State collapse in quantum mechanics

[@deltalima6703]

Gets a little clearer everytime you fill in some more subtleties. Appreciate this rigorous approach very much.

[@ProfessorMdoesScience]

Glad you like it!

[@quantum4everyone]

I like the way you discuss broadly for the cases of degenerate and continuum eigenvalues within the standard Copenhagen interpretation (it should be made clear this postulate is an interpretation). But, the thing missing is an actual description of an experiment that follows these rules. It isn't measuring position over some tiny interval, since we have no device that can do that except STEM, but that operates in a completely different fashion. Measurement of spin one-half via Stern Gerlach, or how it is done with quantum computers is another way, but in both cases, the experiments have an amplification phase to them, where the signal becomes large enough it can be read off. In any case, what is really lacking is one or more clear examples of actual experiments and how they fit into this paradigm. You will find the more you look into them, the more difficult it is to make them fit into this approach. Another useful experiment to consider is momentum measurement, as I think the only practical ways to do this involve actually measuring position and inferring the momentum (time of flight or scintillators, or bubble chamber paths in magnetic fields or Wien filter). One then has a conundrum about fitting such experiments into this paradigm. The only non-position measurement for momentum that I can think of is a Doppler shift of a spectral line, but I don't think that could be done with single quanta. It could only be done with large collections of sources, as found in astronomy. It might be possible to do it as well with a Mossbauer effect of some form, but again, this won't be a single quanta measurement, but a more macroscopic one.

[@ProfessorMdoesScience]

These are all very interesting points. We are certainly taking a mathematical view of quantum mechanics without making direct links to actual experiments. Time permitting, we would certainly like to extend our videos to make these connections.

[@keinbot1236]

Great video, but why can you neglect the absolute value at 8:03 ?

[@snjy1619]

[@geraldpellegrini2782]

If you "measure" the energy of the system the state collapses to an energy eigenstate in which all expectation values are constant in time and there is no longer classical correspondence! Is that correct?

[@angelmendez-rivera351]

Can the state collapse be described as some type of time evolution of the quantum state? One can think of a measurement of an observable A with A|u_n) = λ_n|u_n) as a continuous process that causes the quantum state to evolve from |ψ) = Σ[c_n|u_n)] to |u_n) within some small time interval Δt. So I can imagine a time-dependent ket |Φ(t)) such that |Φ(0)) = |ψ) and |Φ(Δt)) = |u_n). Is there a good mathematical description of a time evolution that describes this ideas? And if yes, is there any physical insight that can be made from said description?

[@ProfessorMdoesScience]

This is a very interesting question, for which there is no answer. Quantum theory does not provide a description of what happens at the point of measurement, and the result is that there are several so-called "interpretations" of this. The standard one is the Copenhagen interpretation, and another famous one is the many-worlds interpretation. However disappointing it may be, for most uses of quantum mechanics, all we need to know what the state of the wave function is before and after the measurement, without the need to know how it really gets there. I hope this motivates you to look into this fascinating topic!

[@schmetterling4477]

There is no such thing as a "state collapse" because there is no such thing as a state in nature. The "state" in quantum mechanics is a description of the ensemble of the system, similar to a probability distribution. It is not a measurable physical quantity that belongs to the individual system. We can only assign it to an infinite number of copies of the system, which, of course, is non-physical. There is no such thing as an ensemble in reality. We make that up as part of the description of the process in the same way we make up "reference frames" in classical mechanics.

[@ajilbabu13]

Is it same as the measurement of an observable and action of the operator on the state

[@ProfessorMdoesScience]

Very good question. No, these two things are not the same. The action of an operator on a state does not "measure" any property of the state, it simply gives you another state: A|psi>=|phi>. By contrast, when you measure A in state |psi>, then you get as a result one of the eigenvalues of A, say lambda_n, and the state after the measurement becomes the associated eigenstate |u_n>. This state is in general different from the state A|psi>.

[@yuchuanwei]

I published a method to teleport a particle. Suppose the wavefunction of a particle has nonzero value only around x=-1000. Now we measure the parity of the state. The wavefunction collapses to an eigen function of the parity. It has 50% chance around x=1000. Next We measure the .particle around x=1000. If we get it, the particle has been moved successfully. If not, it means it is still around x=-1000, repeat the above process. Can you comment on this or even discuss this in your video? (Phys. Rev. E 93, 066103 Appendix II).

[@ProfessorMdoesScience]

Thanks for the reference, will take a look!

[@yuchuanwei]

actually I know nothing more than the short appendix and I am not sure we can move particle faster than light. So I wish to get your comments and I wish more people to discuss it.

