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The Mathematics of Quantum Mechanics [pdf] (uwaterloo.ca)
189 points by sajid 6 months ago | hide | past | web | favorite | 53 comments

Most intro courses on QM that I've seen cover exclusively problems of operators with continuous spectra (like position and momentum). For those, the mathematics of L2 space is critical (and, in my experience, woefully neglected). Because multiple observables are rarely treated, tensor products and entanglement are less important.

For intro to QIT, thoroughly understanding the C^2 space and tensor products is critical. This book appears to be useful for that. Unfortunately, it can be hard to make the jump to the former way of looking at things, without someone explicitly pointing out how to think of L2 as a Hilbert space. Maybe this is just intuitive to some, but it wasn't for me.

And the converse seems to be true, too: you can apparently master the former without having a clue about the latter. As Scott Aaronson points out:

> Today, in the quantum information age, the fact that all the physicists had to learn quantum this way seems increasingly humorous. For example, I've had experts in quantum field theory -- people who've spent years calculating path integrals of mind-boggling complexity -- ask me to explain the Bell inequality to them. That's like Andrew Wiles asking me to explain the Pythagorean Theorem.


This reminds me of Peter Woit, a theoretical physicist who has written a book on "Quantum Theory, Groups and Representations", asking if QM is a probabilistic theory.


By the way, this example illustrates the point I made in another comment: you can have a long career in physics and even become professor of theoretical physics without ever caring about the interpretation of QM.

Edit: I can't resist copying the opening of a recent comment in that thread.


John Baez says:

September 18, 2018 at 4:05 pm

Peter Woit wrote:

"The state of the world is described at a fixed time by a state vector, which evolves unitarily by the Schrodinger equation. No probability here."

And perhaps no physics here, either, unless we say how the state of the world is described by that vector: that is, how we can use the vector to make predictions of experimental results.

(A long and insightful comment by Baez follows)

To link the two approaches, it can be useful to understand the wave function as an object that takes values in a tensor space of C^n. In particular, if we had two particles, both spin-1/2 (C^2), then at the position configuration point {x,y}, the wave function takes a value in C^2 tensor C^2 where the tensoring is not done over the particle numbering 1 and 2 (which does not exist), but done over the set {x,y}. This is how one naturally leads into the choices of bosons vs. fermions.

To say it another way, the configuration of the system remains rooted in position in physical space and it is the value space of the wave function where spin resides. This is made most clear in Bohmian mechanics, where the position of the particle is always defined in the theory, but whether a particle will end up in the spin-z up or down regions does depend on the experimental setup. One can arrange experiments where the particle, due to symmetry for example, will always go up given the same starting point even if one flips the magnetic field in the Stern-Gerlach device so that the experimental conclusion would be spin up in one scenario and spin down in the other. Same exact path happens in both cases, but the spin value conclusion is the opposite. That is to say, while position is determined ahead of time, spin is not. Spin is not a real property of the particle; it is a property of the wave function.

This looks really nice. But can someone recommend something more like "The Practical Experiments of Quantum Mechanics"?

I've read some articles and seen some video courses on Quantum Mechanics, and it seems that there's always way more focus on the mathematics and abstract/simplified physics, and not much description of the actual experiments. I get that the main challenge of teaching it is understanding the mathematics, but it's hard to stay focused when - even though I understand the equations - I'm not sure what the mathematics describes physically. Or when I do understand it, that I have no idea how one would go about preparing the state.

I think I've gotten a half decent understanding of the Stern–Gerlach experiment, but that's about it.

I would love to get a lab manual to accompany Griffith's book on QM. I actually find it bizarre how little information there is, and I have looked. My gut tells me someone with an M.A. in physics, $5,000, and a lot of time could set up a hobby lab in her or his garage to validate everything through QED...

As for interpetation of QM, I don't think that's the problem since one can just set up an experiment based on prediction from the mathematical models. Sure, you will have make multiple runs to show the predicted distribution of the non-deterministic, umm, stuff, but that isn't a problem.

Maybe you will find some introductory stuff on quantum optics interesting. Check Nicolas Gisin's book "Quantum Chance".

