PHYS771 Lecture 9: Quantum (2007) | Hacker News Reader

PHYS771 Lecture 9: Quantum (2007)

scottaaronson.com

109 pointsthunderbong52 comments

For another take on the "computer science" approach to QM, check out Chapter 9 in my book on linear algebra: https://minireference.com/static/tmp/quantum_chapter_excerpt...

I think it's important to understand linear algebra before doing matrix quantum mechanics, this way you can focus on learning the new quantum concepts and understand we are just using vectors and matrices as representations for them.

  - quantum state = vector
  - quantum gate = unitary matrix
  - quantum measurement = set of projection matrices that add to the identity

My book only covers very basic QM—not a full course by any means. For anyone interested in getting into quantum computing, I recommend Thomas G. Wong's book Introduction to Classical and Quantum Computing which you can find here https://www.thomaswong.net/#textbook or Nielsen and Chuang which is a classic.

Nice book, I really like this approach. I also think Townsend's A Modern Approach to Quantum Mechanics is a wonderful book. It's similar to Sakurai but a bit more intelligible for undergrads.

As a person who was taught the “traditional” way, I feel that Aaronson’s disparagement of it misses—and the title tellingly omits—an important reason for it: that it tries to explain why quantum mechanics is mechanics. For that, you actually do need to know Hamilton–Jacobi, and ideally also know how to get the eikonal equation from electrodynamics, perhaps even understand high-frequency (Liouville–Green or, with a bit of abuse of language popular among physicists, JWKB) approximations in general. Same applies somewhat less poignantly if you’re not looking for motivation but are still planning to do QM to things in, y’know, continuous physical space.

The realization, apparently due to quantum computing people, that you can get away with not doing QM to things in continous physical space and still understand quite a lot of fun and useful QM stuff is very much valuable; you absolutely can build good courses that way. It’s just that I think that the pedagogical problem of laying out this general “quantum dynamical systems” approach in a compatible and not terribly redundant way to include the things learned a “quantum mechanics in space” course—which is absolutely required if e.g. you want to move on to QFT in solid-state or high-energy physics—is unsolved and to a large extent even untried. (Susskind’s semi-popular book is a notable exception, but I don’t feel he managed to pull it off.)

And, as usual, calling the traditional approach “historical” is a huge stretch. The standard QM course doesn’t talk about the founders’ (valid but extremely clumsy) approaches to relativistic corrections to the hydrogen spectrum any more than the standard calculus course talks about Newton’s early forays into algebraic and (what would come to be called) tropical geometry. (Or the standard programming course talks about native–bytecode interworking in the AGC, but even the most academic of programming courses are rarely branded as historical.)

> tries to explain why quantum mechanics is mechanics

Curiously though, since the 1800s analytical mechanics, too, began to be seen as a branch of mathematics. But I think that in this case, just as in the case of quantum mechanics, it is still extremely important to see the physical content, and the physical principles, that allow us to apply this or that mathematical framework. The fact that physics is a heavy user of mathematical methods (probably the heaviest of all sciences) does not mean that it reduces to mathematics. Don't lose the forest for the trees.

As someone who was a math and physics major at one point, that's what I consider the biggest missed opportunity in physics education. I was always taught physics as a set of theorems that you apply to some set of random problems. Zero effort given to the context, actual applications, and physical insight.

That kind of put me off physics, and I've been longing ever since for someone to build a course that essentially covers physics historically, starting from the problems physics helped us solve, and that puts modelling and experiment at the center, instead of just teaching recipes.

I feel the like incredibly common idea that all applications derive from theory might be one of the must harmful misconception in education.

I was a math and physics major too. What made it come alive for me was not just learning about experiments, but actually working in the lab.

I believe every person who is allowed to work in a quantum field (heh) should have been able to run at least a trivial quantum experiment.

That is, Scott Aaronson should go sit in some collaborator's lab and set up all the apparatus and analyze all the data, before he gets to spew about "quantum" to the general public. This should also apply to any theoretical physicist- I would love to see String Theorists in a lab setting up a michaelson morley experiment.

I guess this comes down to the distinction between quantum mechanics and quantum information. Just as classical probability and information theory can be explored independently of classical mechanics, quantum probability and information theory make sense to study independently of quantum mechanics. (I think the article serves as a great intro to quantum information, but a rather lacking intro to quantum mechanics.)

What do you think of the theory that space itself is emergent from QM? Something like loop quantum gravity?

In that case perhaps attempts to fit QM into continuous physical space is misguided, similar to attempts to explain planetary motion using epicycles.

