Saturday, July 18, 2020

Understanding Quantum Mechanics #4: It’s not as difficult as you think! (The Bra-Ket)

If you do an image search for “quantum mechanics” you will find a lot of equations that contain things which look like this |Φ> or this |0> or maybe also that <χ|. These things are what it called the “bra-ket” notation. What does this mean? How do you calculate with it? And is quantum mechanics really as difficult as they say? This is what we will talk about today.


I know that quantum mechanics is supposedly impossible to understand, but trust me when I say the difficulty is not in the mathematics. The mathematics of quantum mechanics looks more intimidating than it really is.

To see how it works, let us have a look at how physicists write wave-functions. The wave-function, to remind you, is what we use in quantum mechanics to describe everything. There’s a wave-function for electrons, a wave-function for atoms, a wave-function for Schrödinger’s cat, and so on.

The wave-function is a vector, just like the ones we learned about in school. In a three-dimensional space, you can think of a vector as an arrow pointing from the origin of the coordinate system to any point. You can choose a particularly convenient basis in that space, typically these are three orthogonal vectors, each with a length of one. These basis vectors can be written as columns of numbers which each have one entry that equals one and all other entries equal to zero. You can then write an arbitrary vector as a sum of those basis vectors with coefficients in front of them, say (2,3,0). These coefficients are just numbers and you can collect them in one column. So far, so good.

Now, the wave-function in quantum mechanics is a vector just like that, except it’s not a vector in the space we see around us, but a vector in an abstract mathematical thing called a Hilbert-space. One of the most important differences between the wave-function and vectors that describe directions in space is that the coefficients in quantum mechanics are not real numbers but complex numbers, so they in general have a non-zero imaginary part. These complex numbers can be “conjugated” which is usually denoted with a star superscript and just means you change the sign of the imaginary part.

So the complex numbers make quantum mechanics different from your school math. But the biggest difference is really just the notation. In quantum mechanics, we do not write vectors with arrows. Instead we write them with these funny brackets.

Why? Well, for one because it’s convention. But it’s also a convenient way to keep track of whether a vector is a row or a column vector. The ones we talked about so far are column-vectors. If you have a row-vector instead, you draw the bracket on the other side. You have to watch out here because in quantum mechanics, if you convert a row vector to a column vector, you also have to take the complex conjugate of the coefficients.

This notation was the idea of Paul Dirac and is called the bra-ket notation. The left side, the row vector, is the “bra” and the right side, the column vector, is the “ket”.

You can use this notation for example to write a scalar product conveniently as a “bra-ket”. The scalar product between two vectors is the sum over the products of the coefficients. Again, don’t forget that the bra-vector has complex conjugates on the coefficients.

Now, in quantum mechanics, all the vectors describe probabilities. And usually you chose the basis in your space so that the basis vectors correspond to possible measurement outcomes. The probability of a particular measurement outcome is then the absolute square of the scalar product with the basis-vector that corresponds to the outcome. Since the basis vectors are those which have only zero entries except for one entry which is equal to one, the scalar product of a wave-function with a basis vector is just the coefficient that corresponds to the one non-zero entry.

And the probability is then the absolute square of that coefficient. This prescription for obtaining probabilities from the wave-function is known as “Born’s rule”, named after Max Born. And we know that the probability to get any measurement outcome is equal to one, which means that that the sum over the squared scalar products with all basis vectors has to be one. But this is just the length of the vector. So all wave-functions have length one.

 The scalar product of the wave-function with a basis-vector is also sometimes called a “projection” on that basis-vector. It is called a projection, because it’s the length you get if you project the full wave-function on the direction that corresponds to the basis-vector. Think of it as the vector casting a shadow. You could say in quantum mechanics we only ever observe shadows of the wave-function.

The whole issue with the measurement in quantum mechanics is now that once you do a measurement, and you have projected the wave-function onto one of the basis vectors, then its length will no longer be equal to 1 because the probability of getting this particular measurement outcome may have been smaller than 1. But! once you have measured the state, it is with probability one in one of the basis states. So then you have to choose the measurement outcome that you actually found and stretch the length of the vector back to 1. This is what is called the “measurement update”.

Another thing you can do with these vectors is to multiply one with itself the other way round, so that would be a ket-bra. What you get then is not a single number, as you would get with the scalar product, but a matrix, each element of which is a product of coefficients of the vectors. In quantum mechanics, this thing is called the “density matrix”, and you need it to understand decoherence. We will talk about this some other time, so keep the density matrix in mind.

Having said that, as much as I love doing these videos, if you really want to understand quantum mechanics, you have to do some mathematical exercises on your own. A great place to do this is Brilliant who have been sponsoring this video. Brilliant offers courses with exercise sets on a large variety of topics in science and mathematics. It’s exactly what you need to move from passively watching videos to actively dealing with the subject. The courses on Brilliant that will give you the required background for this video are those on linear algebra and its applications: What is a vector, what is a matrix, what is an eigenvalue, what is a linear transformation? That’s the key to understanding quantum mechanics.

To support this channel and learn more about Brilliant, go to brilliant.org/Sabine, and sign up for free. The first two-hundred people who go to that link will get twenty percent off the annual Premium subscription.

You may think I made it look too easy, but it’s true: Quantum mechanics is pretty much just linear algebra. What makes it difficult is not the mathematics. What makes it difficult is how to interpret the mathematics. The trouble is, you cannot directly observe the wave-function. But you cannot just get rid of it either; you need it to calculate probabilities. But the measurement update has to be done instantaneously and therefore it does not seem to be a physical process. So is the wave-function real? Or is it not? Physicists have debated this back and forth for more than 100 years.

64 comments:

  1. I can remember when I was working on my graduate degree in mathematics, we had a physicist auditing our general topology class. I asked him why he was taking a class in something so abstract. He said, "oh, no, this mathematics is very useful in physics." I'm starting to see why.

    ReplyDelete
    Replies
    1. Actually, it is. For example, in the Singularity Theorems of GR and Global Methods in general. That's just one of the many applications.

