The time evolution operator in quantum mechanics

[@prikarsartam]

nowadays when I'm having some conceptual trouble I'm coming to your channel. Please keep it this way Sir, you're creating beautiful and precise content here. Just one slight thing: I think at 14:26, it would've been better to mention that the operator in the exponent gives the corresponding eigenvalues of the summand |u_i> with the power series expansion of the e^{ (-i/hbar)H(t-t') } and the fact that for a hermitian operator (eg. hamiltonian) H^n |u> = E^n |u> when |u> is an eigenstate of the operator. Thank you by the way!

[@ProfessorMdoesScience]

Glad you find the videos helpful! And yes, I agree that some more detail there would have made the derivation clearer. For completeness, we discuss the general case in the video on functions of operators: kzbin.info/www/bejne/pKKvdpWor7umoZY

[@prikarsartam]

@@ProfessorMdoesScience great!

[@JohnVKaravitis]

Keep 'em coming, and feel free to go both into detail when it comes to typical university material and problems, BUT ALSO please delve into the different intrepretations of QM, and Bell's Inequality. Thank you!

[@ProfessorMdoesScience]

Thanks for the suggestion, we'll take that onboard!

[@stevenschilizzi4104]

These videos sparkle with clarity, to the point of making often complex topics appear simple. Very energizing! Thank you 🙏!

[@ProfessorMdoesScience]

Thanks for watching, glad you like them! :)

[@arjyadebsengupta8159]

Thank you Sir

[@ProfessorMdoesScience]

Thanks for watching!

[@miriamstudyaccount8735]

great video! also love your use of the video description (really helps navigate the videos), and I am actually looking forward to the "Schrödinger vs. Heisenberg picture" video and the "Interaction picture

[@ProfessorMdoesScience]

Thanks for the feedback, and we hope to get to these videos you mention soon!

[@ProfessorMdoesScience]

We have finally published the video on the Schrödinger vs. Heisenberg pictures", you can find it here: kzbin.info/www/bejne/aHWwZneJedKDasU I hope you like it!

[@miriamstudyaccount8735]

@@ProfessorMdoesScience watched it and loved it :) good job

[@adityabaghel1270]

​@@ProfessorMdoesSciencewas looking for the same and this one was also so smooth and clearly explained thanks a lot!!! and wow you care so much of your viewers I'm amazed!!!!!KEEP IT UP

[@iotaphysics909]

awesome explanation...i came here after reading sakurai...and got crystal cleared

[@ProfessorMdoesScience]

Glad to hear you find this helpful! :)

[@WhiskeyTobin]

Thank you so much for this. I was having trouble understanding some of this from JJ Sakurai and my first semester grad QM class alone. Now we're using time evolution with simple harmonic oscillators in the second semester class and this video is a huge help for me to actually feel like I know what I'm doing (test on Monday!) The other Professor M professor's videos on simple harmonic oscillators in QM are also a massive help. Thank you to both of you!

[@ProfessorMdoesScience]

Glad you find them useful! May we ask where are you studying?

[@family-accountemail9111]

@@ProfessorMdoesScience hi professor M , where are you guys from? I am watching these to help my son who has exams including quantum mechanics coming up , these and some other you tubers have been a great help!

[@ProfessorMdoesScience]

@@family-accountemail9111 We are both faculty at the University of Cambridge in the UK. Where does your son study?

[@nasirsiddiqui7573]

wow i wish this existed last year. i was doing a quantum information research project when i was an undergrad and i needed to apply a unitary of the form e^-iHt in discrete time steps to a state vector and had no idea how to. this would have really helped.

[@ProfessorMdoesScience]

Glad you liked it, even though it came a little late to be helpful!

[@rsbenari]

Crystal. Well done, esp the takeaway regarding 'reversing the arrow of time.' Cuts to the quick. Thanks.

[@ProfessorMdoesScience]

Glad you like the video!

[@marlovsk1]

this is gold

[@ProfessorMdoesScience]

Thanks for watching! :)

[@Amandajshack]

REALLY looking forward to that Schrödinger vs. Heisenberg picture video. In fact, I wish it was up now!

[@ProfessorMdoesScience]

Thanks for the comment, it is on our list, and we hope to get there sooner rather than later... we'll let you know when we publish it!

[@ProfessorMdoesScience]

Video finally up! You can find it here: kzbin.info/www/bejne/aHWwZneJedKDasU Hope you like it!

[@guoxinxin693]

So beautiful! Do you guys have plans or have already written a beautiful book like the notes in your video on quantum mechanics? I dream to have such a book. Thanks!

