[Communicating science series]
All hail vector spaces!
Imagine walking into an elementary school classroom and finding kids talking about quantum states. Depicting quantum interactions using diagrams and bra-ket manipulatives, for wave functions. Someday, eh.
While we may never achieve Ernest Rutherford‘s notion of a quantum theory so simple that we can explain it to an untutored barmaid, science communicators and teachers and even well-known physicists have introduced the subject to younger and younger students. Certainly while we all hunker down at home, online resources for exploring the topic are readily available.
When I taught middle school mathematics, the “algebra for all” movement was already underway. I was part of a new program which taught algebra to all 8th graders. Of course with mixed results. Sometimes grappling with “math is hard” and “I’m not good at math” attitudes.
In the end, however, mastery of algebra was evident when students not only talked the vocabulary but used it appropriately, their notational expressions matching their explanation.
So, yesterday, a YouTube video by “rebel” physicist Sabine Hossenfelder caught my attention: “Understanding Quantum Mechanics: It’s not so difficult!” In just 8 minutes, she unpacks the notational framework of quantum mechanics. And then points to the essential mathematical area of study – linear algebra – and online resources.
What is a vector, what is a matrix, what is an eigenvalue, what is a linear transformation?
And there’s the connection with the legacy of “algebra for all” – that algebra’s not so difficult – and quantum literacy.
• YouTube > Sabine Hossenfelder > “Understanding Quantum Mechanics: It’s not so difficult!” (July 18, 2020).
(caption) In this video I explain the most important and omnipresent ingredients of quantum mechanics: what is the wave-function and how do you calculate with it. Much of what makes quantum mechanics difficult is really not the mathematics. In fact, quantum mechanics is one of the easier theories of physics. The mathematics is mostly just linear algebra: vectors, matrices, linear transformations, and so on. You’ve learned most of it in school already! However, the math of quantum mechanics looks funny because physicists use a weird notation, called the bra-ket notation. I tell you how this works, what it’s good for, and how to calculate with it.
Not all physicists touch on the mathematics of quantum theory for the general public. Such books by famous physicists might, at best, have some references in appendices or footnotes for key mathematical stuff.
Other physicists, like Sean Carroll, have been routinely using wave function notation in online public chats and lectures. Even the “1 over square root of 2” coefficient in simple examples.
One of the lessons that was driven home when I was an aerospace engineer was that every specialty, every major program, came with jargon (and lots of acronyms), reflecting compact communications between members of that work environment and customer space. Understanding physics discourse is similar.
What interests me most about Hossenfelder’s presentation is commentary about the mathematical model and reality. That’s where some literacy can help. A better conversation about the necessary simplification in building a model. A better interplay of “the two cultures” in going forward. A better narrative for public policy.
I like the term “measurement update” regarding the (so-called) measurement problem (or wave-function collapse).
Quantum mechanics is pretty much just linear algebra. What makes it difficult is not the mathematics. What makes it difficult is how to interpret the mathematics. The trouble is, you cannot directly observe the wave-function. But you cannot just get rid of it either; you need it to calculate probabilities. But the measurement update has to be done instantaneously [hmm] and therefore it does not seem to be a physical process. So is the wave-function real? Or is it not? Physicists have debated this back and forth for more than 100 years.
 But algebraic concepts were introduced prior to the 8th grade. Most of my 6th graders, for example, learned the basics of equations (within a broader context of problem solving.) Typically with better results than in 8th grade.
And I remember, as a videographer in 2001, witnessing students in a 3rd grade class processing mathematical equations – translating verbal and visual descriptions into mathematical relations. All of them free of any educational rubrics (such as “new math”), fearless in their joy of grasping the language of numbers and expressions. Something which many of their elders (even parents) approached with dread.
Of course, in 6th grade, success depended on a solid grasp of arithmetic. Multiplication tables, for example. And waiting for the bell to ring at the end of class was a good time to call out multiplication challenges.
Another key skill for success was the ability to parse verbal and written descriptions of problems. Word problems. I saw this issue even before I started teaching, while observing high school math classes. The challenge of extracting the relevant information and encoding it into mathematical statements. In middle school, for example, translating various words into addition, subtraction, multiplication, or division; and the corresponding symbols for these operations.
So, a teacher’s toolkit embraced pictures, diagrams, manipulatives, etc. To address how the ways we learn best vary from person to person. There’s no “one size fits all” approach.