      Delete
  2. The state is the equivalence class of the wave function in projective Hilbert Space.

    ReplyDelete
  3. The wave-function defines an objective probabilistic disposition on the observables.

    ReplyDelete
  4. Sabine,

    Dare I suggest that the best place to start, is with Schroedinger's equation applied to the simple harmonic oscillator. You can extract the wave functions as explicit expressions. To begin with you don't even need to introduce Hermite polynomials - just display the first few solutions.

    Then you can discuss such issues as what the wave function means, and why the wave function should be normalised etc.

    I would argue that this approach connects with all the calculus that people learn in school.

    ReplyDelete
  5. Good description. In my senior year of high school I purchased the Feynman Lectures on Physics 3 volume set. I also got the MTW Gravitation. I eagerly read these as best as I could. One thing that god-smacked me was how the Bra-Ket formalism was much the same as how a metric is formed and that the two were in fact intervals or distances. I still have these books. My copy of MTW is held together with glue and duck tape.

    ReplyDelete
  6. Funny how the apparent incompatibility between Quantum Mechanics and General Relativity extends to the notation used. Both deal mostly with the same kind of objects, linear spaces and manifolds, and yet when in QM we use the Dirac notation for the inner product, , in GR we use the Einstein convention a_i b^i. One would think that, after a century of playing with the same mathematical entities, physicists would standardize on a single notation, but alas...

    ReplyDelete
  7. Sabine,

    I think you took quite a bit of a quantum jump in here!

    Personally, I remember going through some other accounts that had similarly rushed through this material in more or less exactly the same manner, may be some 15+ years ago. But it all had left me utterly confused. Finally, it was the first chapter from van Dommelen's book which really clarified the idea to me. (He describes a graph of a function as comprising of an infinity of upward pointing Dirac's delta's, each of which is to be taken as a vector. Expansion into an infinite basis set!)

    Anyway, let me share what should work with most any one (like engineers), what the physicist should explain first, before explaining a wavefunction as a vector in an Hilbert space:

    The first idea to touch upon is: Expansion of a function into a set of other, simpler, functions (the basis functions). Here, the polynomial basis set comes in quite handy. We engineers use it a lot in techniques like FEM (Finite Element Method). Of course, for most direct relevance to QM, you then have to quickly move on to the Fourier expansion. Engineers know it pretty well.

    You can then quickly touch upon the orthogonality of the Fourier basis *functions*. People know the orthogonality of line segments serving as unit vectors. The orthogonality of basis functions has to be pointed out---even if engineers have previously studied this topic.

    It is at this point that you can introduce the idea that the underlying algebraic structure behind expansions, involving functions, is quite similar to that involving vectors as directed line segments.

    Depending on the sophistication or maturity of the audience, you may or may not go over the column-, row- and matrix-notation. (However, personally, I understood the idea of the dual space only after relating it to the column vs. row vectors.) Also realize, topics like the tensor product (and even the Schmidt decomposition) become very easy to grasp once you present them using matrix notation: tensor product as a column vector into a row vector (contrast: the dot product as a row vector into a column vector). The matrix notation also helps in better understanding the idea of an operator acting on the elements of a vector.

    The passage from the matrix notation to the Dirac notation then is very, very smooth.

    Some time by now, the student has already become ready to be told that in QM, you now begin to take the imaginary part too, because the Schrodinger equation comes as two coupled real-valued equations that obey the complex algebra. They already know the Fourier expansion using complex notation. Now they use the full expressions.

    What else then is left of the otherwise formidable-looking Hilbert space? Precious little.

    People may laugh, but the above is what I honestly believe is a *truly* helpful path to an engineer. (I speak from experience.)

    My two cents.

    Best,
    --Ajit

    ReplyDelete
  8. "all the vectors describe probabilities." ?????

    ReplyDelete
  9. Sabine,

    One of the simplest paths to learn QM is the Teaching Company lectures by Ben Schumacher, "Quantum Mechanics: The Physics of the Microscopic World." No math beyond high-school algebra (not even complex numbers!) needed, but he goes into detail on all the main issues. (To anyone who cannot get it from the local library: do not buy it until they have one of their frequent price-cutting sales!)

    Schumacher is a pioneer in quantum information theory, supposedly co-inventor of the term "qubit" along with more substantial contributions.

    Sabine, one of the key points you did not cover is the tensor-product construction for states in multi-component systems. This seemed self-evident to me when I was first exposed to it, but I have never figured out how to explain it simply to others.

    One could also argue that the idea of "quantum amplitude" is more basic than quantum state, especially if you think like Feynman, though in the end they are equivalent.

    All the best,

    Dave

    ReplyDelete
    Replies
    1. @ PhysicistDave,

      Quantum Theory is an application of generalized probability theory (might include incompatible observables). Here one always has states (probability measures on the quantum logic), but they are not always represented by "quantum amplitudes" (Gleason's Theorem shows that they are for the Standard Quantum Logic).

      Delete
    2. Hello, professor.
      the difference
      between quantum theory...
      -and quantum mechanics,
      is .
      Best wishes,

      Delete
  10. David Bailey wrote:
    >I would argue that this approach connects with all the calculus that people learn in school.

    But I think Sabine's point (and also the point of Schumacher's lectures that I mentioned) is that you do not need calculus to understand QM.

    Mathematically, QM is really easy. But, in terms of physical understanding... well, to paraphrase Feynman, if you think you understand QM, you don't.

    ReplyDelete
    Replies
    1. @ PhysicistDave,

      "Feynman, if you think you understand QM, you don't." That's because he didn't take the quantum logic approach which is very easy to understand!

      Delete
    2. Quantum mechanics by itself is not hard to understand. It is a theory of linear vector spaces and operators. What could be simpler? In mathematics a goal is to often convert some system into linear algebra, where upon it is worked much more easily.

      Where quantum mechanics is difficult is its relationship with classical or macroscopic physics. Curiously we don't have Feynman saying, "If you think you understand classical mechanics, you don't."

