Two interacting quantum particles: relative motion

[@KH_Dewantoro]

Dear Prof. M, I really like your channel and I wish to go through all your videos. As I have little background in quantum theory, shall I go through it from the oldest video towards the newer ones? Or perhaps there is a proper order to watch it? I also find the 'background' video mention in the description useful, however also confusing at the same time because it is jumping here and there. Thanks for working on the channel! :D

[@ProfessorMdoesScience]

Thanks for watching! We are working on a website to help guide students better, but in the meantime, the best way to go is to follow the playlists. A good order to get started is the following: 1. The postulates of quantum mechanics, which covers the basic mathematical tools to understand quantum theory: kzbin.info/aero/PL8W2boV7eVfmMcKF-ljTvAJQ2z-vILSxb 2. The quantum harmonic oscillator, which provides a standard example of using the postulates with a much-studied system: kzbin.info/aero/PL8W2boV7eVfmdWs3CsaGfoITHURXvHOGm 3. Angular momentum in quantum mechanics, which is essential to then explore more advanced topics such as the hydrogen atom: kzbin.info/aero/PL8W2boV7eVfmm5SZRjbhOKNziRXy6yIvI These are a good starting point. After those, it would be natural to move to central potentials and the hydrogen atom (although we are still building that playlist). After that, identical particles and more advanced topics such as second quantization, which would prepare you to study areas such as chemistry, condensed matter physics, or particle physics. I hope this helps!

[@Celtics-x4w]

@@ProfessorMdoesScienceh

[@drdca8263]

Great as always :) I hadn't seen the bit about using the reduced mass for the fictitious particle for the relative positions, and I suspect that if I had attempted to break things up like this, I would have gotten confused at that step. Or, before that step, come to think of it. I probably would have just taken the difference between the corresponding momentum operators (or perhaps half of that difference) without weighting by the ratio between the masses, and gotten mixed up that way. so, I guess I would have ended up with, (1/(2 m_1))((P + p)/2)^2 + (1/(2 m_2))((P - p)/2)^2 , and then the cross terms with both P and p wouldn't have canceled out (unless m_1 = m_2), and I would be stuck with a mess, confused why it didn't work out. So, thank you for preventing me from becoming confused in that way! So, in the case with many particles, where the potential only depends on the distances between them, I imagine that the center of mass fictitious particle should still have mass the sum of the masses of the real particles, position the center of mass of the real particles, and momentum the sum of the momenta, then, I suppose pick one of the real particles to kind of leave out (if one of them is much more massive than each of the others, it seems convenient to pick that one), and for each of the other particles, the corresponding fictitious particle should have position equal to ((the position of the actual particle) - (the center of mass position)), and it's momentum should be a weighted linear combination of the momentum of the actual particle and, I suppose the momentum of the fictitious center of mass particle? but I'm not sure what weightings. Well, looking forwards to it. Thanks :)

[@ProfessorMdoesScience]

Glad you found our video helpful! The relative coordinates approach becomes less useful when we have more than two particles, so we typically need to find alternative methods to solve the problem. The quantum problem becomes even more complex when we have multiple *identical* particles, such that the resulting problem (e.g. the solution of the many-electron problem in chemistry or condensed matter physics) is still a topic of active research efforts. We take the initial steps towards this problem in these two more advanced playlists: * Identical particles: kzbin.info/aero/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf * Second quantization: kzbin.info/aero/PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb And moving forward, we hope to look at multiple examples of multi-particle systems.

[@subhajitsadhukhan8521]

Yes yes... I'm really really waiting for your lectures on H atom for some months.... After this video hope it's coming soon... :)

[@ProfessorMdoesScience]

Yes, the videos on central potentials and this video are building towards hydrogen! :)

[@pew6126]

As usual enjoy your videos. Here’s perhaps a slightly difficult problem: chemical bonding from the quantitative POV of physics e.g quantum statistical physics. Usual cases: covalent, hydrogen, ionic, van der waals. It would be nice for example to discuss H2 or O2 ; HCl; H20. Is this feasible? Also interesting to think of bonding as simplest example of “emergence.” I have no idea how to approach these mathematically (I am a mathematician). Are there techniques for handling this relatively rigorously? Would be very appreciative.👍 Guess I’m looking for a clear, rigorous, mathematically feasible approach to molecular orbital theory? Thanks!😊

[@ProfessorMdoesScience]

We are hoping to cover chemistry and condensed matter physics in the future, but will likely take some time before we get there. A key topic to study these subjects is the quantum theory of many identical particles (electrons), and we have a few videos that cover some of the basics in this area: 1. Identical quantum particles: kzbin.info/aero/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf 2. Second quantization: kzbin.info/aero/PL8W2boV7eVfnSqy1fs3CCNALSvnDDd-tb I hope this helps!