[@ProfessorMdoesScience]

Thanks for your kind comments! We don't have plans for a book, but we're working on trying to expand the type of material we provide. Hopefully we'll have some news on this front in the next few months :)

[@davidroux7987]

Your videos are just excellent. Thank you for sharing your hard work.

[@ProfessorMdoesScience]

Thanks for your support! :)

[@tommysullivan]

Thank you!

[@ProfessorMdoesScience]

Thanks for watching!

[@mehdisi9194]

Hi and thank you so much for your great videos on quantum mechanics. If possible, please make some videos about qed. Good luck

[@ProfessorMdoesScience]

Thanks for the suggestion! We are hoping to get to more advanced topics after we've covered the basics of quantum mechanics.

[@paulbk2322]

Thanks to the team once again for the exceedingly useful and lucid discussion 👍👏👌

[@ProfessorMdoesScience]

Thanks for watching, and glad you like it!

[@alijoueizadeh2896]

Thank you for your time and effort.

[@ProfessorMdoesScience]

Glad you like our videos!

[@Muhammadusman52281]

Thank you very much for such a wonderful work

[@ProfessorMdoesScience]

Glad you like it! :)

[@DivisionbyZer0]

Excellent. Curious if the video with the Dyson expansion has been done yet?

[@ProfessorMdoesScience]

Glad you like it! We don't have a video on the Dyson expansion yet, we first need to publish a series of other videos that are necessary background to understand it. But we'll hopefully get there sooner rather than later!

[@vromerob]

hello profesor m, im victor and i was struggling while studying this quantum mechanics, i have a question, you know, quantum states are the properties of a particle, right?. So, if we have two arranged particles with different quantum states (obviusly) lets call it QS(A) and QS(B) if one of them changes their quantum state, it has to evolve to another, instead of QS(A) it has to evolve into a different one, so my question is, the other particle that is interacting with particle A, has to change to ? or not necessarily? because how do you know the past ? I hope you explain me professor M because i cant understand. thank you

[@ProfessorMdoesScience]

Quantum mechanics becomes very interesting when we have more than one particle, particularly if these particles are indistinguishable. We have a whole series of videos covering this topic, so I recommend those as a starting point: kzbin.info/aero/PL8W2boV7eVfnJ6X1ifa_JuOZ-Nq1BjaWf I hope this helps!

[@vromerob]

I understood it bro, but maybe i didnt got it but, lets Say that if we have an Apple, then we cut it by half, the Quantum states has to change? And if this is true, wich particles dhould change?

[@ProfessorMdoesScience]

@@vromerob Yes, the quantum states describing the constituent atoms will in principle change. Take an atom that is at the centre of the apple to start with, it is surrounded by other atoms. After you cut the apple in half, that atom is now at the "edge", so no longer surrounded by atoms in all directions. This means that it experiences a different potential energy, and therefore its state will change. I hope this helps!

[@vromerob]

@@ProfessorMdoesScience thank you very much, but does that mean that the atoms that are surrounding this atom that "changed" should also change their quantum state, I mean, if an atom changes, the atoms that surround it could also because they are experiencing a new quantum state that would be the atom that "had changed", or suddenly they do not necessarily change, if what I say is true, how could we know these atoms were united in the past, do you understand me?

[@ProfessorMdoesScience]

@@vromerob Not sure I completely follow. In general, perturbing the state of an atom will affect all other atoms in the system. However, from a practical point of view, only the closest atoms will have an important effect, as the changes in atoms that are sufficiently far away will be negligible. I hope this helps!

[@sayanjitb]

Around timestamp 2:27, why did you express |psi(t0)> as a linear combination of two other states (similar for |psi(t)>)? Were those any two arbitrary kets or basis kets? PS: One request, can you please upload a video on the "Dyson" series! Thanks

[@sayanjitb]

Another doubt, around (8:10), you mentioned "It doesn't require different times are ordered", what do you mean by this? at 15:39, this expression for the U is only true if the Hamiltonian commutes at two different times, right?

[@ProfessorMdoesScience]

At that stage those two kets are arbitrary, the point is simply to exemplify what we mean by the fact that time evolution is linear. And we do hope to get to the Dyson series, but this is a somewhat more advanced topic so it may take a while before we've built all the necessary knowledge to get there!

[@ProfessorMdoesScience]

These are two different things. At 8:10 what I mean is that we can combine together U operators in such a way that the times in their arguments are not always getting "bigger". For example, imagine we have t1

[@sayanjitb]

@@ProfessorMdoesScience thank you indeed. Is it right to say that for non conservative system if Hamiltonian at two different times commute then we can write U(t,to)= exp( -i/h integration {H(t')} dt') ?