      Delete
    3. Prof. David Edwards wrote to me:
      >That's because [Feynman] didn't take the quantum logic approach which is very easy to understand!

      But you've conceded that Bohmian mechanics, which gives the same experimental predictions as quantum mechanics, does not use quantum logic.

      That is an existence proof that quantum logic is not needed.

      QED.

      More pragmatically, you suggest that QM would be more comprehensible if we started with quantum logic. Do you know of any widely used, successful, introductory text that uses quantum logic to teach QM? Why not if that is the simplest way to understand QM?

      Personally, I will take quantum logic seriously when its proponents start using quantum logic to prove theorems! The fact that they use classical logic in proofs suggests to me that they do not really believe in quantum logic.

      The basic problem of quantum mechanics is that we need separate “measurement postulates” to get from the purely quantum world of the continuous Schrödinger equation to the macroscopic world in which one state is “projected out” by the measurement process and the other states can be ignored.

      I know there are reams and reams of writings “explaining” how this happens (decoherence, quantum logic etc. ad infinitum). But none of it results in a simple explanation of quantum mechanics, starting from scratch, free of separate measurement postulates, where the Schrödinger equation alone is all that rules.

      After all, measurement apparatuses and even human observers should be describable by QM, so the Schrödinger equation alone should suffice.

      The promise of MWI is that it does this: alas, MWI has, in my opinion, insurmountable technical problems.

      And Bohmian mechanics cuts the Gordian knot, but we all know that the way it interfaces with relativity is very kludgy (though it agrees with experiment).

      So, sorry, Dave, but quantum logic and all the other attempts to sweep the problems of QM under the rug are “fool's gold”: they may look pretty, but they do not solve the problem.

      For some reason, it is exceedingly hard for some human beings to admit that there are some basic things about the nature of reality that we humans do not (yet) understand. The weirdness of quantum mechanics should be a rebuke to such misplaced human arrogance.

      Dave

      Delete
    4. The basic problem of quantum mechanics is that we need separate “measurement postulates” to get from the purely quantum world of the continuous Schrödinger equation to the macroscopic world in which one state is “projected out” by the measurement process and the other states can be ignored.

      Hello Dave,

      I have a naive question. It’s more philosophy than science, though.
      First, I have to distinguish possibility and actuality.
      By definition, possibilities are not actualities. But possibilities are real, they exist. In the same way that the future exists. So what’s the difference? Let’s say that the future has no formal actualization. But here’s the more important point: it seems that possibilities (since it seems that QM, in the present discussion, deals essentially with possibilities) must necessarily be defined continuously. They cannot be defined discontinuously, you cannot jump from a possibility to another, right?
      To illustrate my thought: for example the EM spectrum is a continuous representation of the values that photons can take. But of course, actual photons always exhibit particular, restrictive values. You don’t ask a photon to take all these values at once! Every transition from possibility to actuality – i.e every process – implies full determination, hence selection. This, and not that (what can be called formal actualization). You can use the words «choice», «selection», or «projection», at this stage of the process, you’re done.
      By definition, full determination (or actuality) implies a «collapse» among the continuous realm of possibilities. We just have to assume that there are at least two possibilities, or more. The more the possibilities there are, the bigger the selection. For this reason, I tend to think that the «collapse of the wave-function» does not belong to a unique or specific feature of QM, but to a more general or metaphysical aspect of evolution. It is always less than what was possible, but always more or different from what was already realized.

      So, I don’t see really how scientists could do otherwise than to meet this f….g measurement postulate ;-)
      As an aside, I am wondering whether the epistemic/ontic divide does not come from the fact that sometimes people tend to confuse possibility with actuality. They are both real, they just don’t have the same mode of existence.
      But maybe I am wrong about possibilities, and I am ready to change my mind.

      Delete
    5. Thanks for your response, Dave, but my point would be, if you are at university, learning about QM, you will have some calculus background - nobody starts learning QM from scratch! I remember very clearly that the hardest part for me about the transition from school maths/physics to university maths/physics (we had a physics course, even though I was taking chemistry) was those places where the approaches simply didn't connect.

      Also the history of QM is bound up in the double slit experiment, usually explained in terms of waves, and after you have learned calculus, it really helps to see a wave-like formula appear!

      Your approach is a bit like teaching kids basic maths, starting from Peano's axioms - if it sounds great to you, just try it out!

      Delete
    6. Felker wrote to me:

      >By definition, full determination (or actuality) implies a «collapse» among the continuous realm of possibilities. We just have to assume that there are at least two possibilities, or more. The more the possibilities there are, the bigger the selection. For this reason, I tend to think that the «collapse of the wave-function» does not belong to a unique or specific feature of QM, but to a more general or metaphysical aspect of evolution. It is always less than what was possible, but always more or different from what was already realized.

      >So, I don’t see really how scientists could do otherwise than to meet this f….g measurement postulate ;-)

      Hi, Felker. I'm afraid you misunderstand the problem.

      The problem is that the quantum mechanics measurement postulates are weird, really weird. Yes, at some level, any scientific theory has to assume that we van actually observe and measure things. But classical mechanics does not require any special measurement postulates. You give the example of the EM spectrum: sure, any individual example of EM radiation cannot be all possibilities at once. But, if you go to Baskin-Robbins, you also cannot eat all 31 flavors, at least not at the same time.

      That's trivial.

      That is not the issue in quantum mechanics.

      Let me give a silly example I like that helps illustrate the real issue.

      I live in Sacramento and sometimes travel to Los Angeles.

      Suppose next weekend that there is a 25 percent chance that I will travel to LA via car and a 25 percent chance that I will go to LA via plane (and will not use any other mode of transportation, and will not use both).

      I assume that you agree with me that there is then a 50 percent chance that I will go to LA next weekend.

      That is not how life works in the quantum world.

      In the quantum world, the two possibilities can interfere “destructively” so that there is ZERO chance that I will go to LA next weekend.

