Quantum Mechanics: The Theoretical Minimum

The Theoretical Minimum

by Leonard Susskind, Art Friedman

From the bestselling author of The Theoretical Minimum, a DIY introduction to the math and science of quantum physics

First he taught you classical mechanics. Now, physicist Leonard Susskind has teamed up with data engineer Art Friedman to present the theory and associated mathematics of the strange world of quantum mechanics.

In this follow-up to The Theoretical Minimum, Susskind and Friedman provide a lively introduction to this famously difficult field, which attempts to understand the behavior of sub-atomic objects through mathematical abstractions. Unlike other popularizations that shy away from quantum mechanics’ weirdness, Quantum Mechanics embraces the utter strangeness of quantum logic. The authors offer crystal-clear explanations of the principles of quantum states, uncertainty and time dependence, entanglement, and particle and wave states, among other topics, and each chapter includes exercises to ensure mastery of each area. Like The Theoretical Minimum, this volume runs parallel to Susskind’s eponymous Stanford University-hosted continuing education course.

An approachable yet rigorous introduction to a famously difficult topic, Quantum Mechanics provides a tool kit for amateur scientists to learn physics at their own pace.

Genres: Science, Physics, Nonfiction, Mathematics, Quantum Mechanics, Popular Science, Stem

384 pages, Paperback

First published January 1, 2014

About the author

Leonard Susskind is the Felix Bloch Professor of Theoretical Physics at Stanford University. His research interests include string theory, quantum field theory, quantum statistical mechanics and quantum cosmology. He is a member of the National Academy of Sciences, and the American Academy of Arts and Sciences, an associate member of the faculty of Canada's Perimeter Institute for Theoretical Physics, and a distinguished professor of the Korea Institute for Advanced Study.

read more: http://en.wikipedia.org/wiki/Leonard_...

Reviews

[Manny · Rating 5 out of 5]

I've finished volume one and now I'm dying to find out what happens next. Talk about a cliffhanger ending!
__________________________________

I must confess that I didn't enjoy the second volume quite as much as the first, but that mainly shows how high the bar was; this is still the best introduction to quantum mechanics I have ever seen, and if you have some mathematical background (linear algebra, calculus) I can't recommend it too highly. It requires some effort to read, but it's definitely worth it.

The overall plan is extremely well thought out. Most books on quantum mechanics start off by introducing the quantum versions of position and momentum, so that they can get to Heisenberg's Uncertainty Principle as quickly as possible. After reading Susskind and Friedman, I am sure this is a mistake. The problem is that we know what position and momentum mean in classical physics, so it's impossible to read about their quantum analogs without thinking that they're basically the same thing. They aren't: quantum mechanics is much weirder and much more interesting, but you don't immediately notice.

Instead of taking this tired old road, Susskind chooses a completely different starting-point, the concept of spin. Spin is a truly quantum mechanical concept, which is so different from classical angular momentum that there is no possibility of confusing them. If you've read any quantum mechanics at all, you'll know that spin can be either "up" or "down". But how can this make sense, when you stop and think about it? Surely we need three independent directions, corresponding to the x, y and z axes, rather than two directions which, to make things even more counter-intuitive, are oriented along the same axis?

But it does make sense. We start by considering a device that measures an electron's spin. We orient it vertically, and it only ever gives two possible readings: +1, or "up", and -1, or "down". Now we rotate it 90 degrees, so that the two readings instead mean "left" and "right". If the electron was previously in the "up" state, a naive guess might be that we'll now get a zero reading. Wrong! There is no zero reading; we'll get "left" or "right" with equal probability; so "up" must be a combination of "left" and "right", where the two components are in some sense given equal weight. Similarly, when we orient the apparatus along the third axis, we find that "left" and "right" are combinations of "in" and "out". In fact, each of the "up"/"down", "left"/"right" and "in"/"out" pairs can be expressed in terms of any of the others, and there is a simple mathematical way to write down the relationships using matrix algebra and complex numbers. It all works, and you can see why!

I was hooked. Forget that stupid cat, which only appears here as the subject of a couple of ironic jokes. This is the right way to do it.

[WarpDrive · Rating 5 out of 5]

This is a really nice introductory book on Quantum Mechanics - Quantum Mechanics done for real, with some decent detail and good mathematical treatment of some of its most important aspects.

As a list of prerequisites for fully enjoying this book, I would recommend the following:
- read volume 1 (on Classical Mechanics) which is simply fantastic on its own
- basic knowledge of linear algebra, calculus and complex vector (Hilbert) spaces

The author is great at explaining potentially tricky and complex concepts in a simple, concise, lucid, easily understandable way: the book is also supported by a very good choice of accompanying exercises, and there are also the famous accompanying online lectures (The Theoretical Minimum), which are a real pleasure to follow.
But make no mistake: Quantum Mechanics is necessarily deeply mathematical and counter-intuitive (which is part of its beauty), so do not consider this some light bedtime or beach reading, as it does require attention and focus.

