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Quantum mechanics math basics – tasting the notation

[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.[1]

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.[2]

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.[3] 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.[4]


[1] 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.

[2] For example, YouTube > “The Biggest Ideas in the Universe | 8. Entanglement” (May 12, 2020).

[3] 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).

[4] 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

  1. 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.

    Bohr and Heisenberg began to develop a mathematical theory of these quantum phenomena in the 1920s. Heisenberg’s quantum mechanics enumerated all the allowed quantum states, and implicitly assumed that jumps between them are instant — discontinuous, as mathematicians would say. “The notion of instantaneous quantum jumps … became a foundational notion in the Copenhagen interpretation,” historian of science Mara Beller has written.

    Another of the architects of quantum mechanics, the Austrian physicist Erwin Schrödinger, hated that idea. He devised what seemed at first to be an alternative to Heisenberg’s math of discrete quantum states and instant jumps between them. Schrödinger’s theory represented quantum particles in terms of wavelike entities called wave functions, which changed only smoothly and continuously over time, like gentle undulations on the open sea. Things in the real world don’t switch suddenly, in zero time, Schrödinger thought — discontinuous “quantum jumps” were just a figment of the mind. In a 1952 paper called “Are there quantum jumps?,” Schrödinger answered with a firm “no,” his irritation all too evident in the way he called them “quantum jerks.”

  2. 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.

    [from transcript]

    We often say that the particle could be in multiple states simultaneously, which is a gross misinterpretation or misrepresentation of what quantum mechanics actually says. (Which, you know what, we’ll probably get to in a future video.) It’s just that the states of these particles, they don’t exist like normal things exist. Welcome to quantum mechanics! Here we are!

    So we say this particle exists in a superposition of states, but that superposition is one state. It’s not multiple states. It is one existence for that particle. It’s just that existence doesn’t make classical sense. Right? That superposition is a combination of multiple classical states, the kind of states we’d expect a ball to be in.

    A superposition is a single state that just doesn’t make sense to our ape brains.

    Q: Is it like four dimensions if you will? How, like, we get three dimensions, but when you start talking about higher dimensions, we just really struggle with understanding what that’s like and how that works?

    Yeah, that’s exactly it.

    Q: If a person is not experimentally measuring it, then what is a measurement?

    I would say that a measurement is any particle interaction that releases information about a quantum system to the surrounding environment. Now, that environment might be a lab with a person in it, but it doesn’t have to be.

    … we’re trying to define a measurement like this so that quantum mechanics doesn’t sound like magic. Right? It’s not human beings are looking at the situation and we’re like: “We’re going to look at this and see what it’s doing and then it changes.” Like, we’re not affecting the physics of the problem, of the experiment. It’s just happening because of the way that we’re measuring it. In order to measure something about an electron, we have to do something like shoot photons at it.

    And it’s the photon that’s making the measurement happen, not the person.

    Q: So that measurement would happen if a person shot the photon or if the Sun shot the photon?


    We like to view physics as though we’re finding some deeper understanding about the universe. But that’s not really what physics is about. Physics is about making predictions.

  3. 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].

    My goal in this seven(!)-part series is to introduce the strangely beautiful effects of quantum mechanics and explain how they’ve come to influence our everyday world. Each edition will include a guided hike into the quantum mechanical woods where we’ll admire a new — and often surprising — effect.

    Series guidelines

    • No math.

    • No philosophy.

    • Everything we encounter will be experimentally verified.

    What I find fascinating is that “good enough” increasingly isn’t. Much of the technology developed in this century is starting to rely on quantum mechanics – classical mechanics is no longer accurate enough to understand how these inventions work.


    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).


    The wave nature of particles appears everywhere. The iridescence of hummingbird feathers and soap bubbles, the anti-reflection coatings on camera lenses, and the optics of electron microscopes all rely on the wave-like quantum motion of particles.

    Application: optical gyroscope.

  4. 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.

    “Let’s talk about process, let’s talk about how particle physicists analyze data, let’s talk about how they problem-solve,” Torpe [an Illinois high school science teacher who has taught professional development workshops at Fermilab] says. “The ideas of energy and cause and effect fit in naturally, too. I think a good strategy is to find a little bit of particle physics that you find interesting and pop it in here or there.”

