Quantum Matter Animated!

by Jorge Cham

What does it mean for something to be Quantum? I have to confess, I don’t know. My Ph.D was in Robotics and Kinematics, so my neurons are deeply trained to think in terms of classical dynamics. I asked my siblings (two engineers and one architect) what comes to mind for them when they hear the word Quantum, what they remember from college physics, and here is what they said:

– “Quantum Leap!” (the late 80’s TV show)

– “Quantum of Solace!” (the James Bond movie which, incidentally, was filmed in my home country of Panama, even though the movie was set in Bolivia)

– “I don’t remember anything I learned in college”

– “Light acting as a particle instead of a wave?”

The third answer came from my sister, who went to MIT. The fourth came from my brother, who went to Stanford (+1 point for Stanford!).

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I also asked my spouse what comes to mind for her. She said, “Quantum Computing: it’s the next big advance in computers. Transistors the size of atoms.” Clearly, I married someone smarter than me (she also went to Stanford). When I asked if she knew how they worked, she said, “I don’t know how it works.” She also said, “Quantum is related to how time moves more slowly as you approach the speed of light, right?” Nice try, but that’s Relativity (-1 point for Stanford!).

I think the word Quantum has a special power in our collective consciousness. It’s used to convey science-iness, technology, the weirdness of the Physical world. If you Google “Quantum”, most of the top hits are for technology companies that have nothing to do with Quantum Physics (including Quantum Fishing Tackles. I suppose that half the time, you pull up a dead fish).

It’s one of those words that a lot of people have heard of, but very few really understand what it means. Which is why I was excited when Spiros Michalakis and IQIM approached me to produce a series of animations that explore and explain Quantum Information and Matter. Like my previous videos (The Higgs Boson, Dark Matter, Exoplanets), I’d have the chance to interview experts in this field and use their expertise and their voices to learn and to help others learn what amazing things lie just around the corner, beyond our classical understanding of the Universe.

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This will be a big Leap for me (I’m trying to avoid the obvious pun), and a journey of exploration. The first installment goes live today, and you can watch it below. Like Schrödinger’s box, I don’t know what we’ll discover with these videos, but I know there are exciting possibilities inside. This is also going to be a BIG challenge. Understanding and putting Quantum concepts in visual form will be hard. I mean, Hair-pulling hard. Fortunately, I’ve discovered there’s a remedy for that.

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Watch the first installment of this series:

Jorge Cham is the creator of Piled Higher and Deeper (www.phdcomics.com).

CREDITS:

Featuring: Amir Safavi-Naeini and Oskar Painter http://copilot.caltech.edu/

Produced in Partnership with the Institute for Quantum Information and Matter (http://iqim.caltech.edu) at Caltech with funding provided by the National Science Foundation.

Transcription: Noel Dilworth
Thanks to: Spiros Michalakis, John Preskill and Bert Painter

65 thoughts on “Quantum Matter Animated!

  1. Is it still not possible that the laser gave some part of it’s energy to the mirror? Is it possible to detect such small instantaneous rise in temperature (which will be dispersed to the surroundings within fraction of a second as it is maintained at 0K) ?? Because if it is not completely possible to measure such small changes in temperature in such small time then how can we be sure that the red shifted laser is NOT due to the laser giving off it’s energy ?? And if this is the reason then this still does not prove that the mirror was vibrating. It started vibrating only after being hit by the laser. But due to temperature dispersal the mirror was instantly damped and brought again to zero vibrations or ground state.

    • I am not a physicist, but the intuitive answer to your question is that if the laser were imparting energy to the mirror, and that was where the red shift was coming from, then there would still be a corresponding blue shift.

      • Right. When the oscillator is in its quantum ground state, it can absorb energy but cannot emit energy because it is already in its lowest possible energy state. Reflected light can be shifted toward the red (have lower energy than the incident light, because the oscillator absorbed some of the incident energy), but cannot be shifted toward the blue (have higher energy than the incident light). That’s what the experiment found.

        • Just a follow-on to John’s response…

          The inability of the mechanical resonator to give up energy when it is in its lowest energy state seems like an obvious statement (by definition of “lowest energy state”), and so why is the experiment interesting then? All it did was confirm that indeed this energy emission goes away as the object gets colder and colder and approaches its ground (lowest energy) state.

          It is really the fact that the mechanical resonator can absorb energy when it is in the ground state that is interesting. The classical description of the motion of a mechanical object has no way of allowing for this asymmetry in the emission and absorption of energy with the environment; the processes must be symmetric and zero when the object is not moving at temperature=0K. Think of it from the stand-point that the mechanical object isn’t moving when in its classical ground state, and thus it is not doing work on its environment and the environment is not doing work on it.

          That is what makes the quantum description of the ground state of motion interesting; it allows for the asymmetry in the process of emission and absorption of energy by the mechanical resonator to (or from) the environment. I like to make the analogy to the spontaneous emission of light from an atom, in which there is no corresponding spontaneous absorption process of light. A well defined “mode” (think of it as a particular direction and polarization) of light can be described by a similar set of quantum equations as that describing the mechanical resonator, and thus also has a ground state with intrinsic fluctuations. These “zero-point fluctuations” or “vacuum fluctuations” can be thought of as triggers for atomic spontaneous decay and emission of light by the atom, but do not cause the reverse process of spontaneous excitation of the atom.

          [Aside: This used to really mystify me when I first learned about spontaneous (and the related stimulated) emission of atoms. The excellent little book by Allen and Eberly,

          http://books.google.de/books/about/Optical_Resonance_and_Two_Level_Atoms.html?id=1q0ae-XNmWwC&redir_esc=y,

          does a nice job of de-mystifying the vacuum fluctuations.]

          A nice description of the above argument is also given in Aash Clerk’s Physics Viewpoint accompanying article:

          http://physics.aps.org/articles/v5/8

          OJP.

          • Hi Oskar, John, and Paras:

            0. For some odd reason, while fast browsing, I first read Oskar’s reply, and then John’s, and only after both, Paras’ original question. (Oskar’s was the longest and innermost indented reply, and so it sort of first caught the eye in the initial rapid browsing.) Even before going through your respective replies, I had happened to think of what in many ways is the same point as Paras has tried to point out above. … Ok. Let me put the query the way I thought of.

            1. Here is a simple model of the above experimental arrangement, simplified very highly, just for the sake of argument.

            The system here consists of the mechanical oscillator and the light field.

