This is a simple quantum state …
… built from an up |↑> state and a down state |↓>. α and β are complex numbers. The result |➚> is in the middle, oblique. The oblique state is a superposition or the up and down base states.
Making a measurement, you use a (polarization-like) filter. The filter can decide between up or down, along some direction. When the measurement is done, the system is in either the filter’s up or the down state.
Measuring |➚> using a |↑>-or-|↓> apparatus, you will obtain the |↑> result with a probability of |α|2 and |↓> with |β|2. Only the squares of the absolute values are accessible to intuitive interpretation. Or are they?
Roger Penrose argues that these complex numbers can have a tangible interpretation (in his book The Road to Reality). The Riemann sphere and the magic of complex numbers (his words) come to rescue. Up and down could be any abstract states, |↑> and |↓> are just symbols. But in the case of, say, an electron, the arrows stand for its spin aligned up and down an axis in three-dimensional space. The spin is the quantum analogue of angular momentum.
The angle θ at which |➚> is titled visibly, is directly related to a beautiful geometric representation of the ratio of complex numbers α and β – when you invoke stereographic projection (introduced here, here, and here).
In quantum mechanics, you only care about directions. All states are normalized. The probability to measure the electron being in any of the available states has to be 1.
Going from up to down is sort of a rotation. For two arbitrary complex numbers this means:
β = √(1-|α|²)∙eiφ
… where φ is an arbitrary phase angle. Normalization determines the relationship between the magnitudes of α and β. You can whirl around either of them in the complex plane – by some angle φ.
I want to relate α and β to the only angle in the game that we believe to understand intuitively – θ. For up |↑> it is zero, for down |↓> it is 180° (π). If up and down contribute equally, then it has to be 90° (π/2), for symmetry reasons: The arrow is perpendicular to both up and down: |➙>. In a series of measurements with an |↑>-or-|↓> filter, up and down come up with the same probability. (Perpendicular refers to our three-dimensional space, in the two-dimensional space of quantum states, |↑> and |↓> would be called perpendicular to each other.)
θ=π ⇒ |α|=0,|β|=1
θ=π/2 ⇒ |α| = |β|
Can the magnitude of α and β be parameterized in terms of θ? The square root in |β| = √(1-|α|2) hints at one magnitude being a cosine, the other one a sine. Of which angle? θ itself does not work: For both 0 and π the sinθ is zero. We need an angle whose sine and cosine are the same if θ is π/2. Testing simple functions of θ: It is θ/2! This meets all conditions.
Getting back to θ, using trig identities:
|β|² = sin²(θ/2) = ½(1-cosθ) (✺)
What does this mean geometrically? Think of the sphere with a f radius R of 1. Intersect the θ-inclined spin axis with the sphere. This is a point on the sphere. Connect this point with the South Pole – this connection is the projection ray of stereographic projection. Intersect the connection line with the xy-plane – the equatorial plane of the sphere.
In the xy-plane, the intersection point lies on a circle with radius u. (Where exactly on the circle, depends on phase angle φ). Geometry – similar, right-angled triangles – and some trigonometric identities tell us:
R=1 ⇒ u = sinθ / (cosθ + 1) = tan(θ/2)
cosθ = (1-u²)/(1+u²) (✺✺)
Getting back to α and β! A ratio of two complex numbers is again a complex number – described by two real numbers, two parameters. These parameters could be two angles or one length and an angle. Rotating an arrow of length 1 in three-dimensional space needs two angles. Stereographic projection turns this into a length (u) plus an angle (φ) in the equatorial plane.
Plugging in the results from normalization and constraints for special cases of θ (✺) and from the geometry of stereographic projection (✺✺):
(✺✺) ⇒ (1 – cosθ) / (1 + cosθ) = u²
Every complex number u∙eiφ corresponds to a point in the ‘xy’-plane. It can be projected up to the Riemann sphere. Every point on this sphere – a surface – can be described with two parameters. Like u and φ. Or θ and φ.
You can also find θ/2, |β|, and |α| in this image. The angle near the South Pole has to be θ/2 because of the equilateral triangle with two sides being equal to the radius R=1. Because of normalization, |β| and |α| have to be sides in a right-angled triangle whose hypotenuse is 1.
Introductory texts on quantum mechanics often introduce a much more complex system first – one whose state vector has infinite dimensions. The wave function that describes the probability to find an electron orbiting a proton. The electron could be found in any point in space. To tame it, sort of, and normalize it, you need a delta function in space. The probability wave is described by Schrödinger’s ‘wave’ equation that may invoke ‘familiar’ memories of hydrodynamic waves.
Richard Feynman thought otherwise when introducing quantum mechanics. The familiar wave function only shows up very late in Volume III of his Physics Lectures. Feynman starts with a particle that has spin. He conjures up the mathematics of the underlying (Lie) group transformations without ever using these words, in colloquial language and simple math only. He gives you the feeling that it is all about properties of space and rotations, but it is still hard (without looking at Lie groups as purely mathematical structures) where pure math ends and the peculiarities of the physics of our world begins. I recommend to complement any light reading with Frederic Schuller’s rigorous Lectures on the Geometric Anatomy of Theoretical Physics.
Going from rotations in 3D space (about θ) to rotations of a ‘2D complex vector about θ/2’ is the underlying reason of the weirdness of rotations that can be demonstrated by Dirac’s Belt Trick. David Tong‘s lecture on Dynamics and Relativity has also a short and accessible chapter on these so-called spinors – objects with two complex components that can represent vectors in four-dimensional spacetime. A most concise intro to Lie group SU(2) being a double cover of SO(3).
Image created with SageMath, all lines and ‘wire loops’ crafted as individual parameterized curves. It is not possible (yet) to use LaTeX is 3D text in SageMath. So, I am using Unicode characters instead of LaTeX also in the text here. Thanks to WordPress for letting me use inline style tags in the HTML source code here – to format formulas in the text as the drawing’s legend. (I’ve had to change the characters for the the quantum state arrows and angle brackets after publication – the ones I chose for the images do not seem to work on some smartphone browsers.)
All of this was just an excuse to craft this image – and rotate and zoom it. The text does not scale with zooming, which makes for an interesting constraint. I have not annotated the sides of the triangle colored in purple – these are |α| and |β|.
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