[@yuchuanwei]

some one says the speed of the particle here can not be faster than light, because your equipment must cover the long distance. When you order every one to start to measure the parity, the order itself has to travel lower than light. Actually, we can make an appointment that we start at noon tomorrow, and let everyone and his local equipment ready in advance.

[@prikarsartam]

Would you kindly refer a well-approved collection of all postulates of QM, since many places mixup the numbering as well as statements (with quite subtle changes)

[@ProfessorMdoesScience]

There is no universally agreed number or numbering of the postulates, pretty much every textbook makes their own choice. What is important is that, at the end of the day, they all amount to the same. For what it's worth, we have taken the same numbering as Cohen-Tannoudji.

[@prikarsartam]

@@ProfessorMdoesScience oh great. Thanks!

[@mrouzbeh4925]

Hi. I have problem in calculating forth_order of J. Could you please guide me on this? J^4

[@ProfessorMdoesScience]

What do mean by J? There is no J in this video.

[@mrouzbeh4925]

@@ProfessorMdoesScience I khnow. It was my question. I searched a lot but I couldn't find anything. I asked you I will be so thankful if you guide me

[@ProfessorMdoesScience]

I would be happy to help if I can, but I still don't understand your question: what is "J"?

[@mrouzbeh4925]

@@ProfessorMdoesScience angular momentum

[@mrouzbeh4925]

@@ProfessorMdoesScience I am studying lecture note " theory of magnetism" by Carsten Timm. It is in 3ird chapter of this lecture note which I couldn't realize

[@amaljeevk3950]

❤

[@schmetterling4477]

Yeah, sorry to disappoint, again, but that is _not_ what a measurement is in QM. A measurement is an irreversible process. Irreversibility implies that the process has infinite duration. There is no such thing as "the end of the measurement". The measurement has to leave a persistent change in some external system and that physical change can never be undone, hence a measurement lasts from "now" to infinity. If it wasn't like this, then we could never talk about energy Eigenvalues to begin with, the entire concept would be unphysical due to energy-time uncertainty. What really happens, and this is the challenge that nonrelativistic quantum mechanics can not solve because it lacks the necessary field quantization language for it, is that a local quantum system couples to a quantum field of (near) infinite extent and releases a quantum of energy that then gets removed (simply by the infinite volume of the surrounding space) from the system by the physical vacuum. This is the self-consistent interpretation of scattering in quantum field theory, which is mirrored by the "interaction point" vs. "detector volume" language of experimental high energy physics. For bound states we have to work a little harder and solve the contour integrals around the poles of the scattering function, if I remember correctly. Either way, it would be great if theorists would spare a few minutes to explain these things to students with a diagram or two before or after rattling down hundred year old mathematics that thoroughly hides the actual physics behind linear algebra. So, no, we aren't measuring Eigenvalues. We are measuring photons in the vacuum field. Their _physical_ energy gives us the information about the energy of the bound state of the atoms (or, better, the energy differences between bound states that differ in their angular momentum by one). By the time these photons are absorbed by the detector in our spectrometer, they are 100% decoupled from the atoms themselves. Nature provides the irreversibility with relativistic fields and that is just not something the non-relativistic formalism can (or needs to express). It would be nice, anyway, if somebody were to tell the students what the mechanism behind the symbols is. (And, yes, explaining Stern-Gerlach this way is a lot more complicated because now we would have to get into weak measurement theory in addition, so I would stick to the atomic physics example of a measurement which is simple, clean and physically relevant.).

[@ProfessorMdoesScience]

For an elementary course in quantum mechanics, we think the description of state collapse we have is very useful for students who are learning about these topics for the first time. Indeed, the vast majority of standard textbooks and university courses also take this approach, so we are not alone in this. This is not to say that there may not be alternative ways of explaining this that could be clearer, and I would encourage you to do it yourself if you think so!

[@schmetterling4477]

@@ProfessorMdoesScience Yes, and the vast majority of textbooks are wrong in the way they present the material (by which I don't mean that the material that they present is wrong, it's just not connecting to actual physics). I am not a teacher. You are. That's what "professor" means. I don't have that title and I don't get paid to do this right. You do. I am simply an experimentalist who has noticed that there is a deep disconnect between the way we teach QM and the way QM actually is in nature. There are missing pieces. So, no, you can not put this on me. I am only in this world to build particle detectors for you. The teaching of students and interested laymen is your job. You have to decide whether you want to teach physics as it actually is or whether you want to keep teaching it the way "it has been established" by a number of textbook authors (going back almost a century now) who didn't think this through quite as deeply as they should have. I hope you will instead of hiding behind the "it has always been done this way" label.