circles—these are called Villarceau circles, similar to what you’d find in the architecture of Strasbourg Cathedral. But here’s the catch: each of these circles is perceived as just a single point—a spin pointing in a particular direction ($theta, phi$). The psi phase, which determines where we are on that circle, remains hidden from us.

Want to see what these circles look like? I whipped up some visualizations in Mathematica, and here’s what I got: 

X[theta_, phi_, psi_] = Sin[phi/2]*Cos[psi];

Y[theta_, phi_, psi_] = Sin[phi/2]*Sin[psi];

Z[theta_, phi_, psi_] = Cos[phi/2]*Cos[psi + theta];

W[theta_, phi_, psi_] = Cos[phi/2]*Sin[psi + theta];

ParametricPlot3D[Table[

  {X[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p]), 

   Y[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p]), 

   Z[i*Pi/18, Pi/4, p]/(1 - W[i*Pi/18, Pi/4, p])}, {i, 0, 35}], {p, 0,

   2 Pi}, PlotRange -> All, PlotStyle -> White, Background -> Black]

Still not satisfied? I’m currently experimenting with MathMod to generate even better images, but for now, this will have to do.

Wrapping Up

In the next post, we’ll explore how these tori and invisible wheels interact with each other. We’ll dive even deeper into their physical interpretation, so stay tuned!

For those of you curious about 4D visualizations, check out Dimensions Math. Search for "Hopf fibration"—it’s the key to everything we’ve discussed, even though I haven’t officially mentioned it... yet.

Until next time, keep spinning those wheels—visible or not!

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):