Archive for Quantum Mechanics

Cosmology: The Professor’s Old Clothes

Posted in Education, The Universe and Stuff with tags , , , , , , , on January 19, 2018 by telescoper

After spending  a big chunk of yesterday afternoon chatting the cosmic microwave background, yesterday evening I remembered a time when I was trying to explain some of the related concepts to an audience of undergraduate students. As a lecturer you find from time to time that various analogies come to mind that you think will help students understand the physical concepts underpinning what’s going on, and that you hope will complement the way they are developed in a more mathematical language. Sometimes these seem to work well during the lecture, but only afterwards do you find out they didn’t really serve their intended purpose. Sadly it also  sometimes turns out that they can also confuse rather than enlighten…

For instance, the two key ideas behind the production of the cosmic microwave background are recombination and the consequent decoupling of matter and radiation. In the early stages of the Big Bang there was a hot plasma consisting mainly of protons and electrons in an intense radiation field. Since it  was extremely hot back then  the plasma was more-or-less  fully ionized, which is to say that the equilibrium for the formation of neutral hydrogen atoms via

p+e^{-} \rightarrow H+ \gamma

lay firmly to the left hand side. The free electrons scatter radiation very efficiently via Compton  scattering

\gamma +e^{-} \rightarrow \gamma + e^{-}

thus establishing thermal equilibrium between the matter and the radiation field. In effect, the plasma is opaque so that the radiation field acquires an accurate black-body spectrum (as observed). As long as the rate of collisions between electrons and photons remains large the radiation temperature adjusts to that of the matter and equilibrium is preserved because matter and radiation are in good thermal contact.

 

Image credit: James N. Imamura of University of Oregon.

Eventually, however, the temperature falls to a point at which electrons begin to bind with protons to form hydrogen atoms. When this happens the efficiency of scattering falls dramatically and as a consequence the matter and radiation temperatures are no longer coupled together, i.e. decoupling occurs; collisions can longer keep everything in thermal equilibrium. The matter in the Universe then becomes transparent, and the radiation field propagates freely as a kind of relic of the time that it was last in thermal equilibrium. We see that radiation now, heavily redshifted, as the cosmic microwave background.

So far, so good, but I’ve always thought that everyday analogies are useful to explain physics like this so I thought of the following.

When people are young and energetic, they interact very extensively with everyone around them and that process allows them to keep in touch with all the latest trends in clothing, music, books, and so on. As you get older you don’t get about so much , and may even get married (which is just like recombination, not only that it involves the joining together of previously independent entities, but also in the sense that it dramatically  reduces their cross-section for interaction with the outside world).  As time goes on changing trends begin to pass you buy and eventually you become a relic, surrounded by records and books you acquired in the past when you were less introverted, and wearing clothes that went out of fashion years ago.

I’ve used this analogy in the past and students generally find it quite amusing even if it has modest explanatory value. I wasn’t best pleased, however, when a few years ago I set an examination question which asked the students to explain the processes of recombination and decoupling. One answer said

Decoupling explains the state of Prof. Coles’s clothes.

Anyhow, I’m sure there’s more than one reader out there who has had a similar experience with an analogy that wasn’t perhaps as instructive as hoped or which came back to bite you. Feel free to share through the comments box…

Advertisements

The Quantum Mechanics of Voting

Posted in Politics, The Universe and Stuff with tags , , , , , , on June 23, 2017 by telescoper

Now that I’ve finished a marathon session of report-writing I thought I’d take a few minutes out this Friday afternoon, have a cup of tea and pass on a rather silly thought I had the other day about the relationship between Quantum Mechanics (and specifically the behaviour of spin therein) and voting behaviour in elections and referendums.

Gratuitous picture of a Stern-Gerlach experiment

For a start here’s a brief summary of the usual quantum-mechanical context as it relates to, e.g., electrons (rather than elections). Being fermions, electrons possess half-integer spin. This attribute has the property that a measurement of its component in any direction has only two possible values, ±½ in units of Planck’s constant. In the Stern-Gerlach experiment illustrated above, which measures the spin in the vertical direction of silver atoms emerging from a source, the outcome is either “up” or “down”, not some spread of values in between. Silver has a single unpaired electron which is why its atoms behave in this respect in the same way as an individual electron.