    One such practice is taking students through “predict-observe-explain” cycles.


    • Cloud chamber

    • “Sharat [1] 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

    For now, COVID-19 has brought in-person professional development workshops to a halt. But teachers can still access some resources online: CERN’s hands-on learning lab S’Cool LAB (run until recently by Woithe), Perimeter Institute, Fermilab and QuarkNet offer free downloads of their interactive teaching materials.


    [1] Vinita Sharat, STEAM coordinator, Shiv Nadar School Noida, New Delhi.

  5. 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 [1].

    • YouTube > Sabine Hossenfelder > “Chaos: The real problem with quantum mechanics” (May 28, 2022)


    You have probably heard people saying that the problem with quantum mechanics is that it’s non-local or that it’s impossible to understand or that it defies common sense. But the problem is much simpler, it’s that quantum mechanics [QM] is a linear theory and therefore doesn’t correctly reproduce chaos. Physicists have known this for a long time but it’s rarely discussed. In this video I explain what the problem is, what physicists have done to try and solve it, and why that solution doesn’t work.

    [From transcript]

    Saturn has 82 moons. This is one of them, its name is Hyperion. Hyperion has a diameter of about 200 kilometers and its motion is chaotic. It’s not the orbit that’s chaotic, it’s the orientation of the moon on that orbit. It takes Hyperion about 3 weeks to go around Saturn once, and about 5 days to rotate about its own axis. But the orientation of the axis tumbles around erratically every couple of months.

    And that tumbling is chaotic in the technical sense. Even if you measure the position and orientation of Hyperion to utmost precision, you won’t be able to predict what the orientation will be a year later.

    Because quantum mechanics predicts that Hyperion’s chaotic motion shouldn’t last longer than about 20 years [the “Ehrenfest time”]. But it has lasted much longer. So, quantum mechanics has been falsified. Wait what? Yes, and it isn’t even news.

    That quantum mechanics doesn’t correctly reproduce the dynamics of classical, chaotic systems has been known since the 1950s. The particular example with the moon of Saturn comes from the 1990s.

    It can’t possibly be that physicists have known of this problem for 60 years and just ignored it? Indeed, they haven’t exactly ignored it. The have come up with an explanation which goes like this.

    [Explanation recap]

    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].

    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.

    This neatly resolves the problem: Hyperion correctly tumbles around on its orbit chaotically. Hurray. But here’s the thing. This only works if the collapse of the wave-function is a physical process. … you have to actually change something about that blurry quantum state of the moon for it to agree with observations. But the vast majority of physicists today think the collapse of the wave-function isn’t a physical process. Because if it was, then it would have to happen instantaneously everywhere.

    [The double-slit experiment for electrons is noted.]

    But the example with the chaotic motion of Hyperion tells us that we need the measurement collapse to actually be a physical process. Without it, quantum mechanics just doesn’t correctly describe our observations. But then what is this process? No one knows.


    [1] This video basically is a talking-head exposition. Key point visuals might strengthen Hossenfelder’s pitch.

    • Such as:

    • An example of chaos
    • Proposition: QM can explain chaos
    • Posit: Physical processes cannot be instantaneous.
    • Chaos is non-linear
    • The Schrödinger equation is linear
    • Collapse of the wave function is non-linear
    • Collapse of the wave function resolves agreement with observation
    • And that requires a physical process
    • But such a process must be instantaneous over spacetime!

    • 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 [3]. 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.)

    [2] She notes:

    The Schrödinger equation is linear, which just means that no products of the wave-function [such as ψ^2] appear in it. You see, there’s only one Psi on each side. Systems with linear equations like this don’t have chaos. To have chaos you need non-linear equations.

    [3] 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)

    Decoherence is responsible for the transition from the weird world of quantum mechanics to the ordinary world of everyday experience. Each time the environment interacts with a quantum system, it can be seen as a tiny measurement made on the system — a way for the environment to learn something about the quantum system and destroy its “quantumness.” Physicists have gotten very good at shielding their quantum experiments from anything in the environment that would disturb them. But they can’t shield them from gravity.

    Quantum chaos
    Credit: Pixabay/CC0 Public Domain

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