A few articles about the experimental violation of Bell’s inequalities:

https://static.scientificamerican.com/sciam/assets/media/pdf... https://qudev.phys.ethz.ch/phys4/studentspresentations/epr/a... http://drchinese.com/David/Aspect.pdf http://www.rpi.edu/dept/phys/Courses/PHYS4510/AspectNature.p...

Quantum Mechanics, An Experimentalist Approach: https://www.amazon.com/gp/product/110706399X/

If one wants to get into some of the philosophy about it, then try: Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Theory https://www.amazon.com/gp/product/3319658662

This looks really nice. But can someone recommend something more like "The Practical Experiments of Quantum Mechanics"?

You might have a look at https://www.amazon.com/Exploring-Quantum-Physics-through-Pro... . This book stops short of covering molecular beams, though.

Feynman volume III: it's very wordy, and concrete.


You might also like John Bell's original papers. Feynman doesn't do entanglement. Conway's "free will" paper is good aswell.

Famously there is no consensus of how to interpret the equations of quantum mechanics, which naturally restricts introductory discussion to the mathmatical aspects.

I'm not sure the lack of consensus is the reason. One can learn QM and use it for a whole career without ever worrying about these issues.

I'm not sure why do you mean by "introductory discussion" but I think "popular science" books focus mostly on the controversial parts. At the university level, at least in my experience, you study a lot of classical physics, early quantum physics (1900-1925), and mathematical physics before taking a first look at Quantum Mechanics (Schroedinger/Heisenberg). And you may never be exposed to the relativistic extensions.

I meant the discussion in the intro courses; Griffiths just plonks down the Schrodinger equation on page one and you just have to eat it.

In contrast a textbook about thermodynamics will be all steam engines at the start, or maybe talk about a thermometer. Solid, sensible things and no talk about Gibbs free energy or any of the deep theory that exists.

You could start a QM book talking about the double slit experiment, but would have to write a lot of "one interpretation" and "some views say this". Or you could just brazenly pretend there is no controversy and make every reader that knows of it very suspicious indeed.

I’m not familiar with that book. I learnt QM from a 2-volume, 1500 pages book that gives you Schroedinger’s equation 10 pages in. But that book made no claim of being “introductory” and I previously had a “quantum physics” course including the different things that lead to Schroedinger’s equation (Plank, Rutherford, Bohr, etc).

In any case introductory books that discuss the physics and not just the mathematics do also exist, like vol 3 in Feynman’s lectures.

Edit: I’m looking at Griffiths’ book and after defining Schroedinger’s equation in section 1.1 he devotes three pages to discussing its statistical interpretation in section 1.2 and mentions different interpretations, the Bell inequalities and the measurement problem. He comes back to these philosophical issues at the end of the book in a 12-page afterword.

>You could start a QM book talking about the double slit experiment

That is indeed how my favorite QM book starts: Cohen-Tannoudji. See also Landau for one that starts from experiment and not theory.

Yes, but it quickly cuts to the meat :-) (I mean C-T/D/L, that’s the book I was talking about in a sibling comment)

I think there is a ton of consensus on how the equations predict lab results, however, and I would be very happy with that.

For practical uses the "shut up and calculate" interpretation of QM works just fine.

For a recent head-scratcher, though, see https://www.nature.com/articles/s41467-018-05739-8

There was some discussion about that paper here: https://news.ycombinator.com/item?id=18023452

I think it's much less interesting than it seems at first sight.

Related (and clearer): In Defense of a "Single-World" Interpretation of Quantum Mechanics (Jeffrey Bub) https://arxiv.org/abs/1804.03267

Fry: Hey, professor, what are you teaching this semester? Prof. Farnsworth: Same thing I teach every semester, the mathematics of quantum neutrino fields. I made up the title so that no student would dare take it. Fry: [writing] Mathematics of wonton burrito meals... I'll be there! Prof. Farnsworth: Please Fry; I don't know how to teach. I'm a professor.

Semi off-topic, for a more in depth treatment I recommend "Mathematics of Classical and Quantum Physics" by Byron and Fuller.

It's a graduate level text, so probably somewhat dense if you haven't been exposed to the material in some way before, but as a mostly refresher I found it very good.

And, it's a Dover book, so it's very cheap.

I just spent about half an hour reading from the preview on Amazon, and I've got to say I was super impressed with the style—very clear and focused on the most interesting things.

Agreed. I think this was the first book that helped me understand L2 as a Hilbert space.