      The evidence of this sort of behavior at the quantum level is very common and quite conclusive: I'm not asking you to take my claim on faith – look at any solid introductory reference, best would be Ben Schumacher's lectures, but there are also many good texts, generally at a more advanced level than Schumacher's lectures (I like volume III of the Feynman Lectures).

      Most people who have thought about this conclude that this sort of “cancellation effect” occurs because in some sense neither option really happens – neither the trip by car nor by plane – until we somehow observe or measure it. Hence, ordinary macroscopic reality is somehow brought about by measurement.

      Of course, for this to work, measurement in QM must be very, very different from what we ordinarily think of as measurement!

      Now, of course, people have thought of other ways of explaining this weird cancellation effect, this “destructive interference.” I am not going to go into them here except to say that if you explore them, you will find all of them to be pretty weird too.

      What I am trying to get across to you, however, is that merely thinking in terms of ordinary daily experience and waxing philosophical just does not work to understand quantum mechanics. Things happen in quantum mechanics that do not happen in ordinary life.

      But to grasp this, you have to go to the trouble to understand quantum mechanics.

      (cont.)

      Delete
    7. (cont.)

      By the way, to return to my humorous example above, why do we never get such quantum effects in terms of my travels to LA? After all, cars and planes are ultimately just composed of atoms, which must be described by quantum mechanics!

      The answer to that, more or less, is what is known as "decoherence theory."

      Just as the two options can interfere “destructively,” they can also interfere “constructively” to give a chance that I will go to LA this weekend of 100 percent!

      For various technical reasons, for macroscopic phenomena you get a mixture between constructive and destructive interference, so the odds of going to LA are the average of ZERO and 100 percent – i.e., 50 percent, just as it should be.

      So, crazy things are really happening in the everyday, large-scale world too, but the craziness just gets averaged out so we never see it!

      You don't believe that?

      Well, in truth, neither do I and neither do most physicists.

      But that is what the math of quantum mechanics seems to say.

      This is another way of viewing the “measurement problem” is QM.

      Yeah, I think we are missing something. But I do not know what it is.

      Anyway, I hope I have given enough information for you to see that the truth must be more subtle than you suggest.

      Dave

      Delete
    8. Hello Dave, nice explanation, Stefan

      Delete
    9. Dave,

      Sure, I still don’t get it. I am questionning. Thanks for this informative comment and for the references, I will certainly look at them.

      Delete
    10. Hello Dave, if I understood you correctly, you see some basic problems in quantum mechanics and understanding of micro objects.
      Do you think that calculus is part of the solution or part of the problem?
      Stefan

      Delete
    11. Physicist Dave wrote: "...alas, MWI has, in my opinion, insurmountable technical problems." Dave, your opinion prodded me into another reading of Hugh Everett (1957) and Bryce DeWitt (1971). I moved on to Asher Peres. I perused two interesting papers: Can We Undo Quantum Measurements (1980) and When is a Quantum Measurement (1985). In the later read:
      "Everett's work can also be interpreted in a way which is completely compatible with the standard approach." (page 692, AJP, Vol. 54, N. 8). The earlier paper reads: "This result justifies von Neumann's measurement theory, without any hypothesis on the role of the observer." (page 879, Phys. Rev. D22).
      While revisiting these papers, I asked myself: where are those "insurmountable technical problems" ?

      Delete
    12. Gary Alan asked me:
      >While revisiting these papers, I asked myself: where are those "insurmountable technical problems" ?

      There is an extensive literature on what are often known as the “preferred basis” problem and the “probability-measure” problem.

      From time to time over the decades, the MWI guys claim to have solved these problems, but they never manage to convince most physicists they have done so.

      The usual description of MWI as “splitting universes” is also just plain wrong in various ways: for example, in QM the set of basis states is always there; it does not split into new basis states (this would violate unitarity).

      Sabine also has an objection that I am not sure I understand: I think her objection is that MWI does not really explain where the usual measurement postulates (Hermitian operators and all that) come from, but I may be misunderstanding her.

      Anyway, there are good reasons so many physicists reject MWI besides just Ockham's razor.

      Dave

      Delete
    13. Stefan Freund wrote to me:
      >Hello Dave, if I understood you correctly, you see some basic problems in quantum mechanics and understanding of micro objects.

      Well... maybe we misunderstand macro objects – who knows? But, yeah, basically I agree with Einstein and Schrödinger and de Broglie and Wigner and Steve Weinberg and John Bell and Sabine and many other physicists over the decades that somehow our understanding of QM is incomplete.

      Stefan also asked:
      >Do you think that calculus is part of the solution or part of the problem?

      Well, most of the problems show up with simple two-state systems where calculus is not an issue – as Sabine says, linear algebra is enough to demonstrate the problems.

      You want my gut-level guess (my gut is wrong more often than not)? I think Bohmian mechanics and related theories would be fine, except that the way they handle Special Relativity is very kludgy. So, I think it is worth focusing in on that problem – superluminal but Lorentz invariant influences, a block universe approach, super-determinism, a relaxation process in hyper-time, or whatever.

      But lots and lots of people have struggled with approaches like that, and none has (yet) succeeded. So, maybe my “gut” is (once again!) wrong!

      Dave

      Delete
    14. Thanks, Dave, for your reply. As you are aware, I do not believe that the 'preferred basis problem' poses an "insurmountable technical problem." (quoting your phrase). Zurek uses the terminology 'basis ambiguity' arguing that it is settled "by the environment." (arXiv:0707.2832, Relative States and the Environment). Also, rejecting MWI from a perspective utilizing Occam's razor is fruitless, it being a philosophical objection (even John Baez writes: "simplicity is subjective"). Interestingly enough, the Ph.D. of Hugh Everitt claims: "It has the virtue of logical simplicity..." I agree with Steven Weinberg when he writes: "But, different physicists are satisfied with different interpretations." (page 102, Lectures on Quantum Mechanics).

      Delete
    15. Gary Alan,

      Well... “the environment” is not part of the formalism of quantum mechanics! To refer to the basis ambiguity as being resolved by the “environment” is just smuggling measurement postulates back in through the back door. Might as well just stick with the Copenhagen interpretation and accept that “for all practical purposes” QM does not apply to the macro world.

