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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.”

One thought 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.”

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