[HN Gopher] PHYS771 Lecture 9: Quantum (2007)
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PHYS771 Lecture 9: Quantum (2007)
Author : thunderbong
Score : 89 points
Date : 2022-10-13 07:34 UTC (15 hours ago)
(HTM) web link (www.scottaaronson.com)
(TXT) w3m dump (www.scottaaronson.com)
| dekhn wrote:
| 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.
| pfortuny wrote:
| 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
| sigmoid10 wrote:
| 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.
| pfortuny wrote:
| I know. I was trying to stress that it is not novel at all to
| think of "complex probability".
| mananaysiempre wrote:
| 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.)
| sigmoid10 wrote:
| 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.
| mananaysiempre wrote:
| 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.)
| Koshkin wrote:
| > _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.
| davidivadavid wrote:
| 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.
| analog31 wrote:
| 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_.
| qbit42 wrote:
| 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.)
| femto wrote:
| > 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?
| eterevsky wrote:
| 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.
| femto wrote:
| 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?
| mananaysiempre wrote:
| 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-
| Schrodinger'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.
| Attrecomet wrote:
| 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.
| prof-dr-ir wrote:
| Surely that probability should be i x delta(0) x (-i) x
| delta(0) so, you know, your everyday squared Dirac delta
| function.
| femto wrote:
| Thanks!
| goerz wrote:
| 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).
| goerz wrote:
| https://en.m.wikipedia.org/wiki/Born_rule
| whatshisface wrote:
| 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.
| plandis wrote:
| 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.
| dennis_moore wrote:
| Shut up and calculate!
| ivan_ah wrote:
| 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.
| jwuphysics wrote:
| 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.
| tromp wrote:
| > 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?
| ivan_ah wrote:
| 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(ot). If instead you give it a "kick" at zero
| displacement, it will oscillate like sin(ot). 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 a _cos(ot) + b_ sin(ot), where a and b are the
| amplitudes. In practice we usually write a _cos(ot) + b_
| sin(ot), 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 a
| _cos(ot) + b_ sin(ot) as A*cos(ot-ph), 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.
| kgwgk wrote:
| 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.
| Attrecomet wrote:
| Yes, that's how you get things like entanglement and
| interference of states.
| qbit42 wrote:
| 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).
| tgflynn wrote:
| 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.
| kgwgk wrote:
| 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.)
| tgflynn wrote:
| 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.
| eterevsky wrote:
| 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.
| 363849473754 wrote:
| 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 Lowenheim-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"
| tgflynn wrote:
| > 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".
| 363849473754 wrote:
| 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.
| eterevsky wrote:
| 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.
| tgflynn wrote:
| 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.
| eterevsky wrote:
| 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.
| hackandthink wrote:
| 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
| dang wrote:
| Related:
|
| _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)
| ahelwer wrote:
| 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 Schrodinger) 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.
| dekhn wrote:
| 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.
| meltyness wrote:
| 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.
| prof-dr-ir wrote:
| 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.
| dekhn wrote:
| 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.
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