[@ProfessorMdoesScience]

Yes, this looks correct!

[@JohnAlexander-hj2nx]

Hiiiiii, hope You answer me on a video thst was two years ago, when something like an electron or an atom changes their quantum state in a system, the other particles that are connected to them should Also change, , or only changes the one particle that was affected?

[@ProfessorMdoesScience]

In general the state of all particles will change, as they will be interacting. Hope this helps!

[@JohnAlexander-hj2nx]

​@@ProfessorMdoesScienceoh great! , so in the hypotetic case that mmm lets say a glass of water falls and breaks into pieces, in that case ALL THE ATOMS should change, or only the ones whose got separated ( i mean the edges of the glass pieces that were united) I wait your answer. Thanks You!

[@ProfessorMdoesScience]

@@JohnAlexander-hj2nx As I said in my answer already, in principle *all* the atoms will be affected. However, in practice, the change in the atoms that are sufficiently far away from the broken edges will only be affected very weakly, so their change may be negligible.

[@JohnAlexander-hj2nx]

@@ProfessorMdoesScience So if their changes are "insignificant" that means that the ATOMS that are far away changes in the .. lets say "same quantum state", can we say that?

[@JohnAlexander-hj2nx]

@@ProfessorMdoesScience hope your answer teacher thanks you

[@user-uk3yb7qy9b]

I did not understand the proof at 8:20. If t=t'' then t' which is between them is also equal to them, and then we are left with a meaningless equation of identity matrices, isn't that right?

[@ProfessorMdoesScience]

I think we did not specify anywhere that t' had to be between t and t'', and in general we do not assume that, so then the equation should make sense. This arises from the fact that the time dependence as given by the Schrödinger equation is the same going forward or backward. I hope this helps!

[@user-uk3yb7qy9b]

cool! it does, Tnx@@ProfessorMdoesScience

[@paulg444]

U here is asymptotically unitary, it is unitary in the limit as epsilon goes to zero and It goes to zero as epsilon^2.

[@RickyVipy]

A question that i have of this is : Imagine we have two quantum states (A AND B) so if A changes their Quantum state? The quantum state B hss to chsnge it too? Or is not necesarilly?

[@RickyVipy]

Or maybe just because one quantum state of a system changes doesn't mean that the other one should too.

[@ProfessorMdoesScience]

It would be necessary to specify the setup you have in mind in some more detail for a full answer. But in the context of time evolution, then every quantum states evolves in time. As time evolution is driven by the Hamiltonian, it turns out the energy eigenstates have a "trivial" time evolution in that they only acquire an irrelevant overall phase factor (hence they are called stationary states). All other states will have a more complex time evolution. I hope this helps!

[@RickyVipy]

@@ProfessorMdoesScience oh, sure, My setup For this is that if we have maybe a paper and i cut it in half, then in that case the Quantum states that once we're part of the entire paper, dhould change ? Or maybe it stills the same. Oh and if the Quantum states changes, should be ALL, or some ones? I hope You respond me teacher thanks you

[@RickyVipy]

I know it can be a strange question, but maybe you can give me a full answer for this question?

[@ProfessorMdoesScience]

@@RickyVipy The quantum states will in principle change. Going back to your paper example, atoms that were in the centre of the paper before you cut it in half will then be at the edges of the paper after you cut it in half. Therefore, the potential energy they will experience will be different, and they will change in response to that. I hope this helps!

[@WhiskeyTobin]

Any idea when the Schrodinger vs. Heisenberg picture videos are coming? Unfortunately it will probably be too late to be useful for the class I'm taking right now, but so far I'm learning more about QM from these videos than I am from JJ Sakurai. Sometimes it felt like just going through motions until I watched these videos and really started understanding.

[@ProfessorMdoesScience]

We are currently working on the series on hydrogen atom, and will hopefully get to the Schrödinger vs Heisenberg picture after that. We would definitely like to have time/resources to publish more regularly...

[@ProfessorMdoesScience]

Just published the video on Schrödinger vs. Heisenberg: kzbin.info/www/bejne/aHWwZneJedKDasU Hopefully it is still useful!

[@chrisyang0423]

Hello! If I didn't make a mistake, I believe that for the infinitesimal unitary operator part 11:24, if U(ε) = 1 - iεF, then U†(ε) = 1 + iεF. In that case, U(ε)U†(ε) = 1 + ε^2 F^2, so indeed U(ε)U†(ε) = 1 to first order, in accordance with your video on unitary operators, but I was just wondering if there was not a less approximation-y way to proving that the time-evolution operator is unitary? In any case, great video!

