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!