And it all had to be in a safe and low stress learning environment. Stress (including food insecurity) tends to compromise learning.
 For example, YouTube > “The Biggest Ideas in the Universe | 8. Entanglement” (May 12, 2020).
 A model is a “reduction of reality,” which makes “some claim about how our world works.” Without simplification, the chance of getting even close to the target reality or solving the equations becomes problematical. And solutions which match measurement bolster the credibility of the model, with predictions that are “good enough.”
Reference: “The Heisenberg Uncertainty Principle of Social Science Modeling” by Ben Klemens (July 7, 2020).
 Additional excerpts from transcript:
The mathematics of quantum mechanics looks more intimidating than it really is. To see how it works, let us have a look at how physicists write wave-functions. The wave-function, to remind you, is what we use in quantum mechanics to describe everything. There’s a wave-function for electrons, a wave-function for atoms, a wave-function for Schrödinger’s cat, and so on. The wave-function is a vector, just like the ones we learned about in school.
Now, the wave-function in quantum mechanics is … not a vector in the space we see around us, but a vector in an abstract mathematical thing called a Hilbert-space. One of the most important differences between the wave-function and vectors that describe directions in space is that the coefficients in quantum mechanics are not real numbers but complex numbers,
In quantum mechanics, we do not write vectors with arrows. Instead we write them with these funny brackets. Why? Well, for one because it’s convention. But it’s also a convenient way to keep track of whether a vector is a row or a column vector. The ones we talked about so far are column-vectors. If you have a row-vector instead, you draw the bracket on the other side.
This notation was the idea of Paul Dirac and is called the bra-ket notation. The left side, the row vector, is the “bra” and the right side, the column vector, is the “ket”. You can use this notation for example to write a scalar product conveniently as a “bra-ket”. The scalar product between two vectors is the sum over the products of the coefficients.
Now, in quantum mechanics, all the vectors describe probabilities. And usually you chose the basis in your space so that the basis vectors correspond to possible measurement outcomes. The probability of a particular measurement outcome is then the absolute square of the scalar product with the basis-vector that corresponds to the outcome. Since the basis vectors are those which have only zero entries except for one entry which is equal to one, the scalar product of a wave-function with a basis vector is just the coefficient that corresponds to the one non-zero entry. And the probability is then the absolute square of that coefficient.
The whole issue with the measurement in quantum mechanics is now that once you do a measurement, and you have projected the wave-function onto one of the basis vectors, then its length will no longer be equal to 1 because the probability of getting this particular measurement outcome may have been smaller than 1. But once you have measured the state, it is with probability one in one of the basis states. So then you have to choose the measurement outcome that you actually found and stretch the length of the vector back to 1. This is what is called the “measurement update.”
5 thoughts on “Quantum mechanics math basics – tasting the notation”
When I added the “[hmm]” in Hossenfelder’s statement that “… the measurement update has to be done instantaneously and therefore it does not seem to be a physical process,” I was recalling articles from 2019 about quantum jumps not being truly instantaneous (as explored in quantum trajectory theory). As well as my own puzzlement as to how any wave-like (frequency based) interaction could be instantaneous, taking zero time – the notion of discontinuities at the Planck level.
So, what’s the relationship between so-called wave-function collapse (measurement update) and real localized oscillations? The debate as to whether the mathematical model (wave-function) is real. Is that model free of any reduction in reality?
Is the problem a question of information about the quantum state? – collecting enough information without destroying the coherence before measurement. Are quantum “jump” events in fact gradual, coherent processes, albeit extremely fast?
For example, some research at Yale University:
• Physics World > “To catch a quantum jump” (07 Jun 2019) – re quantum trajectory theory.
• Quanta Magazine > “Quantum Leaps, Long Assumed to Be Instantaneous, Take Time” by Philip Ball (June 5, 2019) – An experiment caught a quantum system in the middle of a jump — something the originators of quantum mechanics assumed was impossible.
Nick Lucid at Science Asylum discusses some basics of quantum mechanics.
• YouTube > The Science Asylum > “My Wife Reacts to Quantum Mechanics” (Nov 27, 2020)
I particularly liked his wife’s comment about physicists “using words that don’t mean what we think that they mean.” Sort of shoehorning descriptions into a classical characterization. Like the words wave and spin.