I really liked how the author treats some very important concepts in a very nice, logical and easily understandable way:
- one thing that I already knew, but that I really got to fully appreciate in its full power only by reading this book, is Dirac's notation. Once you get proficient with it, it is amazing how Dirac's notation proves itself very conducive to quick analysis and manipulation of QM problems (problems which would have been otherwise very lengthy and cumbersome to solve with the "standard" mathematical notation). Another proof of Dirac's genius.
- the way the Heisenberg's uncertainty principle is simply proved, starting from the Cauchy-Schwarz inequality, is just brilliant.
- I really loved how the author explains the relationship between the quantum mechanical world and the "classical" world, in particular when dealing with the correspondence between the commutator and the Poisson Bracket, and also when he explains how the shape/size of the wave packet and its relationship with the shape/size of the potential function V(x) determine whether the classical approximation is plausible.
- the part discussing entanglement is really nicely done, and I did like how the author explains the concept of density matrix. Very nice explanation of correlation/entanglement in relation to the corresponding characteristics of the density matrix.

There are a couple of things that I would have liked done a bit differently:
- there is a bit of hand-waving and of notational abuse here and there (which is probably fully excusable considering the amount of conceptual apparatus that the author had to condense in a single book, and considering that the author has tried really hard to simplify the treatment of such concepts as much as possible)
- I think that a bit more rigorous treatment of the concept of complex Hilbert spaces would have been beneficial (I already knew this stuff, having studied Linear Algebra and Functional Analysis, so this was not a problem for me, but this might be an issue for somebody who has never been previously exposed to this stuff)
- Also, a mathematically more rigorous treatment of the concept of wave function would have been beneficial (but again, I think it is difficult to strike the right compromise between level of rigor and intelligibility/length of the book).
- I am not fully satisfied with the way the author treats the very important concept of path integrals; I would have loved more detail

But, apart from these minor things, this book represents a really commendable effort by the author to explain in an approachable way the main conceptual and mathematical apparatus of quantum mechanics.
A very enjoyable book, which provides a great starting point for further study from more advanced sources. Highly recommended to anybody interested in the REAL thing.

[Roy Lotz · Rating 4 out of 5]

We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations.

—Richard Feynman

I’m a bit sad to be finally putting this book down. Now, I can no longer tell friends and coworkers that I’m reading a book about the mathematics of quantum mechanics. Oh well.

I’ve already written a fairly detailed review of the first volume (which can be found here); and since this volume is very similar in spirit and scope to its precursor, I’m not sure how much I have to add. I do think that this volume is, if anything, a bit of an improvement from the first; there are far fewer errors, and I think the content is better organized. But, on the whole, it is of a piece with the first volume, using the identical approach, with that approach’s same strengths and weaknesses. (More on that in a bit.)

It should be said that I am a total newbie to this subject; my only preparation for reading this book was the first volume on classical mechanics. I do not have a strong mathematical background, and the last time I was in a physics course was in high school. So it speaks to the skill of Susskind and Friedman that I was able to follow along as well as I did. (How much I actually understood is, as yet, an open question.) I should say that S & F frequently refer back to Volume I, so I’m not sure how easy it would be to skip that first volume and jump to this one. At the very least, the many references to “stuff we’ve already covered” would be frustrating. But perhaps it’s doable if you are already familiar with the subject.

In a way, I think that novices like me might profit much less from these books than somebody who already has some physics experience. This has to do with Susskind’s approach to teaching the material. Susskind does not discuss the kinds of things one might expect in a book on quantum mechanics—electrons, photons, waves, the double slit experiment, Schrödinger’s cat, and so on. In my review of the first volume, I called Susskind’s approach “mathematical,” for he concentrates almost exclusively on nifty mathematical relationships. But now I’m not so sure that’s quite right, as Susskind often teaches the math in ways that would make a mathematician cringe. (If somebody as mathematically illiterate as me could realize this, it’s got to be pretty egregious.)

Instead, I think the best description of Susskind’s approach might be “notational.” Susskind is a firm believer in the power of good notation; and he uses many tricks and abuses of notation to cut through large swaths of material. He is always creating new symbols and collapsing variables, working to express the mathematical relationship as pithily as possible. (Of course, not all of these notational manipulations are his ideas; he uses Dirac's bra-ket notation throughout, which is admittedly impressive.) There are very serious advantages and disadvantages to this method. The most obvious disadvantage is that it often doesn’t feel like you’re learning physics; it feels like your following along as Susskind performs notational tricks—not a very satisfying feeling. Another serious disadvantage is that, in taking shortcuts through the mathematics, Susskind risks not giving his readers the ability to actually solve problems.

So one might logically ask, if Susskind isn’t very good at explaining either the physics or the mathematics, what is the value of this book? I’ll first answer for myself, and then speculate about how this book might help somebody much more knowledgeable than I am. For myself, I think these books are interesting because, by bracketing the math and physics, it allows Susskind to really get at the logic of the theories. Quantum mechanics, Susskind notes, is difficult to understand because it relies on an entirely different logical base than our familiar classical world; thus, the biggest challenge of learning the material is, as it were, to defeat one’s own intuition. Our intuition can only lead us astray when contemplating the quantum world; and so, by cutting out all the physics and most of the math, Susskind—through some notational prestidigitation—takes the reader to the heart of the logical puzzle, and shows how physicists deal with it.