            The environment consists of the light source, the optical measurements devices, the cooling devices, and then, you know, the lab, the earth, all galaxies in the universe, the dark matter, the dark energy … you get the idea. The environment also includes the mechanical support of the oscillator, which in turn, is connected to the lab, the earth, etc.

            *Only* the system is cooled to 0 K. [Absolutely! 😉 Absolutely, only the system is cooled “to” “0” K!!]

            The measurement consists of only one effect produced by the light-to-the mechanical oscillator interaction: the changes effected to the reflected light.

            This effect, it is experimentally found, indeed is in accordance with the QM predictions. (BTW, in fact, the experiment is much more wonderful than that: it should be valuable in studying the classical-to-QM transitions as well. But that’s just an aside as far this discussion goes.)

            2. Now my question is this: what state of |ignorance> + |stupidity> + |insanity> + |sinfulness> [+ etc…] do I enter into, if I forward the following argument:

            At “0” K, the system gets into such a quantum mode that as far as the *reflection* of the light is concerned, if “I” is the amount of the incident light energy (say per unit time), then only some part of it (i.e. the red-shifted part of it) is found to be reflected.

            However, there still remains an “I – R” amount of energy that the system gives back to the environment via some *experimentally* unmeasured means. If it doesn’t, the first law of thermodynamics would get violated.

            We may wonder, what could be the form taken by such an energy leakage? Given the bare simplicity of the above abstract description as to what the system and environment here respectively consist of, the answer has to be: via some mechanical oscillation modes of the mechanical oscillator that we do not take account of (let alone measure) in this experiment. The leakage would affect the mechanical support of the oscillator, which, please note, lies *outside* of the system.

            [The oscillation modes inside the system may be taken to be quantum-mechanical ones; outside of it, as classical ones. But I won’t enter into debates of where the boundary between the quantum and the classical is to be drawn, etc. As far as this experiment—and this argument—goes, we know that “inside” the system, it has to be treated QMechanically; outside, it’s classical mechanically; and that’s enough, as far as I am concerned!]

            Since the system here is not a passive device but is *actively* being kept cooled down “to” “0” K, it means: it’s the “freeze” sitting in the environment, not to mention the earth and the rest of the universe, which absorb these leaked out vibrations of the mechanical oscillator. The missing energy corresponds to *this* leakage.

            3. Of course, I recognize that my point is subtly different from Paras’. His write-up seems to suggest as if there is an otherwise classically rigid-body oscillator sitting standstill, which begins to vibrate only after being hit by the laser. In contrast, I don’t have this description in mind. He also seems to think rather in terms of a *transient* damping out of the mechanical oscillations. Though I do not rule out transients in the system, that wouldn’t be the simplest model one might suggest here: I would rather think of the situation as if there were a more or less “steady-state” leakage of the missing energy into the environment. Yet, Paras does seem to appreciate the role of the environment—the unmeasured side-effects, so to speak, that the system produces on the environment.

            4. Anyway, I would appreciate it if you could kindly let me know in what final state should I collapse: |ignorance> or |stupidity> or … . And, why 🙂

            [BTW, by formal training, I am just an engineer. And, sorry if my reply is too verbose and had too many diversions…]

            Thanks in advance.

            –Ajit
            [E&OE]

            • About your parts (2) and (3), I think it is easier to think of it this way:

              At the low temperatures the system is subjected to (I really don’t think it even makes sense to say that “only the system is cooled down to 0K”; instead, just say that the system is cooled down to low temperatures is enough), a lot of the system’s constituent particles are in their ground states.

              What is happening in this experiment, is that they are observing that absorption and excitation of constituent particles up from ground states is observed without the corresponding “classical” de-exciting reflection wave that you normally get. This is predicted from the quantum physics.

              The special thing about this experiment, though, is that they are also saying that the entire system itself, a macroscopic body, has a quantum wavefunction just like their microscopic parts. That is the part that is interesting and worth reporting upon. Because, if a macroscopic body has a quantum wavefunction, then it can also do all the rest of the quantum weirdness, and that applies to us humans, the Earth, being able to, say, perform quantum tunnelling.

              Once you see the experiment in this way, it is then obvious that the loss of energy that you perceive, is merely the spontaneous emission of light by the excited particles, and, in this way, they drop back into the ground state of the entire system. This is important, because spontaneous emission is basically undetectable in our case, which is what the experiment observed. The point is that, classically, you are supposed to observe substantial energetic reflection (along with the spontaneous emission that you cannot remove), and you do not observe that in this experiment.

  2. Could you add a link to the paper about the experiment for those readers who want more details about it?

  3. Whenever someone asks me for a book to explain quantum mechanics to laymen, I always point them to this:

    http://www.amazon.com/What-Quantum-Mechanics-Physics-Adventure/dp/0964350440/ref=sr_1_1?ie=UTF8&qid=1370957947&sr=8-1&keywords=What+is+quantum+mechanics

    It’s an illustrated book about the history of quantum mechanics created by Japanese translation students studying english. They chose the topic because they needed to be able to accurately translate relatively technical material. It’s wonderful for answering the questions you raise in the post above.

  4. Great video describing a really interesting experiment. However, it is far from reaching the important lessons from Quantum Mechanics that have shaped the way we see the universe.

    Forget Quantum Computing. I am not saying that Quantum Computing is not sexy or something, but it is not where the paradigm shift is. One of the greatest thing that a Physics undergraduate degree forces you through, is to learn about Condensed Matter Physics.

    You might think that, in contemporary Physics education, they would certainly teach you both Quantum Mechanics, and General Relativity. After all, they are what we call the new world view, that revolutionised how we as a species see ourselves.

    The truth, however, is that, if I did not force them to teach me, they would have ignored General Relativity and just taught me Quantum Mechanics. Lots of it. Without motivation. It is only at the end of the Physics degree do you get to see why it is arranged in the way it is.

    Special Relativity, the one that Einstein published in 1905, is a really easy thing. Yes, it is bizarre, but you can easily teach it, and later on, you can tell students to apply what they have learnt. That it is reducible into small equations that are easy to memorise, is another plus point. General Relativity, on the other hand, is a pain to teach — everybody, mathematician or physicist, would be confused by the initial arguments, the mathematical notation and all that jazz, until you have completed the module. And even after that, some people just never get it (although, luckily, it is simple enough that a large chunk of people actually understand it very fully, to the contrary of Eddington’s bad joke).

