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Quantum superposition and spinors – a saga of electrons

[What’s changed in the last ~100 years]

A recent Scientific American article reminded me that quantum spin underlies the stability of matter – without which there’d be no life. But the article prompted another dive into the “mathematical machinery” describing the quantum state of a single electron or a single photon.

The Stern–Gerlach experiment established that an atomic-scale system has intrinsic quantum properties. In particular, quantization of an electron’s (intrinsic) angular momentum – spin. As such, yet another quantum state with an observable in superposition. In this case, a superposition of up/down spin direction (along an axis).

Quantum superposition is a fundamental principle of quantum mechanics. … that every quantum state can be represented as a sum of two or more other distinct states. [Note reference to the diffusion equation in section on Hamiltonian evolution.]

Vectors and spinors and bispinors, oh my!


Here’s a historical recap of a seminal physics experiment 100 years ago – the Stern-Gerlach experiment [1]. Regarding a property of elementary particles which underlies the stability of matter – the spatial orientation of [intrinsic] angular momentum: the concept of quantum spin (in 4-D Hilbert space). In this case for leptons, e.g., electrons [2].

• Scientific American > Quantum Physics > “100 Years Ago, a Quantum Experiment Explained Why We Don’t Fall through Our Chairs” by Davide Castelvecchi (February 8, 2022) – The basic concept of quantum spin provides an understanding of a vast range of physical phenomena.

Without quite realizing what they were seeing, [Otto] Stern and his fellow physicist and collaborator Walther Gerlach discovered quantum spin: an eternal rotational motion that is intrinsic to elementary particles …

As physicist Wolfgang Pauli would explain in 1927, spin is quite unlike other physical concepts such as velocities or electric fields. Like those quantities, the spin of an electron is often portrayed as an arrow, but it is an arrow that does not exist in our three dimensions of space. Instead it is found in a 4-D mathematical entity called a Hilbert space [3].

Unlike in modern experiments, the displacement of the beams was tiny – about 0.2 millimeter – and had to be spotted with a microscope.

It was only after modern quantum mechanics was founded, beginning in 1925, that physicists realized that the silver atom’s magnetism is produced not by the orbit of its outermost electron but by that electron’s intrinsic spin, which makes it act like a tiny bar magnet.

… to this day, physicists continue to argue about how to interpret the experiment [regarding “quantum superposition” and the measurement problem].


The characterization of an electron as a spinor / Dirac spinor (aka bispinor) is helpful. But following all the math is beyond my ken. Everyday (3D) analogies offer some insight.

Spinors and tensors are associated with continuous geometries of energy density, stress, flux. Kinked energy, as I’ve noted elsewhere. The mathematical framework implies higher dimensional (3-sphere) topology. Which limits an intuitive grasp. Mathematically, the well-defined formalism may be routine; but visualization elusive – the geometric significance still mysterious.[4]

Notes

[1] Wiki notes that “the experiment was performed several years before Uhlenbeck and Goudsmit formulated their hypothesis of the existence of the electron spin.”

The Stern–Gerlach experiment has become a prototype for quantum measurement, demonstrating the observation of a single, real value (eigenvalue) of an initially unknown physical property.

[2] And, as also noted in Wiki’s article, the correspondence between an electron’s spin and angular momentum of the photon – photon polarization.

• Wiki > Spin (physics)

For photons, spin is the quantum-mechanical counterpart of the polarization of light; for electrons, the spin has no classical counterpart.

Spin is described mathematically as a vector for some particles such as photons, and as spinors and bispinors for other particles such as electrons.

[3] Wiki notes:

The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections.

The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy.

[4] Mathematical formalism / representation and reality

Of course, there’s an interesting history even for numbers which were considered fictitious in some sense. These numbers changed our perspective on things, whether or not actually existing or exactly measurable in our everyday “real” landscape. Useful representations (as are infinite series). Numbers like zero. Transcendental numbers (e.g., π and e). Irrational numbers. Imaginary numbers (e.g., the square root of negative one).

Spinors

Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or “spin”, of the electron and other subatomic particles.

Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group).

Regarding the history of irrational numbers, see also my March 16, 2017 comment re Effective Theory and Pi Day.

Terms

• Wiki > Dirac’s bra–ket notation

• Wiki > Spinor > Attempts at intuitive understanding

Several ways of illustrating everyday analogies have been formulated in terms of the plate trick, tangloids and other examples of orientation entanglement.

Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah’s statement that is recounted by Dirac’s biographer Graham Farmelo:

No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense they describe the “square root” of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors.

• Wiki > Bispinor

In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons.

A bispinor is more or less “the same thing” as a Dirac spinor. … That is, the Dirac spinor is a bispinor in the Dirac convention. By contrast, the article below concentrates primarily on the Weyl, or chiral representation, is less focused on the Dirac equation, and more focused on the geometric structure, including the geometry of the Lorentz group.

Bispinors are so called because they are constructed out of two simpler component spinors, the Weyl spinors. Each of the two component spinors transform differently under the two distinct complex-conjugate spin-1/2 representations of the Lorentz group. This pairing is of fundamental importance, as it allows the represented particle to have a mass, carry a charge, and represent the flow of charge as a current, and perhaps most importantly, to carry angular momentum.

Quantum spin — angular what?

A particle by any other name?

What the heck is superposition?

2 thoughts on “Quantum superposition and spinors – a saga of electrons

  1. Here’s a useful conceptual overview of the Stern–Gerlach experiment, with some delightful commentary on the mystery of an electron’s spin – what that means for a so-called point particle. (And in the first video’s description > questions, even a reference to volume 3 of The Feynman Lectures.)

    Video one

    • YouTube > Looking Glass Universe [1] > “What is Spin? | Quantum Mechanics” (Jul 30, 2015)

    Video two – a follow-up video

    • YouTube > Looking Glass Universe > “Is Spin Angular Momentum afterall? (‘What is Spin?’ follow up)” (Sep 7, 2015) – re the meaningfulness of intrinsic angular momentum: analogies such as “spinning without spinning.” (See note #2.)

    (from description of video one)

    12. Find other reasons we don’t believe electrons are actually spinning. (An interesting one is about rotating a spin particle 360 degrees, and not getting back the exact same wavefunction.)

    (from transcript of video one)

    I can measure the up or downness of a particle, so that is an observable. What are it’s eigenstates? Well, this experiment showed that the particle can only be fully up or fully down, and nothing in between, so there are only two eigenstates. We’ll call them spin up and spin down. Now we can apply the quantum mechanical principle that, to fully describe the wavefunction of this particle, we only need to describe it in one basis, so I can fully write the state of this particle in terms of up and down.

    I think this image of the electron spinning, is really more harmful than helpful – not just because it makes all kinds of incorrect predictions, but because feeling like we understand something stops us from asking about what it is.

    But, making this video I realized there’s lots of other physics terms I don’t understand. What is energy? What’s charge? It seems like we define those things by how we measure them, and then same is true for spin, spin is that thing that makes some particles act a bit like magnets.

    I really hope that, in the future as we understand more physics, more of these terms can be understood in terms of deeper physics. There is some hope for spin.

    I told you that spin had to be added to quantum mechanics in an ad hoc way. This is true, but when Dirac tried to merge standard quantum mechanics and relativity, something in his new equation acted a lot like spin. While this was awesome, I don’t feel like it solves the mystery of spin, because while it comes out of the maths, it still doesn’t explain what it is, or why relativity demands it exists. Maybe there are some properties of physics we can never understand, and maybe spin is one of them.

    Notes

    [1] Author: Yoganathan, Mithuna; University of Cambridge, PhD 2021.

    Also, on this YouTube Channel is a video about her personal journey in physics.