Now, for the person with experience in physics, this book may have an added benefit. It has to do with the psychological phenomenon known as “chunking,” wherein information is bundled to ease cognitive load—very desirable, considering how few things we can think about at once. I recently read a book by Steven Pinker on writing, and I think much of what Pinker says is illuminating here. Writers like Susskind, who are experts in their fields, are plagued by what Pinker calls the “curse of knowledge”—which is just the difficulty experts have in speaking about their field to non-experts. Susskind has been working with these equations for years; thus, he has gotten into the habit of “chunking” information together into bundles, for more easy manipulation. This is what all experts do: since you cannot think of that many things at once, you learn to bundle information into packets. (And this, I think, is why this book was often unsatisfying for me, since I was too inexperienced to easily collapse all this new information into little symbols; I needed it spelled out.)

This is why Susskind is so fond of good notation: by collapsing information into symbols, he is doing on paper what he already does in his head. And this is why I think this book might be fascinating for the reader with experience in physics. Once Susskind shows you how to collapse a bunch of information into manipulable packages, it will make thinking about it easier; and, because the information is easier to toy around with, it might be easier to speculate and to come up with new approaches. (I am, of course, just speculating myself here.)

So, to sum up, I think this book, as well as its predecessor, is one with serious flaws and serious merits; ideally, they should be combined with other books, to fill in the large gaps in the reader’s understanding of both the physics and the math. At its worst, reading Susskind can feel like a dry exercise in pattern matching—just some clever notational tricks, completely irrelevant to the physical world. But at its best, when an idea would finally ‘click’, these books make you feel like a kid in an intellectual playground; there are slides and monkey bars and bridges connecting every concept to every other concept. Two things, which look completely unrelated at first, Susskind will reveal, like a magician raising the curtain, to be connected on the deepest level. And that’s really where these two books shine: Susskind shows how the orderly world of classical physics can arise out of the hectic quantum world, too small for our eyes, too foreign for our brains.

(I'd also just like to note something I've realized: Leonard Susskind, from certain angles, looks a good deal like John Malkovich:

Or maybe it's just me.)

[Bernhard · Rating 5 out of 5]

Quantum Mechanics: The Theoretical Minimum is the second book in the Theoretical Minimum series. The first book was about classical mechanics, covering both Newtonian and advanced mechanics, and in a way it felt like a preparation for quantum mechanics. Having read this book, I can safely say that quantum mechanics is even weirder than I thought.

The first half of the book moves at a slow speed. In that sense it reminded me of the first book, just introducing the reader to the necessary tools and mindset for dealing with quantum mechanics, such as linear operators, eigenvectors, eigenvalues, bra-ket notation, among others. In the second half of the book the speed picks up, though not as much as it did in the first book, as we get introduced to entanglement, and then move on to particle dynamics and the harmonic oscillator. Much like its predecessor, this book does a great job at introducing the reader to the subject (with better editing than the first book). This was the first time I read any kind of book that deals with the real physics and mathematics of quantum mechanics, so I don’t know how other books deal with it in this aspect. I can only say that the way the authors prepare the reader for what’s coming up is quite remarkable.

I didn’t remember much of the linear algebra that I had previously learned. The authors do a great job at introducing the reader to this topics. This happens when we get introduced to linear operators, eigenvectors and eigenvalues. I also quite liked the way the authors made the transition from using algebra to using differential equations in the second half of the book, as we get introduced to particle dynamics. Also on a side note, when I was a student at university I was taught that spin is sort of the quantum analog of classical angular momentum, but not really. At the time I found it rather confusing. I’m glad Susskind chose a different route.

It’s quite remarkable that someone who’s not a physicist can read this book and come out of it understanding some parts of quantum mechanics. I can’t recommend it enough to other readers with some mathematical (linear algebra and calculus) and physical background who want to dig a bit deeper than the usual popularizations of quantum mechanics. It presents a step between popular science books and textbooks. Just make sure you read the first book before, as it introduces some key concepts that play a major role in quantum mechanics. The only downside I see in this book is that it may seem somewhat disconnected from experiment and its history, as the double-slit experiment, photoelectric effect, among others, are not mentioned. This is a excellent book, and while I might have enjoyed more the previous book in the series, that was due to the fact that classical mechanics is less abstract than quantum mechanics and easier to digest.

[Lost Planet Airman · Rating 4 out of 5]

I liked this, but then again, I like the maths and the sciences and challenging thinking. This book has plenty, of all three.

You can learn something here, as an amateur physicist, but you may need some elbow grease and brain sweat, to master what this book offers. I didn't apply myself, and I was foundering by the 4th chapter and befuddled by the 10th and final chapter, or "lecture", as the author names his divisions. Not completely befuddled, mind you, but enough to know that I will need an indeterminate number of pencils to go back and master Quantum Mechanics.

"Q" title for the Alphabet Challenge.

[Ivan Vuković · Rating 5 out of 5]

Susskind's Theoretical Minimum series might not be everybody's cup of tea, but I'm really beginning to love this style (way more than the video lectures, I should add!).

It's not your usual popular science with so much hand waving and analogies it almost feels like patronising, but it's not the usual cold and formal textbook style either.

Susskind manages to explain all the core ideas of the theory in a beautiful and coherent way. Some details must, of course, be omitted, but there's no BS and sweeping important things under the rug. Also, the exercises are pretty nice!