    The deal breaker, however, is that the ideas from General Relativity, although a nice help to the other parts of Physics, is very far away from essential. i.e. People can make do without any knowledge of that, and still contribute to the rest of Physics in a proper way. That is not the same with Quantum Mechanics.

    The standard way they teach Quantum Mechanics these days, is to throw the mathematics at you, right at the start. Just write down your energy equation (that you can remember from high school), do your canonical quantisation (which is nothing other than replacing symbols you know about with derivative signs; a monkey can do that), and tack on something magical that we call the wavefunction, and Viola, Quantum Goodness!

    Since there is nothing to actually understand about it, I watched in amusement as everybody around me struggled to understand something out of nothing, congratulating myself for actually knowing the meaninglessness of it all.

    Boy, what do I know?

    The next module, aptly named “Atomic and Molecular Physics”, looked like nothing but applications of the mathematics learnt. It was HORRIBLE to go through, especially since it looked like vocational training — approximation and other calculational techniques that are hardly useful outside higher and higher corrections to the properties of materials that classical physics could have found out about (except quantisation, of course). It was important to have learnt it (not least because it was the first place in which Quantum Entanglement was taught), but it felt like we are just learning tricks instead of ideas.

    Statistical Thermodynamics was better. Building upon Thermal Physics in first year, there was a bit of Quantum effects being shown in action, especially the Quantum Degeneracy pressure that keeps stars the size they are.

    Then BOOM! Condensed Matter Physics (I learnt it under the older name, Solid State Physics). I had to completely rewrite what I thought I had known about Quantum Theory, for it is obvious I knew nothing.

    I am sure you guys have heard of the adage: “When stuff are moving fast, are large, or heavy, General Relativity cannot be neglected. When things are small, Quantum corrections cannot be neglected.” It is still true, but there is a sleight of hand here — we have yet to define what it means to be “large” or “small”.

    In particular, whenever you have a lot of material squeezed into a small space, i.e. high density, it is small. Thus, something can be both large and small at the same time, requiring both General Relativity and Quantum Theory to describe. A black hole is one such object.

    The name “Condensed Matter”, is a really good one. Any liquid or solid, really, is condensed, so condensed, that actually, it is no longer a classical system — the quantum effects DOMINATE. Without incorporating Quantum Theory right into the heart of it all, nothing you calculate even makes sense. And since our first approximations here beat the best classical calculations left-right-centre, there was also no reason to teach the classical approximation techniques either.

    Specifically, notice how, in high school, people teach you that heat and sound are just atoms moving about in different ways? Classical theories can talk about heat propagation and sound propagation and motion. But they are three different islands that don’t even make sense together. So different, that even their mathematical tools are different. But in Quantum Theory, the same mathematics describe all three as one united whole, on the zeroth approximation, and even give you dispersion, which is something classical theories cannot explain without complicated methods.

    After being floored by how it actually is done, the icing on the cake is Transistors. The theory was originally made in order to explain how metals behave, and we talk about a free electron gas, to explain how metals conduct so well. So, it came as a complete shock that any improvement, notice, ANY SIMPLE improvement to the free electron picture, be it Nearly Free Electron model, or the Tight Binding model, energy bands appear. In practical terms, the theory that sought to explain metals, now explains insulators, and even more, predicts the existence of this previously unheard of class of materials, known as semiconductors.

    Indeed, it does even better. It predicts the existence, how to make them, and how they would be useful. It is the first time that Physics THEORY had been faster and earlier than the experimenters at any topic.

    So, yeah, while you are enjoying your computers reading this piece, appreciate the sheer ingenuity and wonder that is brought to you by the Quantum revolution.

    Please alert Jorge to this. He can do wonders with information.

    • “Classical theories can talk about heat propagation and sound propagation and motion. But they are three different islands that don’t even make sense together. So different, that even their mathematical tools are different. But in Quantum Theory, the same mathematics describe all three as one united whole, on the zeroth approximation, and even give you dispersion, which is something classical theories cannot explain without complicated methods. ”

      Sound propagation and bulk motion can be treated the same way, because they are both forms of characteristic wave propagation and show up as eigenvalues of the same equation set. Heat transfer, viscous momentum transfer, and diffusive mass transfer all work basically the same way, because they are closely related effects of the same basic process. All of this can be derived in a unified framework using the principles of classical kinetic theory, because all of it is inherent in the Boltzmann equation.

      It’s true that you need quantized internal energy states to accurately predict something as simple as the temperature dependence of specific heat in a gas. But it seems to me that you are somewhat exaggerating the shortcomings of classical physics.

      • “All of this can be derived in a unified framework using the principles of classical kinetic theory, because all of it is inherent in the Boltzmann equation.”

        I am really doubtful of that. The reason is that the mathematical apparatus is just not the same. For the propagation of heat, you have the heat equation in classical physics, with the propagation constant kappa. For sound, the Harmonic approximation gives rise to a fixed speed of sound, which you later improve upon by adding anharmonic terms so that the speed of sound becomes a variable.

        Those two constants are not the same. Granted, they are dimensionally inconsistent, but the fact is that you have to treat them rather differently. The reason for this discrepancy is that sound propagation exhibits higher frequency dependency, so that it is easier to look at one frequency. Heat, on the other hand, is usually averaged over in the context of classical heat propagation. This makes it really complicated, as you have to average over both spacetime and weight them according to the probabilities of being in so-and-so states. Note that this last thing is also itself temperature dependent, so classical physics is crazy.

        Nothing stops a person from combining the heat and sound contributions in classical physics, but they are like Frankenstein combinations — oh, this contribution is for heat, and that for sound, and this for their interaction. That is very different from truly unified descriptions in Quantum theory, where it is one term, and one term only, that we are looking at.

        Because of that, I do not think I am exaggerating the shortcomings of classical physics. It simply is not a unified framework, although it is frequently possible to push approximations in classical physics to really high orders of accuracy. That, I can give, but not unification.

        And even then, one should notice the tremendous difference in the mathematical methods involved. Yes, both approaches would heavily depend on Fourier analysis, but that is just about their similarities. Instead, a knowledge of the approximation techniques in classical physics is only useful for the continuum free-space approximation of the transport of various quantum objects, whereas proper quantum approximation techniques is frequently simpler than the classical counterparts

        Finally, bulk motion is very different from either of sound nor heat in any case, except the fact that they are all of zero frequency (actually, this is how the normal mode mathematical technique announces its own failure, and there are ways to compensate formally). Luckily, it is seldom a problem that this is happening — after all, bulk motion would, somewhat, be better treated with relativistic methods.