Since it isn't said in the title, this is based on the abstract formulation ie Dirac (material usually covered extensively in QMII courses) and the book is suitable to high school students interested in a QM primer for use in quantum computation yet don't want to delve in physics or mathematics. Also it is mostly self-contained. About 60 pages (13-70) is math background (complex numbers and linear algebra). The QM part is exactly 30 pages (71-101). The final pages are some extra stuff (proofs, trigonometry, etc).

Looks like "Mathematics of Quantum Computing" to me. All the analysis fun is missing.

" Unfortunately, most high school mathematics around the world do not teach linear algebra" Yeah, lol no. Most high-schools do actually around the world. Only in US most don't. Funny, stupid but funny

This is a really terrific and accessible introduction to QM math.

If you get to the end, it is worth noting the similarity of the tensor product and the join operation in a relational database.

This line of thought is very interesting indeed. Understanding the similarity/contrast to the join operator is really at the heart of what quantum is about. It comes down to, what do you mean by "and" ?

Check out the breezy writing in this paper: https://arxiv.org/abs/1602.07618

Also, the Rosetta stone paper (more advanced): https://arxiv.org/abs/0903.0340

"the tensor product and the join operation in a relational database." can you elaborate more. very interesting!

If one column contains the basis vectors of one space, and another column contains the basis vectors of another space, then their cross join will contain all pairs of basis vectors. If you were to take the tensor product of each pair you'd end up with a basis for the tensor product space. This is because the Cartesian product shows up in both places.

I'm not sure if this is what the parent comment is getting at, and I'm not sure I see anything deep lurking there, but that could just be me.

> can you elaborate more

Hope, you'll have to figure it out for yourself. It's not hard. If you just look at the details of how these two operations are defined it should be obvious.

As a PhD in mathematics who converted to development, please, go into the details of what you mean.

I know tensor products. I know SQL joins. I see superficial similarities between the two, but no unifying underlying principle.

The world has had quite enough of nonsense thrown around because it's "obvious". Put your money where your runaway mouth is.

Um, can we tone this down a notch please? I didn't intend for this to be a deep insight, just the observation that both the tensor product and the join operation involve taking all possible pairs of the components of the input data. The only formal difference between the two is that the components of the tensor product are ordered while the components of a join are not (they are a set).

The tensor product is not exactly like the Cartesian product because not all the elements of the tensor product of the Hilbert spaces representing each qbit can be written as the tensor product of the individual qbits.

In the cross join case, all the elements are pairs of elements from the original tables. In the Hilbert space of a composite system only the separable states can be written as the tensor product of states of the subsystems.

"Observation 3.6.5: Interesting two-qubit states. Not every 4-dimensional vector can be written as a Kronecker product of two 2-dimension vectors, e.g., you can have a two-qubit state |Ψ⟩ such that:

|Ψ⟩ = ̸= |ψ⟩|φ⟩, for any one-qubit state |ψ⟩ and |φ⟩

These types of states (called entangled states) are very intriguing and play a fundamental role in quantum mechanics."

Interestingly, I was just about to edit my original comment to say exactly that: there are tables which are exactly analogous to entangled states in that they cannot be written as a cartesian product of two other tables.

But I have to stress that I do not intend this to be a deep insight, just an interesting (IMHO) observation.

It is interesting that the basis of the tensor product of the spaces can be constructed as the tensor product of each possible combination of the basis of the spaces. Among other things, it makes clear that the dimension of Hilbert space for the composite system is the product of the dimensions of the Hilbert spaces. But this can distract us from the most interesting part, which is that the space spanned by these basis vectors is much larger than the simple cartesian product.

Focusing on superficial similarities is a two-edged sword: anchoring on a familiar concept can help or make things more difficult. And it's guaranteed to annoy people! (I'm mostly getting over it, but I still dislike the use of the word "tensor" to refer to multi-dimensional arrays. Tensor has a meaning in geometry and tensor algebra is not about doing linear algebra on 2-d slices of a larger-dimensional object.)

I can’t see see the similarity. SQL join is the categorical fiber product. Tensor product is not a categorical product. It can be related to the product, but not in a trivial way. In particular, I can’t agree with your statement that the tensor product “involves taking all possible pairs of the components of the input data". It’s one way to construct the tensor product, but conceptually it’s not really about that.