I find this perfect because I like studying "in layers" and instead of stopping in order to understand every detail perfectly I prefer several runs over everything, with increasing clarity and understanding in each run (like a breadth-first search as opposed to depth-first, for those who know what I'm talking about).

This style strikes me as amazingly appropriate for the "first run", since it gives you a complete idea of the subject matter and provides a great starting point for further study from more advanced sources.

I have to warn you, though, I'm a physics student and my view surely isn't entirely free of bias, but I highly recommend this to anyone interested in understanding physics, especially to my colleagues, physics students!

(ESPECIALLY if you like studying in layers... But you might not like it if you're a "depth-first" learner.)

[Rama Rao · Rating 5 out of 5]

Learning quantum mechanics from a maestro

This book is designed to meet the needs of a mathematically inclined reader. An undergraduate level physics textbook is perhaps too advanced, and a popular book with no math discusses the principles of quantum reality that is easier to understand, but this book is at the middle level of complexity. This is meant for readers who are interested to know the equations that describes the mechanics of fundamental particles in terms of their position, motion, and energy in spacetime. Math tends to make certain things easy to put in perspective than mere descriptions without equations! The readers are expected to know mathematical concepts such as complex numbers, vector spaces, linear operators, and tensor products, all of which are artistically explained in a series of interludes. Specific concepts of the space of states, time evolution, principles of uncertainty, and quantum entanglement are described at moderate level of complexity, and yet reader-friendly. I recommend doing the exercises at the end of each chapter. I could not answer many of these questions, but it certainly makes you think. That is a learning process.

The biggest challenge is the understanding quantum entanglement because there is no classical analog for a system whose full state description contains no information about its individual parts, and nonlocality (two particles separated at large distances) is difficult to define. The best way to come to terms with these issues is to internalize the mathematics.

Two principles emerge as fundamental, the spin state of quantum particle or qubit. In classical physics, everything can be built out of yes/no (1 or 0) questions. Similarly, in quantum mechanics, every logical question becomes a question about qubits (basic unit of quantum information, two level quantum system, spin up or down, both in a state of superposition). The second principle is the harmonic oscillator. How do particles move in quantum mechanics? We know that fundamental particles have wave-particle duality. It exists in both wave and particle forms. Then how do matter in its wave state can have gravity associated with it? That makes understanding quantum gravity harder. In addition, waves oscillate much like a mass attached to the end of a spring. The oscillators, not masses attached to springs, are imagined as waves, in fact they are the oscillating electric and magnetic fields. For each wavelength, there is a mathematical harmonic oscillator describing the amplitude or strength of the field. For many waves there is a lot of harmonic oscillators all running simultaneously. Fortunately, they all oscillate independently. The higher-energy wave functions oscillate more rapidly and are more spread out. This is the consequence of quantum field theory. Another question is how do quantum states change with the evolution of time? They change so that information describing the system are never erased. This is one of the most fundamental phenomenon that haunts in describing black holes.

This book sticks to the simplest possible quantum system, one with a two-dimensional state space. The algebra is developed from scratch and author Leonard Susskind describes at a very leisurely pace and the quantum reality is described in the simplest context.

[Brian Clegg · Rating 3 out of 5]

I saw this book on the shelves in my local booksellers which are usually reserved for books which are new, interesting and likely to sell a lot of copies. They were right on two out of them, but they were in cloud cuckoo land on the ‘lot of copies’ part (unless we get a ‘Brief History of Time effect’ where lots buy it and don’t read it). This is a new and interesting book, and for the niche it is aimed at it is brilliant – but that is a narrow niche indeed.

Usually there are two kinds of science books. Popular science explains what the discoveries and theories of science, with historical perspective, so that the general reader can get a feel for them – but reading a popular science book on, say, quantum mechanics would not leave you able to solve quantum mechanics problems.

Textbooks, on the other hand, teach the actual science itself, usually with a lot more maths, so that you can indeed do the workings, but they don’t give you any context, and they are inaccessible (and, frankly, highly boring) to most readers.

This book highlights a tiny crack in between the two, a niche where it can do a very interesting job of leading the reader into the actual science, but in a more hand-held and less boring way than a textbook. Because it takes this approach it hasn’t got the context or readability of a popular science book – but it’s far more readable than a textbook. Similarly, it doesn’t have quite enough detail to really ‘do’ the physics – but it takes you well on the way there, so that it would only take a little textbook work to get on top of it.]

The only thing I’d criticise (apart from the narrowness of that niche) is the really irritating attempts at folksy fictional openings to the sections. They don’t work. Stay with what you’re trying to do, guys, don’t try to be entertainers.

For most popular science readers this book simply won’t work. It makes the infamously ‘I started it but couldn’t finish it’ Brief History of Time look highly simplistic and non-mathematical. And for serious physicists, it’s still too limited – though it takes what is in some ways a better approach, giving more emphasis early on to entanglement, than the way quantum physics is traditionally taught. Either for those about to start a university physics course who want some preparation, or for someone who finds popular science explanations too summary and is prepared to take on some quite serious maths (A level maths required as a minimum, I would say) it’s a fascinating addition to the library. For the rest of us, probably best to leave it where it is.