        • I suspect we’re talking past one another a bit here.

          I’m a fluid dynamicist. I’ve studied some advanced solid mechanics and continuum mechanics, but mostly I’m a fluid dynamicist. When you say stuff like “bulk motion is very different from sound”, I think of the underlying physical principles, because in the derived practical equations I use this is not true. But when you say stuff like “the heat equation in classical physics, with the propagation constant kappa”, I think of the phrase “toy equation”. Even in the engineering form of the heat equation, or the Navier-Stokes equations for a linear isotropic fluid, kappa is a coefficient, not a constant (though turbulence modellers generally ignore its thermodynamic derivatives). And it doesn’t show up at all in the Boltzmann equation, unless you do the math and derive it.

          Regarding “unified framework”, I expressed that poorly. Sure, in the engineering equations, first-order fluxes like acoustic propagation and bulk motion are handled differently than second-order fluxes like heat transfer and viscous stress. This is because their behaviours are different, so the simplest reasonably accurate mathematical descriptions of them will unavoidably be different. But it should never be forgotten that they can both be derived from the same statistical mechanical representation.

          It strikes me that what the Boltzmann equation is to fluid mechanics is somewhat analogous to what Schrödinger’s equation is to quantum condensed matter physics (though it isn’t quite as fundamental). The general form isn’t very useful by itself, but specializations and approximations can produce good enough results to translate into engineering equations. The key to the Boltzmann equation (assuming you have enough dimensions to describe all important degrees of freedom) is the collision operator, which could be said to be analogous to the Hamiltonian in the Schrödinger equation. The collision operator describes all interactions between particles and is very difficult to specify exactly for real physical systems, though a number of popular approximations exist. I gather this is a bit different from the quantum-mechanical approach you’re talking about, where a lot of condensed systems can be described surprisingly well with “noninteracting” approaches…

          People have tried to use the Boltzmann equation (with or without quantum effects) to model solids, with mixed success. It seems to be best at fluids, especially gases and plasmas, perhaps because the molecular chaos assumption is difficult to remove.

          Look, I’m not claiming that quantum physics is no better than classical physics. But you seem to be saying “classical physics” when you should be saying “classical engineering approximations”, and then drawing conclusions based on the conflation of the two. Comparing the Schrödinger equation to something like the Navier-Stokes equations, never mind the heat equation, is apples-to-oranges. You can actually derive all of the basic principles of fluid mechanics from Newtonian mechanics, without even referencing electromagnetics, though your accuracy won’t be very good…

          I shouldn’t have gotten involved. I have a segfault to chase down…

          • I better see where you are coming from. You are clearly talking about deeper stuff, and good luck with your segfault.

            However, I do not think that your argument is convincing enough. Yes, it is possible to derive fluid equations and so forth from Newtonian mechanics. The problem still persists, however, that after the derivation (in which kappa turns out to be a derived quantity and actually not a constant), that the treatment of heat and sound needs to be done as stitched patchworks on top of the same fundamentals.

            As you rightly noted, I was saying that you don’t treat it that way in quantum physics, and it is quite important to see how it is actually handled differently.

            Also, the “proper” way to deal with interacting quantum systems is to couple them. For example, phonons and photons, by interacting, means that a proper treatment is to deal with waves that are half-phonon and half-photon and then quantise them yet again. This is completely different from how classical approaches tackle these problems.

            Yet again, I have to reiterate that, I am not saying that you cannot get good results from classical considerations. What I am saying, is that, due to how classical ideas actually arise from quantum fundamentals (namely, that everything classical tends to just be the conflation of modal [as in, most probable] behaviour as the _only_ [or mean, if you are talking about bulk stuff] possibility), the approximation schemes are doomed to complications for little gain. One of which, is the asymmetrical treatment of heat and sound.

            That is, even after you derive the heat and sound from the same underlying bulk motion of continuum mechanics, you still have to treat them separately, whereas quantum physics insists that they are _exactly_ the same thing, just different limits of the same _one_ term in any approximation scheme.

            It is the same thing with fluids. Very few physicists are dealing with Navier-Stokes equation itself, since it is now the preferred game of applied mathematicians. Instead of asking whether Navier-Stokes equations can have solutions for so-and-so kinds of problems, the physicists working on fluids tend to be working, instead, on the quantum corrections that should be added onto Navier-Stokes equations. After all, chaos sets in earlier than Navier-Stokes equations imply, because, near the critical points, modal behaviour is nowhere near the mean behaviour that we should have been focusing upon all this while. Sadly, this is so difficult that we have yet to do something fundamentally good about it.

            In that case, I am not saying that the corresponding classical problems are not important or not good at describing physical systems, but that the quantum world view is very different. And since the fundamental picture clearly needs to be quantum, I merely mean to say that those quantum considerations happen to be even more important than the classical problems.

            • Well, I got led on a merry chase and finally found what was causing the memory error. Turns out it was my fault all along…

              Rather than “the problem still persists” after the derivation, I would say that the problem ARISES in the derivation of transport equations from the fundamentals. The Boltzmann equation doesn’t have separate terms for heat, sound, bulk motion, viscous stress, etc., because it directly describes the molecular motion those things are emergent properties of. It’s not continuum mechanics either; it’s perfectly capable of describing rarefied gases and even free molecular flow.

              Of course quantum mechanics is a much better model than classical physics for condensed matter behaviour, and even some aspects of gas/plasma behaviour. I completely agree with you there. But I maintain that the specific criticism I was responding to, that of classical physics having an inherently fragmented picture of material mechanics, was not accurate, seemingly because of a mismatch in the fundamentality of the descriptions being compared.

            • Sorry, I don’t know why, the comment system won’t allow me to reply to yours.

              I see. That would be totally my ignorance, then.

              However, I would like to point out, to replace the original wrong argument, that the natural ideal gases that we are familiar with, are actually Fermi gases in the high temperature and low density limit. If that were not the case, we would run into what is known as the Gibb’s paradox, in which a classical gas, in the equations, somehow has a lot less pressure than expected. In particular, the ideal gas equation of pV = NkT, would miss out the N which is around 10^24. That makes no sense, until one realises that the quantum indistinguishability (which is basically quantum entanglements, really) needs to be taken into account.

              I hope that little bit, which basically states that, even for dilute gases in which we do not expect quantum effects to be important, turn out to be critically dependent upon quantum ideas nonetheless. Of course, the rest of the system does not require quantum corrections, and there is an easy fudge factor to fix that problem, but it does show how quantum theory is still a vital component of everyday life, not some esoteric correction that only people caring about precise effects can observe. (Which is the underlying point I really wanted to outline, although my choice of example turned out to be wrong.)