The way this is often described in physics textbooks is to say that the operator corresponding to spin in the z-direction has only two eigenstates  (call these ↑ and ↓) ; the act of measurement has to select one of them, not some intermediate state. If the source is thermal then the spins of individual atoms have no preferred direction so 50% turn out to be ↑ and 50% to be ↓ as shown in the cartoon.

Once such measurement has been made, a given particle remains in the same eigenstate, which means that if it is passed through another similar measuring device it will always turn out to have spin pointing in the same direction. If you like, the particle has been `prepared’ in a given state by the act of measurement.

This applies as long as no attempt is made to make a measurement of the spin in a different direction, which is when the fun starts. If we start with a particle in the ↑ state and then pass it through an experiment that measures spin (say) with respect to the x-axis instead of the z-axis then the two allowed eigenstates are then not ↑ and ↓ but ← and →.  A particle that was definitely spin-up would then be forced to decide between spin-left and spin-right (each would have a  50% probability).

Suppose now we took our long-suffering particle that began with spin ↑ after a measurement in the z-direction, then turned out to be spin → when we measured it in the x-direction. What would happen if we repeated the z-measurement? The answer is that “preparing” the particle in the → state destroys the information about the fact that it was previously prepared in the ↑ state –  the outcome of this second z-measurement is that the particle that was previously definitely ↑ now has a 50% chance of being either ↑ or ↓.

So what does all this have to do with voting? It is clear than an election (or a referendum) is very far from a simple act of measurement. During the campaign the various sides of the debate make attempts to prepare a given voter in a given state. In the case of last year’s EU referendum the choice of eigenstates was `Leave’ or `Remain’;  no other possibilities were allowed. The referendum then `prepared’ each voter in one or other of these possibilities.

If voters behaved quantum mechanically each would stay in their chosen state until some other measurement were attempted. But that’s exactly what did happen. Earlier this month there was a General Election. More than two parties were represented, but let’s simplify and assume there were only two options, `Labour’ and `Conservative’.

Now it is true that the `Leave’ camp was dominated by the right wing of the Conservative party, and the majority of Labour voters voted `Remain’, but there were a significant number of Labour Leave voters and a significant number of Tories voted Remain. While these pairs of states are therefore not exactly orthogonal, they are clearly not measuring the same thing so the situation is somewhat analogous to the spin measurement problem.

So along came the General Election result which `prepared’ voters in a state of `Labour’ or `Conservative’, with a slight preference for the latter whereas the earlier referendum had prepared a them in a state of `Leave’ versus `Remain’ with a slight preference for the former. From a quantum mechanical perspective, however, you can further argue that the General Election prepared the voters in such a way that should have erased memories of their vote in the referendum so the previous BrExit vote is now invalid.

There’s only one way to test this quantum-mechanical interpretation of voting patterns, and that is by repeating the EU Referendum…

Wave Mechanics and Large-scale Structure

Posted in Books, Talks and Reviews, The Universe and Stuff with tags , , , on May 24, 2017 by telescoper

I thought I’d share the slides I used for the short talk I gave last Thursday at the Osservatorio Astronomico di Bologna, on the topic of Wave Mechanics and Large-scale Structure. I’ve posted about the general idea underpinning this workhere, and here are some links to references with more details of the cosmological setting, including a couple of papers by myself and Chris Short on some of whose old slides I based the talk.

http://adsabs.harvard.edu/abs/1993ApJ…416L..71W
http://adsabs.harvard.edu/abs/1997PhRvD..55.5997W
http://adsabs.harvard.edu/abs/2002MNRAS.330..421C
http://adsabs.harvard.edu/abs/2003MNRAS.342..176C
http://adsabs.harvard.edu/abs/2006JCAP…12..012S
http://adsabs.harvard.edu/abs/2006JCAP…12..016S
http://adsabs.harvard.edu/abs/2010MNRAS.402.2491J

I had a few problems with the movies during the actual talk, and they probably don’t work in this embedded version. There are a few formatting errors in the slideshare version too, but hopefully you can figure out what’s going on!