[@ProfessorMdoesScience]

Glad you like the video! As we ultimately take epsilon to zero, in for example going to the equation of motion of U, then this approach should be as complete as it gets. However, there are certainly other ways in which the time evolution operator can be introduced, for example that of Sakurai. There the operator is postulated to be unitary to enforce probability conservation through the time evolution (as remember that unitary evolution conserves the norm of vectors). Adding a number of additional requirements, such as composition of time evolutions, and again relying on the infinitesimal operator, leads to the usual form for U. I hope this helps!

[@shrodindsy8992]

hello thank you for the this great content , can you please elaborate about what you meant when your said the correspondence of the vectors is linear ?

[@ProfessorMdoesScience]

Glad you like it! Imagine we have an initial state |psi(t0)>. To figure out what this state is at a later time, |psi(t)>, we can use the Schrödinger equation, which reads: ihbar*d|psi(t)>/dt=H|psi(t)> In the mathematical language of differential equations, this is a linear differential equation in t. To understand what this implies, let |psi1(t)> and |psi2(t)> be two solutions of the equation. If the state at the initial time is |psi(t0)>=a*|psi1(t0)>+b*|psi2(t0)>, then linearity implies that the state at a later time is given by the same linear combination of solutions, in our example it is |psi(t)>=a*|psi1(t)>+b*|psi2(t)>. Any such linear relationship between |psi(t0)> and |psi(t)> can be expressed by a linear operator, so that we can write |psi(t)>=U(t,t0)|psi(t0)>, where U is the linear operator. In this video, we first argue that this operator must exist because of the linear property of time evolution in quantum mechanics, and then proceed to figure out the properties of U, for example finding that is in an exponential function of the Hamiltonian when we have a conservative system. I hope this helps!

[@soumyajyotikundu7147]

Your videos help me a lot❤️ Thanks..

[@ProfessorMdoesScience]

Glad to hear! :)

[@soumyajyotikundu7147]

@@ProfessorMdoesScience 😊😊

[@voidisyinyangvoidisyinyang885]

And in Heisenberg quantum algebra - Moyal Algebra - the Time changes the quantum state! right? thanks. I always wondered how the time operator was implemented. thanks. Study math professor Lou Kauffman. Time is inherently noncommutative. haha. Schroedinger tried but in the end failed.

[@sanjaysrivastava8271]

Psi t0, how first order differential eq. Can have two solutions?

[@theopdiamond6226]

when are the videos on the different pictures coming :( I have exam in my quantum theory modules and those videos would be nice

[@ProfessorMdoesScience]

We hope soon, but we are finding it increasingly difficult to keep up with a regular video schedule as this is only our side project. May not make it for your exam in time, but will let you know anyway when we publish them!

[@ProfessorMdoesScience]

The video comparing the Schrödinger and Heisenberg pictures is now up: kzbin.info/www/bejne/aHWwZneJedKDasU Hopefully still useful for your studies?

[@theopdiamond6226]

@@ProfessorMdoesScience thank you! Sadly my quantum module is over but I'll definitely have a look at it in my spare time as a recap 🙌🙌

[@Barkharawat8979]

Wow♥️♥️

[@ProfessorMdoesScience]

Glad you like it! :)

[@sanjaysrivastava8271]

I think it is breaking the length...

[@amaljeevk3950]

❤

[@davidkincade7161]

Interesting, but "time" is just perception of change....therefore the "linear" equation is subjective and all the equation stuff is a great "framework" and I love this video... but don't think there is some "mechanical operator" that controls things... its your perception of your environment that "creates time". Therefore all these equations are based on your perception. This also explains the "strangeness" of quantum mechanics.... it's all so simple actually. All the math is nice, but it's all based on perception.... "time". People think "time" is a thing... it's not, its perception of change explained by basically day nite cycle sliced and diced through history with clocks and and theoretical ideas... which are useful, but still, not some "universal thing". ... think about that: ancient peoples didn't need all these equations to know about day nite cycle and "the herd of buffalo is two moons away"... to mathematicians time has become a "thing" through this type of explanation.... but time is always subjective: v= d/t........ OK, so t= d/v. Thats an observation... which is kinda obvious... "time" is imbedded in physics and taken as "gospel", but the "t" is perception.... so simple. "Time evolution operator" is your perception.... all the rest is nice mathematical explanation and makes sense.... but everything is based on "time evolution operator" which, to repeat, is "time", which is perceived change.... we know through evolution that energy changes, as he describes i.e time changes..... the whole video is based on "time"- i.e. perception... I hope this is interesting.

[@ProfessorMdoesScience]

Will abstain from commenting on the more philosophical aspects of this, but overall agree with you that what we present is a good mathematical framework to explain how quantum systems behave.