I personally found this chapter very confusing, even with a good working knowledge of quantum computing - specifically the part about paths interfering destructively and canceling each other out. After a lot of thought and asking around I finally figured out what was going on with that diagram and wrote it up in a blog post: https://ahelwer.ca/post/2020-12-06-sum-over-paths/

Basically there are two ways of looking at the model of quantum computation, commonly called the standard (or Schrödinger) and sum-over-paths (or Feynman) methods. In the standard method you keep a 2^n-sized state vector of amplitudes around and multiply it against matrices (gates). In the sum-over-paths method you only focus on specific amplitudes and trace them back through the gates, drawing in all the other amplitudes that contributed to the final amplitude value. In the end it's a basic time/space tradeoff - simulating quantum computers with the standard method takes less time but more space, and the sum-over-paths method takes more time but less space. Another advantage of the sum-over-paths method is you can actually see quantum interference happening when a negative & positive amplitude cancel each other out, which is just sort of swallowed up by the standard method. This is what the diagram is trying to illustrate.

I follow that one can quickly get on-track with this mode of explaining nature, but I still think the rigamarole of dragging students through all the meaningful discoveries since Galileo is useful since understanding the techniques that have been used to imagine those systems of explanation is pretty interesting, if not broadly applicable.

It's naive to think this was the last mystery.

Agreed. Also, there are not "two ways to teach quantum mechanics".

For example, take the hugely popular book of Griffiths: if I remember correctly, equation 1.1 is just the time-dependent Schrodinger equation.

From what I have seen, most teachers do not spend too long on the history of quantum mechanics but they do focus more on the physics of quantum mechanics that this introduction does. The difference is important, and completely ignored in the first paragraph of the linked article.

The more I work in science, the more I go back to the period from slightly before the industrial revolution- the 1750s or so- ending at the beginning of WWII.

Each time I go back I understand just a little bit more about how we got to where we were in 1938. Or how we got to the point that freshman physics students can build a michelson morely interferometer in an afternoon on a benchtop when the original required heroic efforts including a pool of mercury.

Aaronson wrote a great book "Quantum Computing since Democritus" discussing not just quantum mechanics and computing, but also such topics as algorithmic complexity and anthropic principle. It is located somewhere between science and pop-science: it contains quite a few proofs and technical details, while remaining quite approachable for a non-expert.

I disagree that it is “quite approachable for the non-expert” assuming you meant a general audience without a background in high school mathematics. They probably wouldn’t exactly understand the Löwenheim–Skolem theorem or ZFC, or consistency arguments and so forth as presented in the book.

Some parts require more math than probably the general audience will be able to grasp but if they gloss over those sections they could still get the gist of what follows.

I think I’d rephrase it as “quite approachable for a talented high schooler / someone with knowledge in undergraduate mathematics”

> assuming you meant a general audience without a background in high school mathematics

Why assume that ? Isn't the whole point of high school to provide a basic foundation of knowledge that everyone should have ?

The parent comment said “quite approachable for the non-expert” not “quite approachable for the uneducated”.

Depends what “non-expert” means. Because I actually think even if you have an undergraduate knowledge in mathematics then you still may not be able to follow some of his arguments unless you specifically studied mathematical logic. I have a more advanced background in math but not in logic and don’t understand some of his arguments. He sometimes makes loose statements without proof or without providing strong background knowledge to closely follow those things if you don’t already know them. Some of the ZFC stuff I didn’t follow either, but I never really needed to know set theory to the extent Aaronson uses it. I think the statement it’s quite approachable isn’t very accurate. If you want to gloss over it and still get the main idea then it’s good for that but not for rigorously understanding all the mathematical arguments without preexisting background knowledge. It’s between a pop sci book and textbook with a casual tone. It’s a fun book.

You may well be right. I haven't read that book, though I have found much of Aaronson's writing to be quite approachable. But I think your previous comment was an inaccurate way of phrasing those critiques.

Ok, I think I generally agree with your categorisation.

Personally I did study mathematical logic in the University a little bit, but it was 20 years ago and I don't think I remember much about it past what all undergraduates are taught.

Yes, to understand all of the contents in this book you'd need to have knowledge of Math on the level of an undergraduate in a STEM discipline. To me it still meets the definition of "non-experts". It is much more involved than your average pop-science book, but at the same time is much more fun and easy to read than a typical Math college textbook.

Also, about half of the book can be read without any Math background.

I tried reading that book and I can't say I followed everything smoothly. For each chapter, I had to read 10 other articles to better understand the material, though I admit I am not very well versed with the subject matter.

But yeah, it is a very good book, one of its kind. I wish there were more accompanying books written in similar fashion.

Wow. This isn't quantum- it's a mathematical representation of how we currently work with a limited type of quantum states in a limited part of quantum mechanics.

People who study quantum need to know how to set up an actual quantum experiment in the lab, and how to work with hamiltonians.

Scott Aaronson's introduction of Quantum Theory as "a certain generalization of probability theory" is very nice and to the point.