    • YouTube > Looking Glass Universe > “From being terrible at math to a quantum physicist – my journey” (Mar 5, 2019)

    [2] In the comments for this video (from 6 years ago), there’s an interesting perspective:

    The spin of an electron is nothing magical. The electron, according to Dirac, should be described by a 4 component complex wave function [dirac 4-spinor?]. As a wave, it is flowing. The Dirac equation of this wave always causes a turbulence around the electron. Even at rest, the electron carries a rotating energy and momentum flow around it. This generates a real angular momentum around the electron (besides additional orbital angular momentum it may have). As the wave function describes the distribution of electric charge, this turbulence is actually rotating charge, therefore there is a magnetic moment. It is an unfortunate accident of science that some guy said that we should not try to explain spin, and everyone agreed to do so, even after the explanation was given.

    Interesting because of terminology from fluid dynamics: wave, flow, turbulence. But also that, in a sense, there’s “charge without being charged” – that charge is an interaction (the observable) with the Grid, rather than an actual property of an electron. An interaction with “rotating energy and momentum flow [flux],” much like eddies in a superfluid.

    And as a comparison with my April 15, 2019 comment about virtual particles, quoting Wiki re zitterbewegung:

    The apparent paradox in classical physics of a point particle electron having intrinsic angular momentum and magnetic moment can be explained by the formation of virtual photons in the electric field generated [induced?] by the electron. These photons cause the electron to shift about in a jittery fashion (known as zitterbewegung), which results in a net circular motion with precession. This motion produces both the spin and the magnetic moment of the electron.

    Wiki > zitterbewegung

    In physics, the zitterbewegung (“jittery motion” in German) is the predicted rapid oscillatory motion of elementary particles that obey relativistic wave equations. The existence of such motion was first discussed by Gregory Breit in 1928 and later by Erwin Schrödinger in 1930 as a result of analysis of the wave packet solutions of the Dirac equation for relativistic electrons in free space, in which an interference between positive and negative energy states produces what appears to be a fluctuation (up to the speed of light) of the position of an electron around the median, …

    In quantum electrodynamics the negative-energy states are replaced by positron states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron pairs.

  2. The Space.com article noted below starts out with this statement: “[electrons] are considered to be both partially particle-like and partially wave-like, depending on the scenario, according to West Texas A&M University.

    That’s a somewhat problematic legacy characterization [1], which might be better stated as: electrons are localized vibrations (excitations) in the associated electron quantum field. The language of “particles” is a convenient simplification in some contexts.

    Then the article includes some historical recap of discovery and the atomic model, referencing these resources: Khan Academy, Cambridge Coaching.

    • Space.com > “Electrons: Facts about the negative subatomic particles” by Daisy Dobrijevic (March 15, 2022)

    Electrons surround the atomic nucleus in regions of space known as orbitals. According to the educational website Cambridge Coaching, an electron orbital is an area surrounding the nucleus where there is a high probability (over 90%) of finding an electron. “Orbitals are not an exact place but rather an area that includes that exact place” according to Cambridge Coaching. Each orbital can hold up to two electrons, according to the educational website Khan Academy.

    Notes

    [1] Problematic in the sense that: (a) the article does not mention quantum field theory (QFT) or even the notion of fields; and (b) such characterization perpetuates the notion that electrons are both particles and waves.

    • Regarding fields, my post “Point particles RIP” notes that career physicist Art Hobson wrote:

    To find out what textbooks say, I perused the 36 textbooks in my university’s library having the words “quantum mechanics” in their title and published after 1989. 30 implied a universe made of particles that sometimes act like fields, 6 implied the fundamental constituents behaved sometimes like particles and sometimes like fields, and none viewed the universe as made of fields that sometimes appear to be particles. Yet the leading quantum field theorists argue explicitly for the latter view (Refs. 10-18). Something’s amiss here.

    • As physicist Paul Sutter noted, “Every kind of particle that scientists know of, from the electron to a photon, is associated with its own space-time-filling vibrating field.

    • And as physicist Sean Carroll noted, “These days we know it’s all just quantum fields, and both matter and forces arise from the behavior of quantum fields interacting with each other.”

    Photons are excitations in the electromagnetic aka photon field.

    Related posts

    Reality of fields, language of particles – the Standard Model

    QFT – How many fields are there?

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