Tasting the math works better with some personal experience sans the math.
• Ars Technica > “A ‘no math’ (but seven-part) guide to modern quantum mechanics” by Miguel F. Morales, Professor of Physics at the University of Washington in Seattle (1/10/2021) – Kitchen quantum mechanics [article includes images].
• No math.
• No philosophy.
• Everything we encounter will be experimentally verified.
Context: “… the fundamental mystery of quantum mechanics: particles move like waves and hit like particles.”
Analogy: “You can recreate this experience by playing a single note from a synthesizer through a pair of stereo speakers.”
• Classic “kitchen” grade setup
Experiment 1A – the classic double-slit (photon beam slit interference): laser pointer, aluminum foil, razor blades, darkened room, wall.
Experiment 1B – sort of single photon slit interference: darker ambience, weaker laser.
Experiment 1C – different colored lasers.
• Semi-pro lab grade setup
Experiment 2 – beam splitter photon path interference: variable power laser, lens, 2 half-silvered mirrors, 2 normal mirrors (sort of a table-top optical bench).
Application: optical gyroscope.
Particle physics in the classroom – experiencing quantum interactions.
• Symmetry Magazine > “High school teachers, meet particle physics” by Scott Hershberger (1/5/2021 ) – Workshops around the world train science teachers to incorporate particle physics into their classrooms.
• Cloud chamber
• “Sharat  fosters students’ creative sides in her lessons about particle physics by encouraging them to write poems, make videos or choreograph dances to explain the concepts they are studying.”
“We should know the reason of our existence. We should know what we are made of.”
• Online resources
 Vinita Sharat, STEAM coordinator, Shiv Nadar School Noida, New Delhi.
Here’s Sabine Hossenfelder’s take on how chaos exemplifies the debate over the physical reality of the wave-function update process (aka “collapse”) in quantum mechanics. An epistemic vs. ontic problem .
• YouTube > Sabine Hossenfelder > “Chaos: The real problem with quantum mechanics” (May 28, 2022)
1. Entanglement of dust and photons with the moon modifies the moon’s wave function, resulting in (quantum) decoherence. The change of the wave function in time is calculated using the Schrödinger equation, which is linear; and chaos involves non-linear equations .
2. Inability to know the what and the where for gazillions of dust grains and photons requires heuristics and statistical averaging – to agree with what classical Newtonian dynamics predicts.
Problems with this explanation:
A. “One is that it forces you to accept that in the absence of dust and light a moon will not follow Newton’s law of motion.”
B. “The more serious problem is that taking an average isn’t a physical process.”
An analogy to “a classical chaotic process like throwing a die” shows that probability models based on averages do not predict individual outcomes, which should agree with observations.
The wave-function is used to make probabilistic predictions. And the update of Hyperion’s wave function by all those dust grains and photons is indeed a non-linear process.
[The double-slit experiment for electrons is noted.]
 This video basically is a talking-head exposition. Key point visuals might strengthen Hossenfelder’s pitch.
• Such as:
• Here’s another view of Hossenfelder’s reasoning:
After presenting an example of chaos, her argument uses the logic of contradiction:
Premise #1: Suppose QM can mathematically model chaos – “reproduce the dynamics of classical, chaotic systems,” e.g., the chaotic motion of Saturn’s moon Hyperion.
Contradiction: The QM model fails beyond the Ehrenfest time (the Ehrenfest theorem “provides mathematical support to the correspondence principle.”)
Premise #2: The QM model can be fixed (in principle) by incorporating quantum corrections due to entanglement. That is, entangled interactions of the gazillions of dust grains and photons with Hyperion collapse the moon’s wave function into a state of decoherence . That wave-function update is a non-linear process.
Contradiction: That works if the wave-function update is a physical process, but the consensus is that it’s not. And if it was a physical process, then it’s an instantaneous change extended over space. But that’s an action which propagates faster than the speed of light!
(Solving the Schrödinger equation in this case requires a heuristic model of all those dust grains and photons and averaging over them. But averaging is not a physical process either.)
 She notes:
 As noted in this article:
• Quanta Magazine > “An Ultra-Precise Clock Shows How to Link the Quantum World with Gravity” by Katie McCormick (October 25, 2021)
Credit: Pixabay/CC0 Public Domain
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