[@geoffrygifari3377]

The things i noticed around the ideas in this video: 1. If the two masses m1 and m2 are not that far apart, then we have to use μ and the center of mass distance can be quite far from the heavier particle. In the example of the hydrogen atom, electron surrounds a much more massive proton, and in the ground state the wavefunction is spherical, centered at the proton position. Now lets say we have a pair of coulomb-interacting particles, with masses rather close together. I magine even if the ground state is spherical, the center won't be one of the masses. What do you think? 2. I can see a symmetry working here, exchanging the positions r1 and r2. I imagine there will be differemt cases for equal masses and distinct masses. 3. Related to the mathematical physics of tensor product, i now realized that one hilbert space W can be "factorized" in different ways using tensor product (W1/W2 and W com/W relative in this case)

[@ProfessorMdoesScience]

Here are some thoughts: 1. The center of mass is never on one of the two masses, although if one is much more massive than the other then it will be close to the center of that particle. But even in the case of the hydrogen atom, approximating the center of mass with the proton position leads to a small but measurable difference. We'll explore this in more detail when we finally cover hydrogen! 2. If the two particles are the same, then yes, there is a symmetry if we exchange them. In fact, this symmetry is rather subtle in quantum mechanics because identical particles are indistinguishable! We go over this in some detail in this series of videos: kzbin.info/aero/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf 3. Indeed! I hope this helps!

[@richardthomas3577]

👍very clear and to the point - thank you!

[@ProfessorMdoesScience]

Glad you like it!

[@grantholland421]

What is the condition for two interacting particles to be entangled? Does it pertain to their tensor products?

[@ProfessorMdoesScience]

Entangled states can be understood in terms of tensor product states, and we (briefly) discuss it in this video: kzbin.info/www/bejne/oauWY2NsiJd1bLM I hope this helps!

[@5ty717]

Love this site::: consider a series on the imaginary plane v geometric algebra and can you comment on the likelihood and implications that there are no particles in QFT/QM… only in classical mechanics.

[@ProfessorMdoesScience]

Thanks for the suggestions!

[@gianlucafreddi6641]

what program do you use to write digitally?

[@ProfessorMdoesScience]

We use Explain Everything, hope this helps!

[@billbulgari]

Thank you so much for the videos! Do you plan to present the formalism for three-bodies too?

[@ProfessorMdoesScience]

Thanks for watching! Our plan is to move onto many-body physics, where three-body would just be a particular case. Do you have a particular system in mind where the specific three-body discussion would be relevant?

[@billbulgari]

@@ProfessorMdoesScience I am taking Introductory Elementary Particle Physics (we do not use Quantum Field Theory) this semester and we were talking about mesons and baryons the other day. The professor said that the Angular Momentum of a meson is the "Relative" Angular Momentum of its constituents and that we are studying the system of the meson (made up of a quark and an antiquark) in its Center of Mass. He said we are doing "the same" for baryons (made up of three quarks). The problem is he did not explain more about this, so when this video was released I was really hyped. The reason I asked about a three-body system is because of baryons. Of course you could talk about something simpler like Lithium (with some approximation for the Nuclear Potential). Whatever you decide, I am sure would be pretty useful to us students! Thank you so much for these videos on Rigorous Quantum Mechanics!

[@ProfessorMdoesScience]

@@billbulgari Thanks for clarifying, and we'll take your comments into account when designing our videos on many-body physics! And good luck with your studies!

[@DrDeuteron]

@@billbulgarihave you heard of the proton spin crisis? It is so far beyond the addition of 3 spins and orbital angular momenta. Baryons are a mess.

[@jaibhagwa3444]

Thank you for the wonderful lectures. These are really helpful. Can you make a lecture series on physics of ultracold atoms? Like derivation of Gross Pitaevskii Equation and Lieb Liniger model etc.

[@ProfessorMdoesScience]

Thanks for the suggestion! We already have a long list of things to do in the future, and we'll add your request to it!

[@jaibhagwa3444]

@@ProfessorMdoesScience Thanks alot 😊

[@sensorer]

You can edit the description to include a link to the Hydrogen atom video now

[@ProfessorMdoesScience]

Thanks for catching this, updated!!

[@TheWingEmpire]

um, can velocity be defined as a hermitian operator as momentum is? also...even though we are not able to know the velocity of electrons precisely, we do know that it is quite high, so can I use special theory of relativity for the electron and make its mass a variable and find uncertainty in mass?

[@ProfessorMdoesScience]

The usual way to study velocity in quantum mechanics is to consider it as the time derivative of the position operator and then you can find an expression for it from Heisenberg's equation as: vx=dx/dt=(i/hbar)[H,x]. The merging of special relativity with quantum mechanics is a very important topic, and it receives the name of quantum field theory. We hope to get to it after we've finished with the current videos on non-relativistic quantum mechanics. I hope this helps!