[Galen Weitkamp · Rating 5 out of 5]

Spooky Action at a Distance by George Musser
and
Quantum Mechanics the Theoretical Minimum by Leonard Susskind and Art Friedman

review by Galen Weitkamp

Perhaps it’s unfair to compare these two works given the differences in the intent of their authors. George Musser is fascinated by what some identify as the nonlocal nature of the measurement process in quantum theory. It seems to this reader that his intent is not only to inform but to convince the reader of its “reality” and the consequences that “reality” would have for our conception of space and time. Susskind and Friedman, on the other hand, are excited (in the way teachers are) by the power of knowledge. Their intent is to convey that excitement by teaching their readers a little bit of real quantum mechanics.

Both books are for the layperson, although the latter is probably for the more serious layperson (you have to know a little bit of calculus). Both books discuss the key notion behind allegedly nonlocal behavior, namely entangled systems. One book succeeds whereas, in my opinion, the other does not.

Musser is very much concerned with an experiment known as the EPR-experiment. It was an experiment proposed in a 1935 paper by Einstein, Podolsky and Rosen. Two entangled particles are sent in opposite directions, one toward Alice and the other toward Bob. Because they are components of one entangled system, neither particle has a determinate spin. Alice’s component only acquires a spin when she measures it. According to one interpretation of quantum theory, the instant she measures it the state of the whole system collapses and Bob’s particle acquires a spin too, a spin which will be in the opposite direction of Alice’s particle. Einstein pointed out there would be no time for the cause to propagate from Alice’s location to Bob’s and that this “Spooky action at a distance” was counter to the spirit of the special theory of relativity.

Nevertheless, the predictions of quantum theory are born out. The EPR experiment has been carried out countless times. Countless pairs of Alices and Bobs have made simultaneous measurements, compared them and found the results are correlated exactly as quantum mechanics predicts. Musser is convinced this is an important clue to a new conception of space and time, the implications of which are that space or time or both are unreal, nonlocal, emergent or composed from related entities that have no location.

Another interpretation of quantum theory maintains that when Alice measures the spin of her component of the entangled system, she becomes entangled with the system as well. When Bob measures his component he too becomes part of a larger entangled system which includes the two particles, Alice, her brain cells, her notes and records of the experiment as well as his own. Although they are entangled, they cannot compare notes immediately, they must wait until they meet or at least until the messages they send each other have sufficient time to reach their destinations. Suppose they meet as soon as possible and share their information. Since they had to wait, the perturbations caused by their measurements had time to propagate and merge. According to this interpretation of quantum mechanics, this later merger is responsible for the correlations between Alice’s and Bob’s results. The time delay avoids the spooky action at a distance and keeps quantum theory intact. Indeed, Susskind and Friedman neatly demonstrate that no information can be transferred instantaneously across space by any sort of measurement process.

Without a doubt this is a fascinating subject, but without the appropriate background it is difficult to appreciate the subtlety of the phenomenon, understand the problems, the proposed solutions and it is certainly difficult to assess them.

Susskind and Friedman give us a place to start: the concept of quantum spin. When I was a student, most textbooks on quantum mechanics started with continuous systems, free particles and the oscillators. “The Theoretical Minimum” is not a textbook. It’s a guide, rather, for serious amateurs. The emphasis on spin allows the reader to see that quantum theory is about logic and information. Quantum spins are analogous to classical “bits” that can be “on” or “off,” except quantum spins follow a non-Boolean logic. Susskind and Friedman reformulate the EPR-experiment as an attempt to simulate Alice’s and Bob’s measurements with a classical computer. They demonstrate it can’t be done. They also point out, “This is not a problem for quantum mechanics. It’s a problem for simulating quantum mechanics with a classical Boolean computer.”

I found “Spooky Action at a Distance” to be unhelpful, sometimes snide and often slanted. “The Theoretical Minimum was very helpful, kind and honest.

There are a lot of good books on math and physics for the layperson: George Gamow’s One, Two, Three...Infinity, Douglas Hofstadter’s Gödel, Escher, Bach or Roger Penrose’s The Road to Reality. Like Susskind and Friedman, George Gamow just wants to teach and excite the reader. Hofstadter and Penrose also want to teach and stimulate the reader, but like Musser, they also have something they want to sell. Hofstadter will try to convince you that you’re a computer and Penrose wants to convince you that his twistor theory is the way to unify the fields of physics. These books are successful only in so far as they are able to give the reader sufficient knowledge and confidence to think for herself or himself about the subject. In the preface to his book, Penrose says,

“The reader will find that I have not shied away from presenting mathematical formulae, despite dire warnings of the severe reduction in readership that this will entail. I have thought seriously about this question, and have come to the conclusion that what I have to say cannot reasonably be conveyed without a certain amount of mathematical notation and the exploration of genuine mathematical concepts.”

I have the feeling there are people out there who are eager to learn but they find themselves stuck between lay-works that explain nothing and college textbooks that are both boring and beyond their current level of understanding. I applaud authors to encourage us to understand the world better by imparting as accurately as they can what they think they know and I would implore publishers not to refrain from publishing these sorts of works.