      What is really being debated is “in principle” vs. “for all practical purposes.” For all practical purposes, the standard textbook approach certainly works. But it cannot work in principle, because, in principle, the observer too should be described by Schrödinger's equation and you will never “collapse” the wavefunction.

      MWI has the same problem. Sure, for all practical purposes, the “environment” singles out position states. But, in principle, there is no separate environment: the whole idea of “many worlds” is that it applies to the whole world, not just a system separate from the environment.

      There is no environment outside the universe that somehow singles out a preferred basis for the universe!

      What is happening here is that people are easy on their own favored interpretation and accept “for all practical purposes” arguments but hard on other interpretations and insist that other interpretations work “in principle.”

      Like arguments in politics and religion.

      Am I immune? In general, I suppose not. But since I do not believe any of the proposed interpretations of QM, I can cast a jaundiced eye on all of them.

      Dave

      Delete
    16. With respect to the cosmos, the einselection of the classical state of the universe is a problem of quantum gravitation. With the superposition of a single particle state, say the two-slit experiment, the mass-energy of the particle means its gravity field has a source associated with both slits. The gravity field is itself in a superposition of states. For a light enough of a particle the Riemann curvature will be effectively linear as the quadratic terms in the connection are G^2/c^4 times smaller and can be ignored so Γ^2 ≃ 0 and the curvature is R^a_{bcd} ≃ ∂_{[c}Γ^a_{bd]}. I hope that is clear as a commutation of the c and d index, and we might think of this as similar to the curl so R^a_{bcd} ≃ {{∇×Γ}_{cd}}^a_b. So, this is not that different in ways from classical electromagnetism. It is also easily quantized as a linear theory.

      However, with this two-slit experiment something odd happens if we try to measure which slit the particle went through. The quantum state of the particle is entangled with other states and the particle is “collapsed” into being identified with one of the slits. This means our gravitational field associated with the particle is violently adjusted. With this particle the gravitational field is really a superposition of Newtonian gravity fields. However, with he collapse this is lost and the gravitational field is just one Newtonian gravity field. What happens it a gravitating mass is made to disappear? There is a gravitational wave, and there are plenty of videos illustrating how we on Earth would not know about such a disappearance of the sun should it by some means disappear. From a quantum gravity perspective, the same should happen; there would be the emission of a graviton from this system.

      Penrose in his Road to Reality writes about this and on how this might actually be measured. I am surprised there has not been more effort to look at this type of experiment.

      This is a bit different from the usual approach with reservoir of environmental states. It is physically possible that such a graviton is produced in a partial entanglement, though that may not be mandatory. The needle state of the apparatus takes up more of this entanglement. This needle state could be a spin state at one of the slits, where it is in a superposition of spin up and down initially and if the particle enters the slit it goes to spin up and if the particle does not traverse the slit it stays in this superposition. This configuration has some connection with the Bell argument on violation of inequalities. I think Penrose was on to something with this, where the quantum phase could potentially be carried away by a graviton in an “entanglement on the large,” so the graviton has a bit of mass-energy. Penrose suggested a sort of avalanche process by reflecting a quantum state many times; see Road to Reality.

      Delete
    17. @PhysicistDave

      Hello, Dave,

      the differential calculus is part of the problem of understanding micro-objects.

      About 2000 years ago, Ptolemy placed the earth at the center of the universe.
      It was an assumption. Maybe he made it for convenience because we
      observe stars and planets from the earth.
      It was an assumption. But in time, it was forgotten. And after a few years...
      or centuries, everyone believed that the Earth was really at the center of the world.

      Snip

      Around 1685 Newton made the assumption of a continuum. He needed this to describe the orbits
      of the planets. It was a good assumption, since no one has ever seen a planet jump.
      But it was an assumption.

      And today? All the physicists who continue to use this assumption in the sub-micron range.
      How do they actually justify it?

      Continuum is an assumption and in the sub-micrometer range it is a bad one.
      And always leads to new extra rules.

      Whoever may hold to these, let them do so.
      I have no pain with this.
      But it leads nowhere.
      It does not lead to good understanding.

      There are no waves in the double-slit experiment.
      And there are no particles.

      Have fun.
      Stefan

      Delete
    18. Cool,

      -- there is no spoon.


      Thanks,

      Delete
  11. I have to disagree a bit. Sure, the most basic applications of QM, like spin and Stern-Gerlach sequential experiments, are a good start and indeed basic finite dimensional inner product vector spaces are enough. But as soon as you want to introduce anything else, like position and momentum, you will need infinite dimensional Hilbert spaces, i.e., topology (masked by the norm, since it's norm induced.) And things can get quite subtle there. For example, what happens if you try to realize the basic CCR [q,p]=iI in a finite dimensional space? If you take the trace, you get 0=1, i.e., you can't realize them there. Thus, you will need now the representation theory of those relations on infinite dimensional Hilbert spaces and the main result about this which is the Stone-von Neumann theorem. All this is obscured by the bracket notation, which I never recommend.

    Furthermore, it's true that one of the main difficulties of QM is in the physical interpretation of the formalism. But you will never get very far in solving that by using basic math. The best thing is to formulate QM in the most mathematically clear and advanced way, so that the mathematical part of the formalism is exhausted. It surprises me sometimes that many physicists pretending to talk about foundations are not familiar with stuff like Gleason's theorem, the Kochen-Specker theorem, and so on. Only elementary versions of the Bell inequalities are taught to students of undergraduate (and even graduate!) QM. To be honest, I find the Kochen-Specker theorem to be much more important in relation to its consequences for foundational discussions.