              Thanks. It seems, however, that it may be that the “classical atoms” view that is given by Boltzmann equation thus incorporates enough physics to reproduce the important things I was caring about. Interesting.

            • I hate to keep doing this, but… The Gibbs paradox has to do with the definition of entropy. If you don’t assume indistinguishability, you can toggle the entropy up and down by opening and closing a door between two identical reservoirs.

              You can get the correct pressure just fine with classical gas kinetics. But there are other things about gas dynamics that require quantum treatment. The temperature dependence of ideal-gas specific heat in multiatomic substances, for instance, is quite substantial and entirely due to the quantization of internal energy storage modes (at lower temperatures, there usually isn’t enough energy in a collision to excite these modes).

            • Or something like that – I had to look up Gibbs’ paradox, and I’m not completely sure my facile description above is right…

            • Nah, I know the classical gas kinetics can derive the pressure just fine. Why, indeed, I was just teaching my student that elementary derivation.

              But it does mean that both Boltzmann and Gibbs entropy cannot be derived from classical reasoning without the indistinguishability fudge factor. You would have to rely on Sackur–Tetrode entropy (removing all the quantum stuff and replacing them with an unknown constant, of course).

              It might not seem like a big issue at first glance, but it actually is. Other than the fact that entropy of mixing (that you were describing) had to be discontinuously and manually handled, it does also mean that stuff like phase changes go haywire. Again, that is useless to a fluid dynamicist until you want to deal with, say, ice-water mixtures or critical phenomena.

              Or worse, the theory is inconsistent. Judging by how seriously you take the mathematics, it is either screaming at you that you are doing something wrong, or that phenomenology needs to be used (by curve-fitting the unknown constant there, for one).

              Instead, what I wanted to impress upon you is that, instead of deriving the pressure from kinetic theory (actually, what a bad name! It is not a theory, nor does kinetic make sense as its modifier. Instead, classical atomic model would be its rightful name), it is possible to subsume the entirety of classical thermodynamics into the 2nd Law. That is, given the existence and some assumed properties of the entropy, you can construct everything you find in classical thermodynamics, even without statistical thermodynamics. That is, 0th Law and 1st Law, in particular, are theorems if you assume the 2nd Law to be your postulate. Actually, it is even a bit less — you assume parts of 2nd Law, and prove the full form of the 2nd Law with the assumptions. The issue I was referring to, is that, if you take this view, in which pressure is just a derivative of the entropy via Maxwell’s relations, and then you try to construct the statistical thermodynamics from it, you will run into Gibbs’ paradox.

              At the end, there is no need to worry about you dragging the conversation out. Actually, I was still waiting for some insights from you — you have already shown me wrong once, and there is no reason why you cannot teach me more.

  5. I particularly like the statement:
    If you Google “Quantum”, most of the top hits are for technology companies that have nothing to do with Quantum Physics.

  6. Okay, physicist, most of the things in the video are not new to me, but good presentation.
    Commented, though, to point out that the “everything is named after Quantum” is an interesting recurring phenomenon in the USA. Perhaps the largest one was the use of “Radio” in naming things. Radio was the internet on steroids, the “tech stock” of the 1920s bubble. One of the most famous meaningless uses of Radio from the time was the little red wagon called a Radio Flyer. The company just put two hot buzz words together, and created a legendary product.

  7. Pingback: The Webcomic Guide to Quantum Physics | Slackpile

  8. Dear Jorge Cham,
    I enjoyed your cute animation. Since you said you were looking for ways to think about quantum mechanics, I thought the resource list below might be interesting. Please feel free to contact me with questions.
    Best,
    David Liao
    http://qbio.lookatphysics.com/contact.php

    People:
    One of my physics professors from Harvey Mudd College (half-hour east of Caltech) wrote a wonderful book on quantum mechanics for junior physics majors:
    John Townsend, A Modern Approach to Quantum Mechanics, University Press: Sausalito, CA (2000) (http://www.amazon.com/A-Modern-Approach-Quantum-Mechanics/dp/1891389785). The academic pedigree of this book comes through Sakurai’s Modern Quantum Mechanics.

    Get a hold of Professor David Mermin at Cornell. Tell him you are working with Caltech on this animation series, and ask him to walk you through his slides on Bell’s inequalities and the Einstein-Podolsky-Rosen paradox (http://www.lassp.cornell.edu/mermin/spooky-stanford.pdf).

    If you can meet with Sterl Phinney at Caltech, talk to him. He seems to know a lot about a lot, and he’s really fun to be around.

    Fundamental concepts:
    There is a variety of ways to introduce quantum mechanics. The following two flavors can be provide particularly satisfying insight:

    Path-integral formulation — A creative child can tell a bunch of different imaginary stories to explain how a particle got from situation A to another situation B during the course of a day. A mathematician can associate with each story a complex phasor. The phasors can be added (in a vector-like head-to-tail fashion) to obtain an overall complex number for getting from A to B, whose squared magnitude is the overall probability of getting from A to B. The concept of extremized action from classical mechanics (think of light taking the path of least time) is a limiting approximation of the quantum-mechanical path-integral formulation. For this brief description, I skipped a variety of details. This perspective is attributed to Richard Feynman.

    State vector, operators — An older, more traditional description of quantum mechanics centers around the state vector (often denoted |psi>). “All that can be described” about an entity of interest is hypothetically abstracted as a vector from a vector space of all possible descriptions that can be associated with the entity. It is hypothesized that the outcomes of measurements correspond to [real] eigenvalues of [Hermitian] operators that can act on the state vector, and that when it is appropriate to describe an entity using one single eigenstate of an operator, this means that observation corresponding that operator will without doubt yield the corresponding eigenvalue as the measured result.
    Note: State |psi> is *not* wavefunction psi(x). psi(x) = , which is a *representation* of |psi> in terms of linear weighting coefficients for adding up basis states |x>.