[Woflmao · Rating 4 out of 5]

The motivation for this book is to explain the fundamental ideas of quantum mechanics (QM) in such a way that readers with very little mathematical knowledge (some calculus, vectors in 3-space) are able to understand it. In this, the book is mostly successful.
As someone with a broader mathematical background who learned some QM before, I was mostly interested in the pedagogical angle the book would take. Most physics books introduce QM from a historical point of view, reviewing the problems leading up to QM, the first (thought) experiments and develop the wave/particle duality. None of this here! QM is introduced in an axiomatic way, and the first and for a large part only example is the simplest possible, a two-spin system. While discussing this example in great detail, but unfortunately many students have trouble learning something that is not motivated for them. In these cases, the lack of motivation might counterbalance the advantage of the more efficient presentation. The book discusses quantum entanglement comparatively early on, and along these lines introduces tensor products of vector spaces and operators. I have to admit that I found the very long chapter on entanglement somewhat tedious and repetitive, so I did not read it too thoroughly. For the uninitiated (regarding tensor products), however, this chapter might prove very helpful in getting acquainted with this for beginners rather abstract and hard to understand concept. In the last sections, vector spaces of infinite dimension are introduced (position, momentum of a particle), and here the overly simple approach turns out to be insufficient. Accordingly, these chapters read more like a popular science book than a textbook. Nevertheless, they can convey the essential ideas, but without fully supplying the mathematical tools needed to deal with them quantitatively.

[Alexandra · Rating 4 out of 5]

I am putting this book on hold for now. I started reading it during a long voyage and it was very interesting and invigorating, like a good workout for the brain. But as soon as I got back to work, I lost the stamina necessary to keep up with the book.
It is hard. The math is real and you have to learn a whole new vocabulary to follow the explanations.
But even if I read only a half of this, I got what I wanted originally. My big question to quantum mechanics was something like "Don't they see how similar their laws are to the ordinary statistics?" This book explains quite clearly that probabilities is all they got and that's the intent. It dives a little bit in philosophical aspects (like whether we just can't measure the quantum world because of our instruments or the uncertainty of the outcome is how the quantum world works), but mostly it focuses on the math.
I now understand what the Schrödinger's equation and Heisenberg's uncertainty principle are about. Most of my knowledge on those two iconic principles came out of Sci-Fi and popular science books (like Hawking's or Brian Green's) and it was really nice to follow the math explanation which demystified them.
Overall, I wish popular science books had a bit more math...
Also, now I am puzzled with the new question - why do statistics and probability theory work in our universe? What makes them work? What if they didn't work? How would our world look in this case? Fun Sci-Fi questions and I am grateful for this book to make my brain tick that way.
I am planning to return to it during the next long vacation, but even if I never do, it was still worth going through the first half.

[Alexander Telfar · Rating 2 out of 5]

I didn't like this book.

I think it approches teching QM entirely the wrong way.
- They start from math and show that it can be used to describe QM. This seems backwards to me. We should learn about quantum phenomena, the data, and then stumble on some math than can help us describe that data.

Also, I think their interpretation of QM doesn't make any sense.
- Waves of probability, spin just 'is', entanglement and spooky action at a distnace, observables are treated differently, ... I mean come on.

But sure.
There was some interesting math. And I did learn a bit about QM.

[Chris · Rating 5 out of 5]

Excellent, effective, and entertaining. Compared to the first volume, The Theoretical Minimum: What You Need to Know to Start Doing Physics, this one has much more accurate editing and typesetting, along with a better co-author. This series of books is destined to be classic.

[Dale · Rating 5 out of 5]

An excellent continuation of The Theoretical Minimum, which is a prerequisite to reading this book.

I say here that I've "read" this book, but so far I've only skimmed to get the lay of the land. Because this is not a bullshit popularized treatment of quantum mechanics, it takes actual work and concentration to make your way through it and really understand the mathematics. So this is going to be my background reading for some time to come.

[Christel · Rating 5 out of 5]

Leonard Susskind puts quantum mechanics in the most entertaining and relatable terms. All you need to understand all this is calculus, linear algebra, statistics and probability. I haven't got all my linear algebra down, but so far haven't run into anything I couldn't figure out within 5 mins of googling.

[David Pérez · Rating 5 out of 5]

Recomanable 100% per assentar conceptes

[Fernando Pestana da Costa · Rating 5 out of 5]

The year 2025 is UNESCO’s International Year of Quantum Science and Technology, instituted to commemorate the 100th anniversary of the birth of Quantum Mechanics. Having studied a bit of the stuff in my now distant undergraduate years and having kept an interest in the mathematical side of the subject, I thought it would be a good idea to read in 2025 some more advanced topics in Quantum Theory of which I know nothing about. So, together with a couple of friends we started a reading seminar on Talagrand’s book What is a Quantum Field Theory? The book is great, at least down to the point where lack of time due to the usual load of professional activities forced us to drop studying it. But in order not to entirely miss the spirit of the Quantum Year, I decided to read this book of the "Theoretical Minimum" series by Leonard Susskind. It is a high-level popularization book requiring a (small) minimum level of mathematical background (1st year undergraduate math is enough) and it has really nice examples and careful explanations of the concepts. If you are interested in learning anew (or refreshing your knowledge of) a theoretical minimum of Quantum Mechanics and do not have the mathematical background or the time to go deeper, this book is certainly my first choice.