> SQL join is the categorical fiber product

You are now beyond the limits of my understanding. I never grokked category theory.

SQL join is a Cartesian product. The Tensor product is analogous to a Cartesian product except that the inputs and outputs are ordered tuples instead of sets. That's all I was getting at.

When you ask someone to explain something, and the response that you get is merely "it should be obvious," what it achieves is a) you have just been told that you are a stupid idiot and b) you are in fact so stupid that you are not worth educating. The remark isn't quite so biting if it's paired with an actual explanation, but by itself, it serves little purpose other than to insult the enquirer.

I disagree. In some cases, such a question is evidence that you have not thought about the problem at all. In this case, the connection is so obvious and so trivial that it's analogous to the observation that, say, there is a connection between multiplication and the areas of rectangles. If you ask me to explain that on HN I think it's not unreasonable for me to tell you to just go think about it some more.

You are the kind of person who made me flip out trying to do abstract algebra homework. When you're trying to get help figuring out a problem, and everyone just keeps responding "group actions are the magic hammer," it is absolutely no help whatsoever. Clearly, the problem is that I was taught group actions very badly, and I needed someone to sit down and explain them to me so that I could see why they would be useful. But nope, too busy being smug to actually explain anything.

As for HN, a lot of people are going to be self-educated in advanced topics. That means there's going to be holes in people's knowledge, especially as it relates to alternative theoretical treatments.

I totally sympathize. But in this case, I really don't see how anyone could fail to miss the connection if they just looked at the description of tensor product in the book, and the description of join in Wikipedia. I don't think that is an unreasonable amount of homework to expect people to do.

Way too basic.

No infinite dimensional spaces. No discussion of self-adjoint operators

These are essential mathematical components of QM.

Looks like a nice introduction with lots of concrete examples although while reading through it I spotted a couple of minor things that are poorly worded/wrong or other things are omitted which are rather interesting from a quantum computing perspective.

Given that it's targeted at kids, the fact that they managed to word it at all is a feat. The same goes for anything that was omitted.

Not gonna argue with this, I just wanted to point out that it's not without mistakes. For Students of the 12th grade it's super well done. For those more concerned with correctness and completeness: For example saying a basis is a _finite_ set of vectors is not generally correct. And I would like to have seen that the Pauli matrices generate all single Qubit rotations. And saying that a bra in itself is simply the conjugate transpose of a state or that <bra|ket> would simply be a multiplication ommits all the cases in which the state is not a simple numerical vector in some basis but perhaps a function of many variables. So read it with caution, but read it nonetheless.

This looks like a very good introduction to quantum mechanics, though I wish the author had also added some physical motivation for why the definitions are the way they are, by referring to experimental results and such.

Not an intro to QM. Just reviews some of the math used in it.

Who is this for, exactly? It's introductory material for the "Quantum Cryptography School for Young Students", but what is that?

One quote:

"We’ll use special mathematics – complex numbers and linear algebra (vectors and matrices). Unfortunately, most high school mathematics curricula around the world do not teach linear algebra. It’s not very complicated. It’s really just a different and clever way to add and multiply numbers together, but it’s a very powerful tool."

Looks like a summer school program at University of Waterloo:

"The Quantum Cryptography School for Young Students (QCSYS) is a unique, eight-day enrichment program for students hosted by the Institute for Quantum Computing (IQC) at the University of Waterloo. QCSYS will run August 10-17, 2018 with students arriving August 9 and departing August 18.

The school offers an interesting blend of lectures, hands-on experiments and group work focused on quantum cryptography"


From their FAQ [1]:

>Can I still apply if I am currently in grade 12?

>Yes. If you are currently in grade 12 and you are attending university or college in the fall 2018 you are still eligible to attend the Quantum Cryptography School for Young Students (QCSYS).

So, high school students. Material looks accessible enough and well presented.

[1]: https://uwaterloo.ca/institute-for-quantum-computing/program...

It's for anyone who wants to understand quantum mechanics, but particularly for people who want to understand quantum computation.

If you don't know what quantum computation is or why it's important, then you have a lot of catching up to do.

love the color and layout. really great for readability which you don't find in many online textbooks

Highly recommended:


Absolutely beautiful way to present quantum mechanics.

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