A couple of interesting things to read:
- As someone interested in lost ancient literature and also science, I really liked this news article from Nature about progress in reading scrolls excavated from Herculaneum. The area around the Bay of Naples was a quite the spot for posh Roman families, and when Vesuvius erupted in 79 CE, whole villas, complete with their libraries of books on papyrus scrolls, were buried and flash-cooked under pyroclastic flows. Those scrolls now look like lump charcoal, but with modern x-ray techniques (CT scanning using the beam from a synchrotron) plus machine learning, it is now possible to virtually unroll the scrolls and decipher the writing, because the ink has enough x-ray contrast with the carbonized papyrus to be detected. There is reason to believe that there are more scrolls out there still buried, and there are lots of other books and scrolls out there that are too delicate or damaged to be handled and read the normal way. It's great to see this approach starting to succeed.
- I've written about metalenses before - using nanostructured surfaces for precise control of optical wavefronts to make ultrathin optical elements with special properties. This extended news item from Harvard about this paper is a nice piece of writing. With techniques now developed to make dielectric metalenses over considerably larger areas (100 mm silica wafers), these funky lenses can now start to be applied to astronomy. Nifty.
Suppose I have a single electron roughly localized at the origin. It has spin-1/2, meaning that, if there are no other constraints, if I choose to make a measurement of the electron spin along some particular axis, I will find that with 50/50 probability the component of the angular momentum of the electron is \(\pm \hbar/2\) along that axis. Suppose that I pick a \(z\) axis and do the measurement, finding that the electron is "spin-up" along \(z\). Because the electron has a magnetic dipole moment, that means that the magnetic field at some distance \(r\) away from the origin should be the field from a magnetic dipole along \(z\).
Now suppose I make another measurement of the spin, this time along the \(x\) axis. I have a 50/50 chance of finding the electron spin up/down along \(x\). After that measurement, the magnetic field at the same location \(r\) away from the origin should be the field from a magnetic dipole along \(x\). It makes physical sense that the magnetic field at location \(r\) can only "know" that a measurement was done at the origin on a timescale \(r/c\). (Note: A truly correct treatment of this situation would seem to require QED, because the spin is entangled with the electromagnetic field via its magnetic moment; likewise one would really need to discuss in detail what it means to measure the spin state at the origin and what it means to measure the magnetic field locally. Proper descriptions of detectors and measurements are really necessary.)
To highlight how subtle the situation is, suppose the spin at the origin is initially half of an EPR pair, so that it's in a spin singlet with a second spin near Alpha Centauri, so that the total spin of the two is zero. Now a measurement of \(s_{z}\) at the origin determines the state of \(s_{z}\) at Alpha Centauri, and the magnetic field near that dipole at Alpha Centauri should be consistent with that. Thinking about all of the subtleties here has been a good exercise for me in remembering how the seemingly simple statements we make when we teach this stuff can be implicitly very complicated.
5 comments:
Indeed, measurement in quantum mechanics and how it can lead to these apparent paradoxes is quite astounding, and it’s good for those of us who are used to it, and take it for granted, to be reminded every now and then.
I feel that the argument and gedanken experiment you describe should be able to be applied more generally to any entangled EPR state of a binary degree of freedom that obeys the same algebra (e.g. light polarization). Is there a particular reason why you’ve been pondering the magnetic realization?
PPP, I have been thinking about how to explain how spin (intrinsic angular momentum) leads to a (usually thought of as classical) magnetic field far away from the spin itself. I think that's the key point of the thought experiment - how does the "collapse"/projective measurement of the spin state *here* affect the magnetic field *over there*.
BTW, beyond assertion or derivations that assume something about spatial wavefunctions in the Dirac equation, I'm still looking for a good, clear explanation of how the (apparently pointlike) electron spin (as an intrinsic angular momentum) leads to a magnetic moment, if anyone has suggested reading out there.
The later part of your post is close to a discussion of Bell's theorem for two particles. I've been advocating for some years that we can and should think about QFT as about noisy fields, so that events happen in our devices because of the surrounding fields; if we change the surrounding noisy fields then the statistics of events in our devices change.
If you can spare 18 minutes to see how that plays out for photons, I've just uploaded a video to YouTube, https://youtu.be/GFkv1O4hQmM, "Bell's Theorem for Noisy Fields", which shows something of how that kind of perspective plays out. It's very different from saying that two particles fly to different ends of an experimental apparatus. There's a suggestion for an experiment, starting at 13'34": although what I suggest is inevitably a theorist's idea, a large part of how I approach QM and QFT is in terms of data&signal analysis applied to actually recorded measurement results, which I hope keeps everything solidly down to earth.
This is based on two articles in JPhysA 2006 and in Annals of Physics 2020.
Doug, I agree that measurement in quantum mechanics is a very interesting topic. At the beginning, you write "I know what the correct answer must be". Could you specify what is the question, to which you know the correct answer? Thanks!
Anon@12:06, apologies for my stream-of-consciousness writing. Here's a phrasing of one specific question: "Is there some effective EM radiation that is emitted by measuring the spin at the origin that corresponds to 'resetting' the B-field to look like that from an appropriately oriented dipole?" For the EPR pair case: "Does measuring the spin at the origin somehow result in changes to the B-field near Alpha Centauri?"
I think the correct answer to the former question is: "If we properly define the apparatus used to do the spin measurement, and the apparatus/method used to measure the magnetic field nearby, and we properly track the quantum state of the EM field, then nothing spooky happens. The spin + field + measurement apparatus always end up in consistent states. In some sense, measuring B near the spin is actually a measurement of the spin, because the spin is entangled with the EM field." The correct answer to the latter question is: "The spin1+electromagnetic field + spin2 are all entangled; as in the prior question, if we treat everything in detail, nothing spooky (at least nothing spookier than usual Bell physics) happens, and we would always find the spins+fields+measurement apparatuses in consistent states." Basically, it's misleading to talk about measuring B from a spin without acknowledging that the spin is entangled w/ the EM field.
Post a Comment