    Risky vocabulary:
    It is important to be aware of verbal shortcuts that are used to make quantum seem more conceptually accessible in the short term that, unfortunately, also make quantum much more difficult to understand fundamentally in the long term:

    There is no motion in any energy eigenstate (ground state or otherwise). Words such as “vibration” and “zooming around” are only euphemistically associated with any *individual* energy eigenstate. As an example, the Born-Oppenheimer approximation for solving the time-independent Schrödinger equation by separating the electronic and nuclear degrees of freedom is often justified using a story that involves the phrase “the light electrons are whizzing around as the nuclei faster than the massive nuclei are slowly vibrating around their equilibrium positions.” This is shorthand for saying that the curvature term associated with the nuclear coordinates is ignored as the first term in a perturbative expansion because it is suppressed by the ratio of the nuclear mass, M, to the electron mass, m, (for details, http://www.math.vt.edu/people/hagedorn/).

    Even though the Heisenberg relationship is often described using phrases such as “not knowing how we disturbed a particle by looking at it,” a more fundamentally satisfying understanding is obtained by seeing that some operators don’t commute. Because some pairings of operators, such as position and momentum, don’t share eigenvectors, it is impossible for an entity to simultaneously be in an eigenvector for one operator, say, x position, while also being in an eigenvector for the other operator, in this example, x momentum. Having the momentum well defined (being in an eigenvector for momentum) corresponds to being unable to associate one particularly narrow range of position eigenvalues with the entity. This is essentially the Fourier cartoon you used in the animation (narrowness in space corresponds to less specificity in frequency/wavelength and vice versa).

    Beware of popular reports of the experimental observation of a wavefunction. Pull up the abstract from the underlying peer-reviewed manuscripts. I bet that the wavefunction has not been directly observed. Instead, the squared-magnitude (probability distribution) has been inferred from a large collection of individual experiments. As an example, a recent work inferring the nodal structure (radii where probability of finding electron around an atomic core vanishes) became popularized as direct observation of the wavefunction, which is not the claim in the original authors’ abstract.

    • Hi David,

      By and large, a good write-up. But, still…

      1. A minor point:

      “State |psi> is *not* wavefunction psi(x). psi(x) = , which is a *representation* of |psi> in terms of linear weighting coefficients for adding up basis states |x>”

      Did you miss something on the right-hand side of the equality sign? In any case, guess you could streamline the line a bit here.

      2. A major point:

      “There is no motion in any energy eigenstate.”

      And, just, how do you know?

      [And, oh, BTW, you could expand this question to include any other eigenstates as well.]

      Anyway, nice to see your level of enthusiasm and interest for these *conceptual* matters as well. Coming from a physics PhD, it is only to be appreciated.

      Best,

      –Ajit
      [E&OE]

      • Thank you for your reply. Hope the following is helpful!

        1) Thank you for catching the typo in the sentence, “psi(x) = , which is a *representation* of |psi> in terms of linear weighting coefficients for adding up basis states |x>.” This sentence should, instead, read, “psi(x) is a *representation* of |psi> in terms of linear weighting coefficients for adding up basis states |x>.” I don’t know how to edit my post to correct this sentence.

        2) You asked how it is possible to know that there is no motion in an energy eigenstate. Below, I include two ways to respond. The abstruse response is an actual answer and points to the insight you are seeking. If you look closely, you will see that the graphical response is not an actual answer. Instead, it is a fun exercise for “feeling the intuition” that energy eigenstates do not have motion. Both responses are important (many physicists enjoy both casual “proofs” and fluffy intuition).

        Abstruse response:
        We argue that an object that is completely described by one energy eigenstate has no motion. An energy eigenstate is a solution to the time-INDEpendent Schrödinger equation. It’s very “boring.” The only thing that happens to it, according to the time-DEpendent Schrödinger equation is, a rotation of its overall complex phase. This phase does not appear in expectation values, and so all expectation values are constants with time. To obtain motion, it is necessary to have a superposition of more than one state corresponding to at least more than one energy eigenvalue. In such circumstances, at least some of the complex phases will rotate at different time frequencies, allowing *relative* phases between states in the superposition to change with time.

        I am not claiming that experimental systems that people abstract using energy eigenstates will never turn out, following additional research, to have any aspect of motion. I am saying that the *abstraction* of a single energy eigenstate itself (without reference to whether the abstraction corresponds to anything empirically familiar) is a conceptual structure that contains no concept of motion (save for the rotating overall phase factor).

        The mathematics described above are very similar to the mathematics that describe the propagation of waves in elastic media. A pure frequency standing wave always has the same shape (though it might be vertically scaled and upside down). A combination of standing waves of different frequencies does not always maintain the same shape.

        Graphical response:
        Go to http://crisco.seas.harvard.edu/projects/quantum.html and play with the simulator. Now set the applet to use a Harmonic potential, and try to sketch, using the “Function editor,” the ground state from http://en.wikipedia.org/wiki/File:HarmOsziFunktionen.png
        You might want to turn on the display of the Potential energy function to ensure an accurate width for the state you are sketching. Run the simulation. Notice that the function doesn’t move very much (or in the case that you sketched the ground state with perfect accuracy, it shouldn’t move at all). Now, sketch a different state that doesn’t look like any one of the energy eigenstates in the Wikipedia image above. This should generate motion (to some extent looking like a probability mound bouncing back and forth in the well).

        You can also look at the animations at http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator and see that the energy eigenstate examples (panels C,D,E, and F) merely rotate in complex space (red and blue get exchanged with each other), but the overall spatial probability distribution is unchanged.

        3) You asked whether one would assert absence of motion for other eigenstates.

        Not as a general blanket statement. The reason that energy eigenstates have no motion is that they are eigenstates, specifically, of the Hamiltonian. Yes, in some examples, it is possible for an eigenstate of another operator to have no motion (i.e. when that state is both an eigenstate of another operator, as well as of the Hamiltonian).

        • Cool.

          Your abstruse response really wasn’t so abstruse. But anyway, my point concering the quantum eigenstates was somewhat like this.

          To continue on the same classical mechanics example as you took, consider, for instance, a plucked guitar string. The pure frequency standing wave is “standing” only in a secondary sense—in the sense that the peaks are not moving along the length of the string. Yet, physically, the elements of the string *are* experiencing motion, and thus the string *is* in motion, whether you choose to view it as an up-and-down motion, or, applying a bit of mathematics, view it as a superposition of “leftward” and “rightward” moving waves.