    I think that a bit more of math than just basic linear algebra won't hurt anybody :-)

    ReplyDelete
    Replies
    1. Thanks aleazk,
      (hope I got it right)
      I agree, I, actually, don't see more - or less -
      math hurting anyone.
      and that's a good thing.
      - nice comment.
      I'm also hoping to reply to
      'Physics Dave'.
      - when it comes to
      ancient teachings,
      - there's supposed to be some ' seven deadly sins'.
      floating around somewhere.
      if so, (when it comes to scientists or the scientific community)
      I would rank, right around the top, arrogance.
      Thanks Dave, for reminding us.
      Best,

      Delete
  12. The problem with Sabine's suggestion is that its hard to motivate the actual numerical results for most of "the usual suspect" exactly solvable problems, even the hydrogen atom, without using calculus (usually differential equations, but you can avoid that with path integrals).

    Its the same reason a big push in quantum field theory in the 60s was fruitless. I refer to the attempt to actually explain things using data like Regge trajectories and S-matrix theory, along with unitarity. You have to know the answer first.

    Sabine is correct that you can explain things like commutation and non-commutation using matrices, can calculate with them, and that's the essence of QM. But realistically you want people to understand how it relates to the usual "toy" systems (even if like the hydrogen atom they are not really toys) with pictures in ordinary 3D space, like hydrogen orbitals.

    People not already **extrmely** comfortable with abstract spaces will find her suggestion confusing.

    ReplyDelete
    Replies
    1. dtvmcdonald,

      I guess you are expressing the same thoughts as mine. When I got to university, matrices were something I new about vaguely, but they also didn't really connect with school physics (things may be different now) and I only gradually learned how useful they could be for handling rotations etc. Matrices may seem obvious in hind sight, but while they reside in the head as just a set of arbitrary rules to manipulate rectangular arrays of numbers, they just produce a headache! They need to be rooted by examples from the physics and maths that people know.

      Maybe Sabine's approach would work well for super bright students (like I am sure Sabine was!), who have already read beyond the current level, but even I ended up with a first class degree from a good university.

      I always felt that our university physics course could have been better taught by introducing abstraction very gradually.

      As another example of a conceptual jump, consider integrals. At school these were invariably integrals of one variable, and they could be evaluated to an answer.

      The jump to line, volume, and surface integrals was taken for granted, but I spent a lot of effort over the Christmas break gradually picking over my notes to make sense of them! I remember suddenly realising that it was mostly not necessary nor possible to evaluate the damn things!

      Delete
    2. If you have ever taught physics you know the most elementary physics is hard to teach. Even F = ma stuff and solving problems such as a block sliding down an inclined plane is confusing to many students. It is really hard to keep a majority of students on board tracking things right.

      A popularization of physics, or a blog discussion of some aspect of physics, will require additional reading or study. For those not familiar with quantum mechanics, they will be far from any mastery of the subject after reading and listening to this blog. However, it can serve as a springboard to further reading and study.

      Delete
    3. I'd recommend Susskind's Stanford lectures in physics available on YouTube. QM is one subject he covers well. It is geared for those who want more mathematical substance than popular articles and books typically provide. More math and less metaphor. A nice middle ground. Also, his book "The Theoretical Minimum" attempts the same.

      Delete
    4. Susskind's online free courses were very helpful to me as a latecomer amateur. Specially designed, Susskind said with black humour, for people who do not have much time [left]. Each course is of about ten 2-hour lectures. I have followed all the courses except one. I already had a degree in Maths and Stats but the lecture format was just what I needed.

      I have had MTW since I was about 25 years old but never had time to read it as my job did not involve physics. When I retired I also bought Feynman vol 3. But to start learning QM I bought 'QM for Dummies' ... and it was very slow going. I only really followed that book after following the online lectures. Likewise I would not start to tackle string theory direct from the book 'string theory for dummies'.

      Despite bad press here about string theory and SUSY, I really enjoyed both of those courses, and also the Standard Model course.

      Before I took the QM lectures I followed the Bell's/ Entanglement lectures. There is a web page available showing links to the whole set of courses.


      Referring to Lawrence's comments on teaching F = ma, I have been using this in the last month or two in connection with negative masses. That formula is set in the context of a pervasive general environment of positive masses and that is taken for granted in the formula. But a negative mass will be attracted to a positive mass (in Newton's Laws) so the acceleration given by ma will be identical for positive and negative masses. That is, F = |m|a. If the pervasive background were to be negative mass (obviously not in our area of the universe) then negative mass would be repelled from negative mass and so the formula would be F = -|m|a. This IMO ought to be the applicable formula in a region purely of dark energy in Jamie Farnes's model.

      But if you use F ~ q1 q2 for the Coulomb force, then you should use F = |m|a for the acceleration due to the coloumb force which is fine for positive masses. And if one were to find the Coulomb force acceleration in a negative mass / dark energy region then IMO one would need F ~ q1 q2 followed by F = - |m|a. But in the same region, if you use F ~ mM and F = ma in sequence IMO one would not use the modulus as the acceleration is between the two masses rather than based on background pervasive mass. Maybe.

      Austin Fearnley

      Delete
    5. Negative mass is a bit odd. With gravity this does result in positive mass repelled from negative mass, but negative mass attracted to positive. So, an + mass and - mass of equal magnitude will form an mutually accelerated pair. However, this is a net nothing and so nothing is accelerating away, which means nothing is really occurring.

      Negative mass, or negative energy, does happen with the Klein-Gordon equation with potential V = ½m^2φ^2 for a tachyon. The tachyon has imaginary mass, which means this potential energy is negative. Physics of this sort is strange and it has some relevance, though we may never directly measure anything of this sort. Negative energy leads to things such as closed timelike curves and other strange things. The anti-de Sitter spacetime has a cosmological constant Λ < 0 and it contains superstrings where the bosonic string has tachyons. Yet the AdS is not spacetime we observe. Hence, it may be relevant only as a sort of gadget.

      Delete
    6. Thanks very much. I am writing a paper and I reckon you may have just pushed back the completion date...

      I wrote a paper two years ago (so did J. Farnes, but professionally in his case) on negative mass as the cause of dark matter and dark energy. I recently wrote a paper on time reversal for antiparticles as by-passing Bells inequalities. Now I want to put them together in one paper about the nature of properties of antiparticles.