(Further Reading)

These notes of Greg Kuperberg are more accessible than Caves, Fuchs, and Schack (at least for me):

"An introduction to quantum probability, quantum mechanics, and quantum computation"

https://www.math.ucdavis.edu/~greg/intro.pdf

Quantum mechanics as a generalization of probability (2007) - https://news.ycombinator.com/item?id=8377680 - Sept 2014 (79 comments)

New straighforward approach to teaching quantum mechanics - https://news.ycombinator.com/item?id=4319276 - July 2012 (55 comments)

Quantum mechanics for mathematicians - https://news.ycombinator.com/item?id=83594 - Nov 2007 (12 comments)

My introduction to quantum was through chemistry- the teacher wrote E Psi = H Psi on the board and we all said "can't you just cancel the Psi" and that's how we leanred about operators and tensors.

Meanwhile Griffith is out there like a boss, slapping a second order partial differential equation on like page 1 of chapter 1 telling you to just do it, bro.

Shut up and calculate!

> In other words, we can always ask, what if we don't know which quantum state we have? For example, what if we have a 1/2 probability of state1 and a 1/2 probability of state2? This gives us what's called a mixed state, which is the most general kind of state in quantum mechanics.

Here one might wonder if pure quantum states could be mixed not with mere probabilities, but with something more general like amplitudes?

Superpositions and probabilistic mixtures are two different ways to combine |0> and |1>.

The superposition a|0> + b|1> describes a linear combination of the two solutions |0> and |1> to some differential equation, but there is no notion of "mixture". A good analogy to understand superpositions is the solutions to simple harmonic motion (like mass attached to a spring). If you start the system from stretched state and zero velocity, it will oscillate like cos(ωt). If instead you give it a "kick" at zero displacement, it will oscillate like sin(ωt). Since any other combination is possible (e.g. kick while stretching), the most general solution to the equation of motion of the mass spring system is acos(ωt) + bsin(ωt), where a and b are the amplitudes. In practice we usually write acos(ωt) + bsin(ωt), so there is nothing "fancy" going on for superpositions: they are just linear combinations of the possible states. (You don't hear anyone talking about a mass-spring system oscillating like cos and like sin at the same time, do you?)

Off topic note for completeness: in physics class we rewrite acos(ωt) + bsin(ωt) as A*cos(ωt-φ), but it's the so you don't see the cos and sin separately, but they are there.

As for "mixtures" those represent our state of knowledge, or rather ignorance. The mixture of 50% |0> and 50% |1> is represented as a density matrix, which has the same "statistics" under measurement. It's hard to do matrices in plain text so I'll cut if off here... but you can look it up.

An even simpler analogy: The difference between a mixture and a superposition is the difference between going either in the North direction or in the West direction with 50/50 probability and going in the Northwest direction.

Yes, that's how you get things like entanglement and interference of states.

Can you explain? That's not how I interpret either. Every quantum state can be realized as a classical mixture of pure quantum states (i.e. as a density matrix).

No, the fact that what you said is not true is basically the core concept of quantum mechanics.

EDIT: As another comment pointed out I was wrong about this, please ignore.

In what sense is that not true?

It seems true at least if we accept that a pure state is a mixture of itself with nothing else.

Any quantum state CAN be represented by a density matrix. That's what density matrices are for. That quantum state may be pure or may be a mixture. (A mixture density matrix could also be a partial trace representing a subsystem of a larger system instead of a true mixture of pure states.)

Yes, I you're right. I misinterpreted "pure quantum states" as "basis states" (I'm not much in the habit of thinking about QM these days). I believe the "(i.e. as a density matrix)", which makes my mistake more obvious, was added after I commented.

Complex measures are a thing in mathematics since… well, possibly since measure theory was. The thing is expressing this in a way which is understandable by a physicist (measure spaces are not exactly “simple”).

https://web2.ph.utexas.edu/~gsudama/pub/1994_004.pdf

Measure theory itself only became a formal branch of mathematics around the turn of the 20th century and complex measures developed pretty much in the fairway of quantum mechanics (which appeared at the same time). This is in stark contrast to e.g. non-euclidean geometry, which had already been well formalised into its own branch decades before the invention of GR.

Riemannian geometry in general (and not just geometry of specific homogeneous spaces like the hyperbolic plane) was still somewhat recent and hard at the time or at least had that reputation. I’ve heard that the notion of parallel transport on Riemannian manifolds, for one, was developed as a consequence of people making sense of GR, before that Christoffel symbols were apparently just ... a formal thing.

(The general theory of connections on bundles certainly postdates GR although it does predate Yang–Mills.)

To some extent, yes. But the point was that Einstein was probably only able to develop GR thanks to work by people like Riemann, who investigated these things not knowing where they'd lead half a century later. For QM it was almost the other way around and people really had to come up with and formalise a whole new branch of mathematics to make sense of the early experiments. Sometimes physics follows math and sometimes math follows physics. For QM it was closer to the latter.