          The issue with the eigenstates in QM is more complicated, only because of the Copenhagen/every other orthodoxy in the mainstream QM. The mainstream QM in principle looks down on the idea of any hidden variables—including those local hidden variables which still might be capable of violating the Bell inequalities. They are against the latter idea, in principle—even if the hidden variables aren’t meant to be “classical.” Leave aside a few foundations-related journal, the mainstream QM community, on the whole, refuses to seriously entertain any idea of any kind of a hidden variable—and that’s exactly the way in which the relativists threw the aether out of physics. … I was not only curious to see what your inclinations with respect to this issue are, but also to learn the specific points with which the mainstream QM community comes to view this particular manifestation of the underlying issue. In particular, do they (even if epistemologically only wrongly) cite any principle as they proceed to wipe out every form of motion out of the eigenstates, or is it just a dogma. (I do think that it is just a dogma.)

          Anyway, thanks for your detailed and neatly explanatory replies. … Allow me to come back to you also later in future, by personal email, infrequently, just to check out with you how you might present some complicated ideas esp. from QM. (It’s a personal project of mine to understand the mainstream QM really well, and to more fully develop a new approach for explaining the quantum phenomena.)

          Best,

          –Ajit
          [E&OE]

          • Ah, I see better where you are coming from.

            You are wondering what explanations someone might give for focusing on mainstream QM interpretations and de-emphasizing hidden variables perspectives. Off the top of my head, I can imagine what people might generally say. I can also rattle off a couple thoughts as to why my attention does not wander much into the world of hidden variables.

            Anticipated general responses
            (0) I imagine usual responses would refer to Occam’s Razor and/or the Church of the Flying Spaghetti Monster. People might say that Occam’s Razor (or something along the same lines) is a fundamental aesthetic aspect of the Western idea of “science.” I am not saying these references directly address the most logically reasoned versions of the concerns you might be raising.

            (0.1) I think some professional scientists are laid back conceptual cleanliness. It doesn’t bother them enough to “beat” the idea of motion in eigenstates out from students in QM. I know a couple professional scientists who are OK with letting students think that electrons are whizzing around molecules.

            Personal thoughts
            (1) I don’t necessarily “believe” mainstream QM in a religious sense, but it feels natural (for my psychology). My gut feelings of certainty about existence of things somewhat vanish unless I am directly looking at them, touching them, and concentrating with my mind to force them “into existence” through brutal attention. People like to sensationalize mainstream QM by saying that it has counterintuitive indeterminacy. At the end of the day, what offends one person’s intuitions can be instinctively natural for someone else. I hear that mainstream QM is also “OK” for people who hold Eastern belief systems (I’m atheistish, so I don’t personally know).

            (2) Mainstream QM has a particular pedagogical value. It offers an exercise in making reasoned deductions while resisting the urge to rely on (some) inborn intellectual instincts. I think it’s good for learning that we sometimes confuse [1] the subjective experience of *projecting* a well-defined, deterministic mental image of the dynamics of a system onto a mental blank stage representing reality with, instead, [2] the supposed process of directly perceiving and “being unified with” reality. Yes, philosophy courses can be valuable too, but in physics you can also learn to calculate the photospectra* of atoms and describe the properties of semiconductors and electronic consumer goods.

            * Surprisingly difficult to do in a fully QM treatment at the undergraduate level. Perturbing the atom with a classical oscillating electric field is *not* kosher. It’s much more satisfying to quantize the EM field.

            Does any of this mean that mainstream QM is true? No. No scientific theory is ever “true” (quotation marks refer to mock hippee existential gesture).

            Best,
            David Liao

            P.S. I am happy to share my email address with you–how do I do that? Does this commenting platform share my address (sorry, not used to this system)?

            • Hi David,

              1. Re. Hidden variables.

              Philosophically, I believe in “hidden variables” to the same extent (i.e. to the 100% extent) and for the same basic reason that I believe that a trrain continues to exist after it enters a tunnel and before it emerges out of the same. Lady Diana *could* suffer an accident inside a tunnel, you know… (I mean, she would have continued to exist even after entering that tunnel—whether observed by those paparazzis or not. That is, per my philosophical beliefs…)

              Physics-wise, I (mostly) care for only those hidden variables which appear in *my* (fledgling) approach to QM (which I have still to develop to the extent that I could publish some additional papers). I mostly don’t care for hidden variables of any other specifically physics kinds. Mostly. Out of the limitations of time at hand.

              2. Oh yes, (IMO) electrons do actually whiz around. Each of them theoretically can do so anywhere in the universe, but practically speaking, each whizzes mostly around its “home” nucleus.

              3. About mysticism: Check out J.M. Marin (DOI: 10.1088/0143-0807/30/4/014). Mysticism was alive and kicking in the *Western* culture even at a time that Fritjof Capra was not even born. The East could probably lay claim to the earliest and also a very highly mature development of mysticism, but then, I (honestly) am not sure to whom should go the credit for its fullest possible development: to the ancient mystics of India, or to Immanuel Kant in the West. I am inclined to believe that at least in terms of rigour, Kant definitely beat the Eastern mystics. And that, therefore, his might be taken as the fullest possible development. Accordingly, between the two, I am inclined to despise Kant even more.

              4. About my email ID. This should be human readable (no dollars, brackets, braces, spaces, etc.): a j 1 7 5 t p $AT$ ya h oo [DOT} co >DOT< in . Thanks.

              Best,

              –Ajit
              [E&OE]

    • Entering this comment for the third time now (and so removing bquote tags)–ARJ

      Hi David,

      By and large, a good write-up. But, still…

      1. A minor point:

      >> “State |psi> is *not* wavefunction psi(x). psi(x) = , which is a *representation* of |psi> in terms of linear weighting coefficients for adding up basis states |x>”

      Did you miss something on the right-hand side of the equality sign? In any case, guess you could streamline the line a bit here.

      2. A major point:

      >> “There is no motion in any energy eigenstate.”

      And, just how do you know?

      [And, oh, BTW, you could expand this question to include any other eigenstates as well.]

      Anyway, nice to see your level of enthusiasm and interest for these *conceptual* matters as well. Coming from a physics PhD, it is only to be appreciated.

      Best,

      –Ajit
      [E&OE]

  9. Great idea for doing this. Just a hint for getting more non physicists involved: talk at least half as fast as you do, people need time to absorbe and self explain, othewise no matter how simple it is, they lose you at the beginning.

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    • Mankei,

      Interesting. You seem to be having been fun thinking about this field for quite some time.

      Anyway, here is a couple of questions for you (and for others from this field):

      (i) Is it possible to make a mechanical oscillator/beam detectably interact with single photons at a time (i.e. statistically very high chance of only one photon at a time in the system)? [For instance, an oscillator consisting of the tip of a small triangle protruding out of a single layers of atoms as in a graphene sheet? … I am just guessing wildly for a possible and suitable oscillator here.] Note, for single photons, it won’t be an _oscillator_ in the usual sense of the term. However, any mechanical device that mechanically responds (i.e. bends), would be enough.]