[Taylor · Rating 4 out of 5]

linear algebra everywhere

[Diego · Rating 4 out of 5]

gives a great introduction to the basics of QM

exercises are **chef’s kiss** and very fun!

Very energy demanding though — you can tell by my more than 1yr to finish!!!

[William Schram · Rating 5 out of 5]

This book is a continuation of The Theoretical Minimum which covered Classical Mechanics in physics. Using simple language and explaining the terminology, Susskind and Friedman go through the basics of Quantum Mechanics. They talk about eigenvectors and eigenvalues, bra-ket notation, wave forms, uncertainty and linear operators. Even more is covered, but you get the basic idea I hope.

This book leaves some things to the reader, for instance, you might have to prove some sort of mathematical theorem or something, but all in all this book is more of a guide you by the hand sort of book. Don't get me wrong. The book is challenging, and it does expect you to build on your imparted knowledge, but it goes step by step logically in a manner that I understood.

As was explained in the book, Quantum Mechanics deals with very small things, things that we as a species are not equipped to understand. This book teaches abstract mathematical theorems and techniques that work in describing quantum behavior. This makes it difficult, because there is no intuition for it. A baseball player may not know the kinematic equations that describe a baseball flying through the air, but the human brain can make predictions and move the arm accordingly. This is not the same for quantum mechanics.

I enjoyed it, but it isn't a book that you read through. It requires you to study it and understand it first. So hopefully one of these days I will have the time and inclination to do both. Five out of five nonetheless.

[James F · Rating 4 out of 5]

The sequel to The Theoretical Minimum, which dealt with classical physics, this book explains the basics of quantum theory in a simple (but not oversimplified) way beginning with spin states and working through the Schrödinger equation, combinations of states, entanglement, and the uncertainty principle. The first half of the book introduces the mathematics of complex vector spaces in a very understandable way (I had never studied linear algebra at all, even with real vectors, and I had no trouble following the authors' explanations of eigenvectors, Hermitean operators, and so forth.) As with the first book, however, the second half seems much more rushed; the explanation of combining operators with tensor products and outer products wasn't nearly as clear and detailed as I needed to follow the argument (they rely a lot on the notation making things obvious, but this only works for HOW to do the calculations, not WHY the equations work), and they assume some things that they haven't explained at all, such as methods of solving certain differential equations. I suppose it is hard for physicists writing for a lay audience to remember that math that seems obvious to them hasn't actually been covered. In any case, they do better in this regard than Penrose. I was struck by how much more sense quantum theory makes with the equations than trying to visualize it with analogies.

[Alex Nelson · Rating 5 out of 5]

Considering this was written for a lay audience (or, perhaps more accurately, a mathematically savvy but not-too-knowledgeable audience), it's quite a remarkable book.

The focus is on finite dimensional systems (i.e., quantum information) initially. This is a similar approach, if I recall correctly, that Isham takes in his Lectures on Quantum Theory: Mathematical and Structural Foundations.

Susskind nicely avoids the problematic terminology physicists have with dimensions. The free particle in one spatial dimension, well, we use L2(R)...which as a mathematical "phase space" is infinite-dimensional. But physicists call it "finite dimensional", for consistencies' sake ;) Field theories are then "infinite dimensional". This confused me when I first learned QFT and QM. Susskind avoids this quite well, clearly and coherently guiding the reader past similar pitfalls.

[Morgan · Rating 5 out of 5]

This is a fantastic introduction to quantum mechanics. It goes over vector and wavefunction formalisms, and discusses almost all the math you need to understand (you'll need calculus before reading this book).

The book starts from a discussion of electron spins, and develops quantum mechanics to explain how spin changes on measurements or if placed in a magnetic field. The Schroedinger equation is derived and explained, and then applied to harmonic oscillators in general.

This book is engaging, well organized, and has some great examples. I won't say it's easy, but it's definitely the best introduction to the subject I've found.

[G.R. Reader · Rating 3 out of 5]

Susskind shows you how beautiful quantum mechanics is. He does a great job - it's almost impossible to believe that something as cool and elegant as this wouldn't be true. Well... string theory is even cooler and even more elegant, so that must be true too. You see?

Okay Len, nice try. But I wasn't born yesterday.

[Alessandro Piovaccari · Rating 5 out of 5]

Best book to learn about Quantum Mechanics

If you don't know anything about Quantum Mechanics, this books is a great place to start. If you already studied it, it is a great tool to refresh it and get an even better level of intuition. Complement it with the free video you can find online.

[Mark · Rating 5 out of 5]

What a delight! Susskind presents a streamlined mathematical introduction to quantum mechanics, with minimal digressions to show how the mathematics leads to the "weirdness" of quantum results.

Expects a solid knowledge of linear algebra. (And a normal amount of calculus.)

[Jeremy Sawruk · Rating 5 out of 5]

Good followup to my Stanford course on quantum mechanics.

[Henry · Rating 5 out of 5]

So I got told, the other day, that I have a profound propensity for saying a lot without actually saying anything. That in mind and given the logical nature of this topic, I’m going to try and keep my thoughts on this one clear and concise…

Holy Moly. This was a full on read. For clarification I have limited physics understanding and a basic grasp of Mathematics, all of which contributed to the long reading process. But boy was the payoff rewarding. Some of the vernacular was very new, often at times feeling akin to the early chapters of learning a new language - but as things pieced themselves together and the whole picture starts to make somewhat more sense, the feeling of understanding something as abstract and complex as quantum physics was like railing the cleanest line of Columbian marching powder off the clearest mirror panelled table self-reflecting the dumb image of yourself back at you, but you no longer care because the rush hits you like ten tonnes of adrenoleeen mixed with a concord fuselages worth of kerosene - it’s punchy.