[@TheWingEmpire]

@@ProfessorMdoesScience thank you very much, it is really helpful

[@DrDeuteron]

@@TheWingEmpirewhen you understand quantum mechanics, you won’t care about velocity

[@CrookedRoadsMusic]

The math here is beyond me, but I have a question. Can quantum systems of more than two particles be solved? And also, is it fair to say that since particles are ultimately fields, all particles are always interacting with all other particles (to some extent)? Thank you.

[@ProfessorMdoesScience]

Not analytically, only numerically. And even the latter often require using high performance computing infrastructure and physical approximations to the equations to be solved. Indeed, one could say that most theoretical chemists and physicists have spent the last 100 years attempting to find solutions to the equations of quantum mechanics for more than two particles. This is also what the other Professor M does for a living :)

[@CrookedRoadsMusic]

Thank you! And solving means getting a single wave equation that describes the two particles as a system, telling us the states of each particle? Would solving it for three particles be of practical or mainly theoretical value? Would you then have to start over for four particles, or is there a hope of discovering some key that makes bigger systems in general solvable? What's the "end game?" :) @@ProfessorMdoesScience

[@ProfessorMdoesScience]

@@CrookedRoadsMusic This is taking us into a huge field of physics and chemistry, but in short there are quite a few different strategies to finding approximate solutions to these equations. A very popular one is "density functional theory", which replaces the many-particle wave function by the simpler density. By contrast, Quantum Monte Carlo techniques directly work with the many-particle wave function, but only attempt to sample it (rather than fully calculating it). And there are many many other possible approaches. I hope this gets you started into reading more about this fascinating world!

[@vperez4796]

I have heard that Uncle Albert didn't believe in particles. Tsh... XClent lecture, very good presentation.

[@ProfessorMdoesScience]

:) And glad you like it!

[@HarshMathur1990]

Hi, The videos are very helpful and comprehensive. If possible can you make some videos in irreducible spherical tensors and the Wignart-Eckart Theorem?

[@ProfessorMdoesScience]

Glad you find them helpful! We are planning on exploring angular momentum further (spin angular momentum, addition of angular momentum) and also on exploring the use of group theory in physics; so we should get to this topic at some point! :)

[@HarshMathur1990]

@@ProfessorMdoesScience Thank you. looking forward to it :)

[@DrDeuteron]

@@HarshMathur1990if you really want to master it, check out: Schur Weyl Duality Robinson Schensted correspondance It’s amazing, and non trivial.

[@evan_dv]

Can you guys do a video on Feynman path integrals?

[@ProfessorMdoesScience]

Thanks for the suggestion! We certainly plan to move to more advanced topics (such as path integrals) after we've covered the fundamentals of quantum mechanics.

[@jacobvandijk6525]

@ 19:00 "They don't interact with each other." BUT BEWARE, whenever there is a potential involved, there ALWAYS is another particle! For the hydrogen atom we have an electron AND a proton. Two interacting particles! Only a free particle does not interact. So don't get confused by this somewhat strange marketing statement: @ 0:53

[@DrDeuteron]

I disagree. When you move to reduced mass, the proton becomes nothing more than the origin. This allows analytic solution. Just don’t forget that this means the proton is also in an orbital. So I guess I do agree with you. But seriously, the nucleus always gets short shrift in atomic physics.

[@jacobvandijk6525]

@@DrDeuteron They interact. That's it.

[@senkum80]

do you guys have any patreon or buymeacoffee?

[@ProfessorMdoesScience]

Unfortunately we don't, but we are working on a webiste to share more material beyond the videos, and that will provide an opportunity to support us. Hopefully it will come live over the next few months!

[@bear2s232]

I have a question: how to add the spin degree of freedom in to the center of mass method?

[@bear2s232]

After thinking about it, I think it actually not related to the spin degree of freedom. This method is kind of like sperate the space Hillbert space into two parts and not related to the spin Hillbert space. But I have another question: how to keep the anti-asymmetry for a two electron system

[@achalvinod7097]

A small request, could you please make video concerning the degenerate case of the time independent perturbation theory :)

[@ProfessorMdoesScience]

Thanks for the suggestion! We are planning a full series on perturbation theory, which should include the degenerate case. We'll let you know when we publish those videos!

[@achalvinod7097]

@@ProfessorMdoesScience That's really great. Looking forward to it! :)

[@faizanraza-se9dc]

Very great..use animations to attract more viewers

[@ProfessorMdoesScience]

Thanks for watching and thanks for the suggestion!

[@garvitmakkar]

where are you guys please upload more

[@ProfessorMdoesScience]

We'll be back very soon!

[@Upgradezz]

Liked+subscribed , plus you are sweet!

[@ProfessorMdoesScience]

Thanks for subbing!

[@herpederpe4320]

:D