Reflections on Quantum Backflow

Posted in Cute Problems, The Universe and Stuff with tags , , , , on November 10, 2016 by telescoper

Yesterday afternoon I attended a very interesting physics seminar by the splendidly-named Gandalf Lechner of the School of Mathematics here at Cardiff University. The topic was one I’d never thought about before, called quantum backflow. I went to the talk because I was intrigued by the abstract which had been circulated previously by email, the first part of which reads:

Suppose you are standing at a bus stop in the hope of catching a bus, but are unsure if the bus has passed the stop already. In that situation, common sense tells you that the longer you have to wait, the more likely it is that the bus has not passed the stop already. While this common sense intuition is perfectly accurate if you are waiting for a classical bus, waiting for a quantum bus is quite different: For a quantum bus, the probability of finding it to your left on measuring its position may increase with time, although the bus is moving from left to right with certainty. This peculiar quantum effect is known as backflow.

To be a little more precise about this, imagine you are standing at the origin (x=0). In the classical version of the situation you know that the bus is moving with some constant definite (but unknown) positive velocity v. In other words you know that it is moving from left to right, but you don’t know with what speed v or at what time t0 or from what position (x0<0) it set out. A little thought, (perhaps with the aid of some toy examples where you assign a probability distribution to v, t0 and x0) will convince you that the resulting probability distribution for moves from left to right with time in such a way that the probability of the bus still being to the left of the observer, L(t), represented by the proportion of the overall distribution that lies at x<0 generally decreases with time. Note that this is not what it says in the second sentence of the abstract; no doubt a deliberate mistake was put in to test the reader!

If we then stretch our imagination and suppose that the bus is not described by classical mechanics but by quantum mechanics then things change a bit.  If we insist that it is travelling from left to right then that means that the momentum-space representation of the wave function must be cut off for p<0 (corresponding to negative velocities). Assume that the bus is  a “free particle” described by the relevant Schrödinger equation.One can then calculate the evolution of the position-space wave function. Remember that these two representations of the wave function are just related by a Fourier transform. Solving the Schrödinger equation for the time evolution of the spatial wave function (with appropriately-chosen initial conditions) allows one to calculate how the probability of finding the particle at a given value of evolves with time. In contrast to the classical case, it is possible for the corresponding L(t) does not always decrease with time.

To put all this another way, the probability current in the classical case is always directed from left to right, but in the quantum case that isn’t necessarily true. One can see how this happens by thinking about what the wave function actually looks like: an imposed cutoff in momentum can imply a spatial wave function that is rather wiggly which means the probability distribution is wiggly too, but the detailed shape changes with time. As these wiggles pass the origin the area under the probability distribution to the left of the observer can go up as well as down. The particle may be going from left to right, but the associated probability flux can behave in a more complicated fashion, sometimes going in the opposite direction.

Another other way of thinking about it is that the particle velocity corresponds to the phase velocity of the wave function but the probability flux is controlled by the group velocity

For a more technical discussion of this phenomenon see this review article. The exact nature of the effect is dependent on the precise form of the initial conditions chosen and there are some quantum systems for which no backflow happens at all. The effect has never been detected experimentally, but a recent paper has suggested that it might be measured. Here is the abstract:

Quantum backflow is a classically forbidden effect consisting in a negative flux for states with negligible negative momentum components. It has never been observed experimentally so far. We derive a general relation that connects backflow with a critical value of the particle density, paving the way for the detection of backflow by a density measurement. To this end, we propose an explicit scheme with Bose-Einstein condensates, at reach with current experimental technologies. Remarkably, the application of a positive momentum kick, via a Bragg pulse, to a condensate with a positive velocity may cause a current flow in the negative direction.

Fascinating!

 

 

 

 

Can single-world interpretations of quantum theory be self-consistent?

Posted in The Universe and Stuff with tags , on May 4, 2016 by telescoper

I saw a provocative-looking paper on the arXiv the other day (by Daniela Frauchiger and Renato Renner)  with the title Single-world interpretations of quantum theory cannot be self-consistent. No doubting what the authors think!