Susskind articulates the absurdity of the kooky nature of our subterranean quantum world beautifully, or as best as I imagine possible, for the layman or those looking to expand their physics understanding. I certainly better understand the strange phenomenon of Quantum Tunnelling, but won’t lie that whilst my understanding of Entanglement is improved, I still feel a little confused on some of its defining features. Namely the whole ‘faster than light travel’ thing and how the coupling of two interlinked particles does not break such boundaries between the communication of state alternations between one particle and another. I know he proves it with math later on in whichever chapter it was 7/8 I think, but I couldn’t wrap my head around how the maths translated into truth in the real world?

Other notable discussion points, were the use of matrices and integrals, used to describe the way in which these sub-atomic particles move and behave, as well as breaking down the Heisenberg Uncertainty principle and the wide-ranging application of the Schrödinger Equation(s) to which I now think pop culture undervalues the integral importance of his contribution to the field as a result of the popularity of his simultaneously dead and alive cat experiment.

There are many things that I’m still not clear on. I got a little ‘entangled’ in how the commutators worked in the last chapter and am still largely unclear on what the link is, if there even is a direct one between quantum behaviours and classical behaviours for various fundamental means of observational measuring of things such as velocities, momenta or the energy of particles. And I still can’t remember what the exact usefulness of Eigen-values and Eigen-Vectors is, whilst also remaining confused on the difference between orthogonal and orthonormal basis state vectors? Who knows.

To sum up, this read was enlightening, and a great intro to an otherwise daunting and complicatedly off putting realm of knowledge, ultimately ending up being kind of fun. I would definitely say it helps to have at least a grounding in some Maths; a bit of Integration, Differentiation, Trigonometry, Advanced Algebra & Probability should do the trick. But don’t be put off by the “general wishy-washiness” of the strange world that is quantum physics. This book shows it is graspable and oh so satisfying once you start to do so. I think the reason I have enjoyed it so much, is in large part due to the rudimentary tenets of the indeterminism that govern the quantum world. The abstract, absurd and unpredictable rules that make this world seem so far removed from the relativistic predictability of our own are actually not as alien as we might think. I have found the chaotic and random behaviours of our quantum building blocks to offer a weirdly pertinent reflection into the true lack of control we have in our own lives, and just how absurdly unpredictable the true nature of our existence is. Maybe the quantum scale isn’t so different from our own?

[Sujith Ravindran · Rating 5 out of 5]

Quantum mechanics is a strange subject. Several great scientists including Einstein thought that the theory is perhaps not correct. I have seen Prof. Penrose mentioning that quantum mechanics is mathematically inconsistent, especially collapse of wave function. I think Prof. Schrodinger himself was upset about this. However, there is a majority who believes that quantum mechanics is more fundamental than any other theories. Keeping aside all the controversies Prof. Susskind in his own elegant way summarizes the power of quantum mechanics and makes it look logical.

There are many topics which may hook you like the derivation of Heisenberg’s uncertainty principle, correlation between commutators and Poisson’s bracket, entanglement and its weirdness, Schrodinger equations and Hamiltonian; and so on. Please note Dirac’s notation is extensively used in this book and it may take some time to get used to it.

I think it is important to read his first book on classical mechanics before reading this so as to have the basics correct. Also, I suggest listening to his lectures on quantum entanglements and quantum mechanics before getting into this. Mathematical interludes are provided wherever it is necessary. But, a preliminary understanding on linear algebra will be handy. As mentioned in the preface the effort of this book is to make a difficult subject as simple as possible but not simpler.

As in his first book his ability to describe complex facts without losing its rigour is exceptional. I have not read any other textbooks on Quantum Mechanics, so I may not be in a position to compare. But, for a layman like me the book is exceptional and his classes are brilliant. Having said that it is not a book for light reading. It has to be approached with a lot of attention and focus. Recommended if you are serious in knowing the quantum world with all its weirdness.

[Krzyś Dz · Rating 4 out of 5]

Jeśli przeczytałeś kilka popularnonaukowych książek o kwantach i chciałbyś teraz poznać ich podstawy matematyczne, to to jest książka dla ciebie. Autor w łopatologiczny sposób na podstawie wektorów i macierzy tłumaczy czym są stany kwantowej, ich wartości czy obserwable. Pierwsza połowa książki była dla mnie bardzo zrozumiała, aczkolwiek wymagała skupienia i przypomnienia podstaw algebry liniowej. W drugiej części pojawiło się już dużo całek i pochodnych, czego już tak dobrze nie weryfikowałem. Jednakże wszystkie przekształcenia w dalszym ciągu były wykonywane zrozumiale. Książka daje bardzo dobry pogląd na to jak zasady rządzące światem kwantów można przedstawić w języku matematyki. Powoduje to jednak, że nie jest ona łatwa. Zainteresowanym jednak zdecydowanie polecam