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The Mathematics of Quantum Mechanics [pdf]
- mcguire 8y agoWho 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."
- mishurov 8y agoNice quote. Matrix arithmetic is a bit different thing than linear algebra.
- lisper 8y agoIt'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.
- jlgray 8y agoLooks 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" https://uwaterloo.ca/institute-for-quantum-computing/programs/quantum-cryptography-school-young-students https://uwaterloo.ca/institute-for-quantum-computing/program...
- deleted 8y ago[deleted]
- forgotpwd16 8y agoFrom 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/programs/quantum-cryptography-school-young-students/qcsys-frequently-asked-questions https://uwaterloo.ca/institute-for-quantum-computing/program...
- lisper 8y agoThis 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.
- hamad 8y ago"the tensor product and the join operation in a relational database." can you elaborate more. very interesting!
- lisper 8y ago> 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.
- gunnihinn 8y agoAs 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.
- deleted 8y ago[deleted]
- lisper 8y agoUm, 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).
- forgotpwd16 8y agoSince 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).
- codethief 8y agoThis 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.
- grigjd3 8y agoNot an intro to QM. Just reviews some of the math used in it.
- audunw 8y agoThis 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.
- SiempreViernes 8y agoFamously there is no consensus of how to interpret the equations of quantum mechanics, which naturally restricts introductory discussion to the mathmatical aspects.
- forkandwait 8y agoI think there is a ton of consensus on how the equations predict lab results, however, and I would be very happy with that.
- jabl 8y agoFor 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 https://www.nature.com/articles/s41467-018-05739-8
- kgwgk 8y agoThere was some discussion about that paper here: https://news.ycombinator.com/item?id=18023452 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 https://arxiv.org/abs/1804.03267
- Escapado 8y agoLooks 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.
- whatshisface 8y agoGiven 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.
- Escapado 8y agoNot 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.
- paulpauper 8y agolove the color and layout. really great for readability which you don't find in many online textbooks
- oytis 8y agoLooks like "Mathematics of Quantum Computing" to me. All the analysis fun is missing.
- 77pt77 8y agoWay too basic. No infinite dimensional spaces. No discussion of self-adjoint operators These are essential mathematical components of QM.
- jabl 8y agoSemi 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.
- monktastic1 8y agoAgreed. I think this was the first book that helped me understand L2 as a Hilbert space.
- westoncb 8y agoI 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.
- diebir 8y agoHighly recommended: https://www.amazon.com/Lectures-Mechanics-Mathematics-Students-Mathematical/dp/082184699X https://www.amazon.com/Lectures-Mechanics-Mathematics-Studen... Absolutely beautiful way to present quantum mechanics.
- monktastic1 8y agoMost 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. https://www.scottaaronson.com/democritus/lec9.html https://www.scottaaronson.com/democritus/lec9.html
- kgwgk 8y agoThis 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. https://www.math.columbia.edu/~woit/wordpress/?p=10533 https://www.math.columbia.edu/~woit/wordpress/?p=10533 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)
- jostylr 8y agoTo 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.
- donpdonp 8y agoFry: 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.
- unnouinceput 8y ago" 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