      In a recent comment near here I suggested to Terry that 'his' Glashow Cube model of antiparticles which has antiparticles occupying ijk=-1 space in quaternion maths, could maybe be expressed easier using geometric algebra which might use a negative trivector for antiparticle space.

      Your information makes me wonder if antiparticle space is similar to AdS space while particle space is similar to dS space.

      If the universe is treated as a particle, then the universe has its own time dimension and direction. IMO particles have their own time dimensions, too. If antiparticles have negative mass then they repel each other gravitationally and cannot form large gravitationally-bound clumps. Hence the universe's thermodynamic time has to correspond to particle time rather than antiparticle time. The runaway accelerated pair is unlikely to happen when the negative mass is microscopic but the positive mass is macroscopic. Instead the runaway effect leads, nicely IMO, to dark matter. And, even when both types of mass are microscopic a computer simulation still showed dark matter and dark energy effects.

      So in our dS universe we can have a dS particle and an AdS antiparticle. I suspect that in an AdS universe an AdS antiparticle would behave exactly like matter and a dS particle would behave like an antiparticle. So we cannot know if we are matter in a matter-like universe or are we antimatter in an antimatter-like universe.

      That means that the negative masses in our universe would eventually stop repelling and later start attracting once they reached (if possible) the AdS universe. However, I do not believe that is possible. Maybe a firewall is in the way. Can one travel smoothly from a dS universe to an AdS universe as one moves slowly through an S^3 double cover space. The dS universe and the AdS universe would have macroscopic time dimensions pointing in opposite directions, I guess. Maybe they would both have to suffer independent end-of-universe events before meeting.

      Austin Fearnley

      Delete
    7. The relationship between de Sitter and anti-de Sitter spacetime is respectively that between a single sheet hyperboloid outside a light cone and the two hyperboloids inside the two cones.

      There are for the dS three ways to partition this into space plus time. The first is a spatial surface at the throat or minimal radius portion of the hyperboloid, The dS spacetime in this situation is a closed sphere tensored with ℝ^± for time measured by the positive and negative reals. Some consider this to be a biverse, and even Hawking wrote a bit on this. Another partition is along a null pair of lines. Consider a null plane that contains the origin of the light cone, and also cuts the hyperboloid equally at a line that is parallel to the light cone. This has topology ℝ^{n-1}×ℝ for the spatial and temporal surfaces. The spatial surfaces for t → ±∞ converse to these null asymptotes. This partition again defines what might be called a biverse. The foliation of spatial surfaces wraps around the hyperboid and these are flat spatial surfaces. Then finally a plane that is timelike can partition as well. The spatial surfaces are hyperboloids. This geometry is a bit more complicated, and where the meaning of time and space are flipped, similar to what occurs with a black hole interior. This connects with the FLRW k = 0 for the null case and k = ±1 for the spherical and hyperbolic spatial situations.

      This biverse is probably not the observable universe. The observable universe is probably best described by FLRW with a minimal radius at a Planck scale or with an inflationary Λ ≃ 10^{70}m^{-2}. This de Sitter biverse is maybe more closely matched with the inflationary spacetime, where vacuum transitions generate FLRW cosmologies. This dS spacetime has a huge vacuum corresponding to Λ ≃ 10^{70}m^{-2}. This has a cosmological horizon with a Planck length in radius.

      The anti-de Sitter spacetimes are in a pair, they may in a sense be entangled pairs in the same sense the biverse may be a sort of entanglement. This has Λ ≤ 0 and may in face be on the scale of -1/ℓ_p^2. This spacetime has topology S×ℝ^{n-1}, where the circle is time. This runs into some trouble with the idea of antiparticle-like spacetimes. Yet, what is usually worked with is a conformal patch with timelike boundaries that avoids CTCs. It is tempting to see the AdS and dS as different quantum states of the vacuum, one positive and the other negative, The dS vacuum is not eternally stable, and maybe it is absorbed into an AdS. Velinkin also showed the inflationary spacetime is not past eternal, which also might suggest something about the biverse construction, where the null boundary the spatial surface converges to.

      Dark matter most likely not with negative mass. This is, as I remember, what Farnes proposed, not possible. I recall not long afterwards numerical simulations demonstrated a configuration of galactic halos with negative mass are not stable. Sorry for lack of references; I would need to look that up and time is too limited.

      Delete
    8. Daid Bailey,
      I personally find it hard to read about someone breaking their head over such things .
      Really,
      My friend,

      Truly, Any Matrix...
      Is ad hoc, meaning for this purpose only ,or primarily. (Useful at this moment).

      I've seen, and examined matrices from thousands of years ago.

      Ultimately,
      - it comes down to this..

      We wake up in the morning.
      ... Work our way into a bathroom,(somewhere)
      splash some cold water in our face and look in the mirror.

      That's pretty much it,
      - don't break your head over any of this shit.

      -. talk to you
      later.

      Best,

      Delete
    9. Lawrence

      Once more, thank you. Your extended comments are, and will be, very useful and are much appreciated. I followed some of it and can try to follow the rest later. It has given me more ideas, and maybe antiparticles have the geometry of the other part of a dS biverse and not hat of an AdS space, but maybe for another thread ...

      On the theme of learning QM. I followed Susskind's Entanglement course first as that was my interest at that time. I am interested in AdS for the moment but there is no very quick course AFAIK. I did follow Susskind's GR and Cosmology courses too. I was surprised how easy Cosmology was explained (but not so easy to keep in my head afterwards), maybe the easiest of all his courses. And entanglement keeps occurring even in w.r.t. spacetime metrics.

      Dark matter and dark energy with negative mass: I had heard of a negative comment on Farnes paper but did not think it was conclusive. My own 1D simulation worked fine IMO. I have since been looking at effects of negative mass combined with electric charge.

      Austin Fearnley

      Delete
  13. Hi Sabine,. !!!
    Hope your day is good.
    -and all those you know.
    Please don't publish this, there's really no point.
    I was trying to access your blog, and what came up was something known as
    Quantum Dream Inc.
    In a very quick peruse,
    I found it quite interesting.
    Perhaps you might as well.