I know. I was trying to stress that it is not novel at all to think of “complex probability”.

> Well, there's a reason you never hear the weather forecaster talk about a -20% chance of rain tomorrow -- it really does make as little sense as it sounds.

Maybe negative probabilities do make sense?

1 = the event is certain to happen (state moves from A->B)

0 = the event is certain not to happen (state does not change)

-1 = the event is certain to unhappen (state moves from B->A)

If this is the case, to say something is "quantum" is another way of saying the thing has time symmetry and arrow of time can move in either direction.

This gels with experience. Microscopic processes are time symmetric, hence quantum. In the macroscopic domain (where there are lots of independent particles involved?), the reversal of state is less likely to happen, the arrow of time becomes apparent and so the quantum becomes less apparent (aka decoherence is occurring).

Is there any sense in this interpretation?

I think you're on your way to coming up with the idea of an expected value. The additional step of reasoning to get there is to think about what number we'd use to describe an outcome that had a 50% probability of going "forwards" and a 50% probability of going "backwards." If 0 makes sense as an answer to that, you're doing a probability-weighted average of outcomes. Quantum mechanics is full of expected values, and you can interpret wavefunctions just fine with them.

In the QM formalism, we come up with these operators that can be used to compute expected values from wavefunctions, by doing a calculation that, if it were involving vectors, would be written like E[A] = x^T* Ax, where x is the wavefunction, x^T* is its transpose and conjugate, and A is a matrix designed to pull out an expected value when used in that way. The demand that the results of this calculation be real for every possible wavefunction give us the property that A is a hermitian (A = the transpose and complex conjugate of A, it's like being a symmetric matrix), and from there we know that A can always be diagonalized. If A can always be diagonalized, we can always write x in a basis that diagonalizes it, in which case x^T* Ax becomes something that just conjugate-squares the magnitude in front of each eigenvector and multiplies it by something on the diagonal of the now-diagonalized A. If you go back to the original definition of expected values as a probability-weighted average, the conjugate-squared terms are playing the role of probabilities and the eigenvalues on the diagonal of the matrix are playing the role of outcomes.

I'm not a quantum scientist, but I don't think this is the correct view amplitude. If a configuration has amplitude -1 for electron located in some place, it doesn't mean that there's a positron there.

I am a quantum physicist and I can confirm that’s not how probability amplitudes work . The amplitudes are just complex numbers. Remember that complex numbers describe oscillations. A number r * exp(i phi) is an oscillation with amplitude r and current phase phi. Quantum states are wave functions, hence the complex numbers describing the oscillations. Amplitude -1 is just the bottom of the oscillation. Then on top of that you have that probabilities are the squares of the amplitude (for which I’m not sure if have a concise explanation).

https://en.m.wikipedia.org/wiki/Born_rule

I'm not a quantum physicist either. The amplitude would have to be j or -j to get a probability of -1 (probability = amplitude squared?). Does +/-j correspond to a positron?

No. The probability of an event is never, ever negative, even in quantum. We use the complex scalar product, c x c^*, to get probabilities, and those are necessarily positive even for complex numbers. The same is true for the generalization to functions required in quantum mechanics.

If your wave function at point (0,0,0) were i x delta(0) (i.e. a point-like particle perfectly located at the three-dimensional origin), the probability of finding the particle there isn't i^2, but i x i^* = i x (-i) = 1.

Surely that probability should be i x delta(0) x (-i) x delta(0) so, you know, your everyday squared Dirac delta function.

I secretly did the integration over space to go from the probability density to the probability, so the deltas are integrated out :P

Doing calculations over text-based comments is so painful that I gloss over a lot, in the expectation that this isn't the right place to talk about anything but vague ideas anyway.

Thanks!

No, any electron amplitude with absolute value (squared) equal to one corresponds to a probability density of 1 to find an electron. You can have a positron wavefunction if you have a positron, but take care that you don’t have both, or they will (even if with a vanishingly small probability) meet and, by participating in a process with energy comparable to mass, boot you right out of nonrelativistic quantum mechanics into quantum field theory.

I hate to tell people to learn A before they can try B, but here I really, really don’t know of a way to start thinking about QFT with its positrons and annihilation and so on without getting hopelessly confused unless you’re already comfortable with normal wavefunctions-and-Schrödinger’s-equation QM. You’ll still be confused even if you are comfortable with it, mind you, it’s just that then you’ll at least have a ghost of a chance of getting your confusion down to manageable levels.

> ghost of a chance ... down to manageable levels

Argh. Manageable as in non-divergent, so take a small step back to regularize your confusion, and get haunted by a <https://en.wikipedia.org/wiki/Ghost_(physics)>.

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