      (ii) If such a mechanical device (say an oscillator) is taken “to” “0” K, does/would/will it continue to show the red/blue asymmetrical behavior? [Esp. for Mankei] What do you expect?

      –Ajit
      [E&OE]

      • (i) In theory it’s possible, there have been a few recent theoretical papers on “single-photon optomechanics” that explore what would happen, but experimentally it’s probably very, very hard. Current experiments of this sort use laser beams with ~1e15 photons per second.

        (i) I have no idea what would happen then, because my math and my intuition always assume the laser beam to be very strong. Other people might be able to answer you better.

        • Hi Mankei,

          1. Thanks for supplying what obviously is a very efficient search string. (The ones I tried weren’t even half as efficient!) … Very interesting results!

          2. Other people: Assuming that the gradual emergence of the red-blue asymmetry with the decreasing temperatures (near the absolute zero) continues to be shown even as the *light flux* is reduced down to (say) the single-photon levels, then, how might Mankei’s current model/maths be reconciled with that (as of now hypothetical) observation?

          I thought of the single-photon version essentially only in order to remove the idea of “noise” entirely out of the picture.

          If there is no possibility of any noise at all, and *if* the asymmetry is still observed, wouldn’t it form a sufficient evidence to demonstrate the large-scale *quantum* nature of the mechanical oscillator (including the possibilities of a transfer of a quantum state to a large-scale device)? Or would there still remain some source of a doubt?

          –Ajit
          [E&OE]

    • Hi Mankei,

      We also thought about the issue you brought up in arxiv:1306.2699. See, for instance, a recent paper we published with Yanbei Chen and Farid Khalili (http://pra.aps.org/abstract/PRA/v86/i3/e033840).

      I would consider that our experiment measured both the sum AND difference of the red and blue sideband powers. The DIFFERENCE is indeed, as shown in your arxiv post mentioned above, due to the quantum noise of the light field measuring the mechanics. The noise power of the mechanics is in the SUM of the red and blue sidebands. Our experimental data was plotted as the ratio of the red and blue sidebands, which depends upon both the sum and difference of the sidebands powers, and looks very much different than what would be expected even for a semi-classical picture in which the light is quantized and the motion not.

      OJP.

      • I guess we’ve already exchanged emails and come to a consensus, but just to recap, I agree that, through your calibrations, you’ve inferred zero-point mechanical motion and your result is consistent with quantum theory. The word “quantum” of course literally means something discrete and one could argue you haven’t observed “quantum” motion yet, but that’d be nitpicking.

        • And to clarify, the asymmetry itself is not proof of zero-point mechanical motion or anything quantum. The mechanical energy was obtained from the SUM of the sidebands (as Oskar said), and the asymmetry was used as a *calibration* to compare the mechanical energy with the optical vacuum noise.

          • So me and my boyfriend are going on a two hour car ride together in a few wes03&#82ek; I know that’s not a a long time, but I still feel like we’ll need some conversation material on the ride. Last time we took the ride, there were a few awkward silences and I just want to make sure that for most of it, we have something to chat about. Are there any car games that you guys know of that force people to talk?.

    • Hi Mankei,

      Thanks for your response. There are two main claims in your manuscript, 1) centers around the interpretation of our result, 2) is a strong claim about classical stochastic processes being the source of our observed asymmetry.

      In response to 1), the different interpretations of the result (and in particular, the relation between the optical vacuum noise and the zero-point motion) have been considered previously in great depth by our colleagues at IQIM (Haixing Miao and Yanbei Chen) and in Russia (Farid Khalili). I would like to point you to this paper: http://pra.aps.org/abstract/PRA/v86/i3/e033840.

      In response to 2), you claim to “show that a classical stochastic model, without any reference to quantum mechanics, can also reproduce this asymmetry”. We also consider this possibility in a follow-up paper which came out last year (http://arxiv.org/abs/1210.2671), where we show a derivation exactly analogous to what you’ve shown, and then go to great lengths to experimentally rule out classical noise as the source of asymmetry (by varying the probe power and showing that the asymmetry doesn’t change, and by carefully characterizing the properties of our lasers).

      More generally, there are fundamental limits as to what can be claimed regarding `quantum-ness’ in any measurement involving only measurements of Gaussian noise. To date there have been 5 measurements of quantum effects in the field of optomechanics, our paper being the first one (the others are Brahms PRL 2012, Brooks Nature 2012, Purdy Science 2013, and Safavi-Naeini Nature 2013 (in press)). Unfortunately, all of these measurements are based on continuous measurement of Gaussian noise. There are several groups working hard on observing stronger quantum effects (as O’Connell Nature 2010 did in an circuit QED system), but we are still some months away from that.

      Best, Amir

      • Actually, I’d like to make that 6 papers – last week Cindy Regal’s group released this beautiful paper on arXiv: http://arxiv.org/abs/1306.1268.
        Here as well, the `quantum-ness’ can only be inferred after careful calibration of the classical noise in the system, since the measurement is based on continuous measurement of Gaussian noise.

      • Actually I’d like to make that 7 papers – I forgot about the result from 2008 from Dan Stamper-Kurn’s group: Murch, et al. Nature Physics, 4, 561 (2008).

    • hi sonni,thaaks for sharing this recipe..i have never imagine i could make this dish one day until just now i decided to give it a try..oh my god, it taste fabulous!! going to cook this again and again..

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  15. I get very annoyed whenever somebody uses the phrases “quantum jump” or “quantum leap” to imply a BIG change in some domain (such as “our new Thangomizer represents a quantum jump in Yoyodyne’s capabilities). A quantum jump is the SMALLEST POSSIBLE state change in quantum mechanics, so when somebody claims their product represents a “quantum leap,” I mentally translate that as “smallest possible degree of incremental improvement over their previous product!”

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  17. Is it that higher red shift and lower blue shift indicates constant shrinking of the mirror? If that is true then do we expect red shift to die down say we keep the mirror at 0K for long enough time?

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  21. Pingback: Human consciousness is simply a state of matter, like a solid or liquid – but quantum | Tucson Pool Saz: Tech - Gaming - News

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  23. No. No, this shall not stand. Have you no heart, sir? You have a family now, as do I. You simply cannot go around throwing the videogaming equivalent of heneri-lacod crack at folks.Shame! SHAME!

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