    Best wishes,
    Love Your Work

    ReplyDelete
  14. May you explain the nature of Hilbert space?
    If particle has coordinates (2,3,0) in that space, what it is means in the real space?

    ReplyDelete
    Replies
    1. A priori: Nothing. Depends on what the basis describes. Could be spin or momentum or position eigenstates. Anything, really.

      Delete
    2. Alrighty,then.
      First, and foremost.
      Hi Sabine !!! I hope your day is going well.

      Secondly, Alex.

      Your question is so vague that it is very hard to understand.
      It seems like two, first what is the nature of a Hilbert space.?
      Secondly, a particle (wierd). coordinates of 2, 3, 0.
      You must be more specific.
      1) what particle are you talking about? lol.
      The nature of the Hilbert space, is taking the euclidean two dimensional and progressing to the three-dimensional (Hilbert space.)
      Fantastic mathematics, really.
      But, it's just mathematics. that exist in the complex plane.

      Now, while crossing the line between the complex plane and the real, or (Reale')
      Is not easy, ... It has been done before,
      Now, these coordinates 2, 3, 0 for particle in a Hilbert space ?
      (I've seen more than one configuration of Hilbert space).
      Without a larger orientation, they mean nothing to me. (It could be Barach,)
      Apologies, I'm tired.
      In the future the more specific you can be helps me understand more.
      Best

      Sabine,
      Love your work





      Delete
  15. Quantum physics is pure dual information statistics. Measurement is one a physical action and the only known initial condition. Summa summarum: many measurements reveal and create statistics at the same time. Partially interactions become as data of research - human measurements. But we can consider interactions being self-measurements of the universe and intrinsically the system is complete and (super)deterministic or self-organized and conscious.

    Some false statements? How justified?

    ReplyDelete
  16. The real talent that a physicist needs is the ability to understand what the math is saying; it is the ability to draw the results of these equations into a correct model of reality. Oftentimes, even the best in the physics trade have failed to interpret their math into a correct model of reality.

    As a countering example, Julian Seymour Schwinger spent many months improving his Nobel Prize winning math and theory of electrodynamics that complied with experimental results to 11 decimal places.

    ReplyDelete
  17. Eusa,
    Please be more specific.

    - and, as a friend,
    put the pipe down.
    and let it cool off a while.

    conversations later. perhaps.

    Best wishes,

    ReplyDelete
    Replies
    1. Your advice is internally contradictory. ;)

      Being specific demands intensive research.

      Delete
    2. Eusa, my friend,
      Being specific, actually,
      doesn't 'demand' anything.
      Being specific does, however, require that you use the simplest and most direct terms (technical or not) - and that your rant follow a coherent
      train of thought (logic, even...)

      Better luck
      next time.

      Delete
    3. Anyhow. I prefer being specific in research papers to arguing in blog comments.

      Delete
    4. My friend,
      Show me your papers.

      Delete
    5. Can not. In that branch I still have inconsistencies to get rid of.

      Delete
    6. Cool, man.

      Whenever you do, I'll be around. (some where )

      - it's always nice to talk to someone who thinks outside the box.
      Best wishes

      Delete
  18. One interesting perspective which I discovered a few years ago is Koopman-von Neumann mechanics. Basically it's an attempt to rewrite classical mechanics using the formalism of quantum mechanics. They even use the same formalism - bra's, kets & linear operators. Starting from the averaged Newton laws they showed is that if you assume that momentum and position operators commute you obtain classical statisical mechanics. And if you assume that they don't commute, then you obtain quantum mechanics!

    More recent work sheds light on the measurement problem. Basically, non-selective measurements don't disturb the classical motion but they do for quantum motion. They narrow the problem down to the dynamics of phase. It's well known that there is a phase in the quantum theory. But in KvN mechanics there is also a classical phase. However it's decoupled from the classical motion.

    ReplyDelete
    Replies
    1. My Friend,

      Please don't take this the wrong way, but
      ... nice try .

      Delete
  19. I came across some really nice 3D animations of many aspects of Quantum Mechanics on Youtube. These were created by Eugene Khuteryansky, and mostly seem to be about 4 or 5 years old. They seem like a really good complement to the verbal descriptions of QM phenomena.

    ReplyDelete
    Replies
    1. Thanks. I saw a couple of these videos, and they seem to be really good.

      Also, it was only today that I came across the "QuVis" simulations developed at U. of St. Andrews. The simulations here are interactive, and though the UI is relatively simple, their underlying design seems to have been done thoughtfully. The modules are on a slightly more advanced / theoretical side as compared to the "PhET" simulations from U. Colorado at Boulder.

      --Ajit

      Delete
  20. For what it is worth, I would like to "throw my hat into the ring" and make a few suggestions regarding useful references. A compact, introductory, textbook 'Quantum Mechanics in Simple Matrix Form' (Jordan, 1985) utilizes only 2X2 matrices (Preface: "assumes only basic algebra"). Jordan also wrote a more advanced text 'Linear Operators For Quantum Mechanics' (1969, Preface: "entirely conventional view--von Neumann"). From an historical perspective, it is interesting to note, Dirac utilized his Bra-C-Ket notation beginning in the third edition (1947) of his celebrated 'Principles of Quantum Mechanics'. Notice, too, Dirac changed the meaning of the word 'state' beginning in his second edition (1935), as the first edition (1930) utilized only relativistic notion of 'state.'
    Continue reading with Asher Peres: "What is a State Vector ?"
    (1984, American Journal Physics, Volume 52).

    ReplyDelete
  21. Thank you, Sabine! Your explanantion of <|> really helps!

    ReplyDelete
  22. But where do these rules come from?

    ReplyDelete

PLEASE READ THE COMMENT RULES BEFORE COMMENTING.

Comment moderation on this blog is turned on.
Submitted comments will only appear after manual approval, which can take up to 24 hours.
Comments posted as "Unknown" go straight to junk. You may have to click on the orange-white blogger icon next to your name to change to a different account.