Here’s the abstract:

 According to quantum theory, a measurement may have multiple possible outcomes. Single-world interpretations assert that, nevertheless, only one of them “really” occurs. Here we propose a gedankenexperiment where quantum theory is applied to model an experimenter who herself uses quantum theory. We find that, in such a scenario, no single-world interpretation can be logically consistent. This conclusion extends to deterministic hidden-variable theories, such as Bohmian mechanics, for they impose a single-world interpretation.

Since this is a subject we’ve had interesting debates about on this blog I thought I’d post a link to it here and see if anyone would like to respond through the comments. I haven’t had time to read it thoroughly yet, but I do have a bit of train travel to do tomorrow…

 

Do Primordial Fluctuations have a Quantum Origin?

Posted in The Universe and Stuff with tags , , , , , , on October 21, 2015 by telescoper

A quick lunchtime post containing a confession and a question, both inspired by an interesting paper I found recently on the arXiv with the abstract:

We investigate the quantumness of primordial cosmological fluctuations and its detectability. The quantum discord of inflationary perturbations is calculated for an arbitrary splitting of the system, and shown to be very large on super-Hubble scales. This entails the presence of large quantum correlations, due to the entangled production of particles with opposite momentums during inflation. To determine how this is reflected at the observational level, we study whether quantum correlators can be reproduced by a non-discordant state, i.e. a state with vanishing discord that contains classical correlations only. We demonstrate that this can be done for the power spectrum, the price to pay being twofold: first, large errors in other two-point correlation functions, that cannot however be detected since hidden in the decaying mode; second, the presence of intrinsic non-Gaussianity the detectability of which remains to be determined but which could possibly rule out a non-discordant description of the Cosmic Microwave Background. If one abandons the idea that perturbations should be modeled by Quantum Mechanics and wants to use a classical stochastic formalism instead, we show that any two-point correlators on super-Hubble scales can exactly be reproduced regardless of the squeezing of the system. The later becomes important only for higher order correlation functions, that can be accurately reproduced only in the strong squeezing regime.

I won’t comment on the use of the word “quantumness” nor the plural “momentums”….

My confession is that I’ve never really followed the logic that connects the appearance of classical fluctuations to the quantum description of fields in models of the early Universe. People have pointed me to papers that claim to spell this out, but they all seem to miss the important business of what it means to “become classical” in the cosmological setting. My question, therefore, is can anyone please point me to a book or a paper that addresses this issue rigorously?

Please let me know through the comments box, which you can also use to comment on the paper itself…

Quantum Madness

Posted in The Universe and Stuff with tags , , on September 18, 2015 by telescoper

A very busy day lies in store so I only have time for a quick morning visit to the blog. If you enjoyed the recent guest post on the “hidden variables” interpretation of Quantum Mechanics, then you will probably enjoy reading a paper that recently appeared on the arXiv with the abstract:

Motivated by some recent news, a journalist asks a group of physicists: “What’s the meaning of the violation of Bell’s inequality?” One physicist answers: “It means that non-locality is an established fact”. Another says: “There is no non-locality; the message is that measurement outcomes are irreducibly random”. A third one says: “It cannot be answered simply on purely physical grounds, the answer requires an act of metaphysical judgement”. Puzzled by the answers, the journalist keeps asking questions about quantum theory: “What is teleported in quantum teleportation?” “How does a quantum computer really work?” Shockingly, for each of these questions, the journalist obtains a variety of answers which, in many cases, are mutually exclusive. At the end of the day, the journalist asks: “How do you plan to make progress if, after 90 years of quantum theory, you still don’t know what it means? How can you possibly identify the physical principles of quantum theory or expand quantum theory into gravity if you don’t agree on what quantum theory is about?” Here we argue that it is becoming urgent to solve this too long lasting problem. For that, we point out that the interpretations of quantum theory are, essentially, of two types and that these two types are so radically different that there must be experiments that, when analyzed outside the framework of quantum theory, lead to different empirically testable predictions. Arguably, even if these experiments do not end the discussion, they will add new elements to the list of strange properties that some interpretations must have, therefore they will indirectly support those interpretations that do not need to have all these strange properties.

You can download a PDF of the full paper here. It’s a short piece, but with a very good list of references for further reading.