Density operator for pure quantum states

💻 Book a 1:1 session: docs.google.com/forms/d/e/1FA...
📚 We have mostly been doing quantum mechanics using state vectors called kets. In this video we introduce the density operator, which provides an alternative but equivalent way of doing quantum mechanics. The obvious question is: why bother? In this video we only look at pure states to have a nice introduction to the ideas behind the density operator. However, to see the full power of the density operator we need to look at systems for which we only have incomplete information that is described by mixed states, such as those that feature in quantum statistical mechanics.
🐦 Follow me on Twitter: / profmscience
⏮️ BACKGROUND
Dirac notation in state space: • Dirac notation: state ...
Operators in quantum mechanics: • Operators in quantum m...
Eigenvalues and eigenstates: • Eigenvalues and eigens...
Projection operators: • Projection operators i...
Pure and mixed states: • Pure vs. mixed quantum...
Measurements | Maths: • Measurements in quantu...
Schrödinger equation: • The Schrödinger equation
⏭️ WHAT NEXT?
Density operator for mixed states: • Density operator for m...
~
Director and writer: BM
Producer and designer: MC

[@shreyaspradhan8546]

Damn this was great. This 16 minute video has content I would end up spending 4 hours reading. And it was explained so well too. I'm going to binge all your videos soon :P. I really hope this channel gets more popular, it deserves to.

[@ProfessorMdoesScience]

Thanks for your support! :)

[@saadatsalman4578]

Seriously one Heck of a video, If you could keep on providing Such quality content for sure the channel is going to grow exponentially and in a year or so it would be one of the go to channels among the physics student community. Kudos to the team....

[@ProfessorMdoesScience]

Thanks so much! This type of feedback definitely motivates us to keep going!

[@rahulmathew4970]

It is already for me, only that this channel is very new and is lacking in content in accordance with my syllabus

[@attackontitan9834]

@@ProfessorMdoesScience just talk a bit slow sir 🙏🙏🙏🙏

[@armalify]

Many thanks Prof. Nowadays, science needs people like YOU.

[@ProfessorMdoesScience]

Thanks for your support!

[@zairamartinez4274]

Thanks you! I was reading Nielsen's book chapter 2, and I was kind of lost with this topic. REALLY THANK YOU.

[@ProfessorMdoesScience]

Glad you found it useful! :)

[@prikarsartam]

All your derivations and explanations are very lucid and clear to follow. I am sure a whole bunch of generations would hugely be benefited from your channel. Thank you from my side!

[@ProfessorMdoesScience]

Thanks for your kind words! :)

[@alexbyrne1437]

Perfect speed! Surprised this isn't more popular yet. Thanks!

[@ProfessorMdoesScience]

Thanks! We hope we'll get there sometime :)

[@ernek89]

Very nice explanation of these derivations. Thank you!!!!

[@ProfessorMdoesScience]

Thanks for watching, glad you like the video!

[@fernandojimenezmotte2024]

Thank You very much for the very neat, didactical, clean mathematical explanation, I enjoyed a lot. It would be a great idea for example to use this mathematical/physics base applied to a problem, for example in nuclear physics, or chemistry, etc... Would You create videos with applications? (for example applied to the detonation of a nuclear bomb, what happened in such a process with the Eigenstates of the Quantum Mechanical Process?, nuclear fission of Uranium, etc....). You mentioned about the macroscopical system of N particles defined by Avogadro and the Quantum System States, an application to the Hydrogen Atom...

[@ProfessorMdoesScience]

We are indeed hoping to move on to applications of quantum mechanics, including atomic, molecular, and materials physics. However, we first one to finish with the fundamentals of quantum physics, and we still have quite a few topics to cover... :)

[@hephzibah1048]

This is highly explanatory. I really commend your efforts to simplify this.

[@ProfessorMdoesScience]

Glad you like this! :)

[@eatyoualiveinc]

This was extremely helpful. Thank you so much for these videos!

[@abejacob566]

Great video! Clear language, and clean graphics. Really helped me understand the density operator.

[@ProfessorMdoesScience]

Thanks!

[@kals2007]

Excellent presentation.

[@ProfessorMdoesScience]

Thanks! :)

[@TheFirstCreator90000]

You are saving me, thank you very much!

[@ProfessorMdoesScience]

Glad to hear you find it helpful!

[@syedrumman3920]

You guys are great!!

[@attackontitan9834]

You're kind of Sheldon...Hope You win Nobel..Thanks for these videos ❤❤❤

[@ProfessorMdoesScience]

😄

[@mithleshdewangan6787]

awesome, you must be the professor of my university.

[@ProfessorMdoesScience]

This is the beauty of KZbin, educational resources can be shared without boundaries! :)

[@theaceman989]

What a great explanation! We didn't thematize the density operator in our quantum mechanics course and I was curios to learn about it because of the fact that I will have statistical quantum mechanics next semester. Thanks for this video, I will definitely watch the one about mixed states. You've earned yourself another subscriber ;)

[@ProfessorMdoesScience]

Glad you like the video! The density operator will certainly prove useful in your quantum statistical mechanics course :)

[@jaydutta4355]

Fantastic for learning and going beyond.

[@ProfessorMdoesScience]

Glad you like it! :)

[@AdrienLegendre]

Good video. Simple presentation but also complete and informative.

[@ProfessorMdoesScience]

Glad you like it!

[@ashisharya1872]

Great explanation for density operator for pure states. Waiting for your next video on density operator for the mixed states.

[@ProfessorMdoesScience]

Thanks! Working on it!

[@ProfessorMdoesScience]

We have now uploaded the video on mixed states here: kzbin.info/www/bejne/mJiThICPaZWkbrM You can also find the full playlist here: kzbin.info/aero/PL8W2boV7eVflL73N8668N0EQUnID1XaEU I hope you like this!

[@VC-dm7jp]

Thank you so much, you just made my day.

[@ProfessorMdoesScience]

Glad you like it! :)

[@shubhamvernekar6547]

Again a very helpful video, Thanks a lot!

[@ProfessorMdoesScience]

Glad you find it helpful!

[@ashfakhalikm2444]

amazing!!

[@ProfessorMdoesScience]

Thanks for watching!

[@xtnovak]

Very nice presented! Thanks

[@ProfessorMdoesScience]

Thank you too!

[@LifeIzBeautiful10]

Great stuff!

[@ProfessorMdoesScience]

Glad you like it!

[@bobdorsett6572]

Great explanation of the density matrix for a pure state! Thank you! I hope you will soon be able to add explanation of mixed state density matrices.

[@ProfessorMdoesScience]

Thanks! Working on the mixed states video (and quite a few others), and should hopefully come out reasonably soon.

[@bobdorsett6572]

@@ProfessorMdoesScience Super! I’m looking forward to them.

[@ProfessorMdoesScience]

We have now published the video on the density operator for mixed states: kzbin.info/www/bejne/mJiThICPaZWkbrM Hope you like it!

[@aniksingharoy59]

Thank You Professor 🤝.

[@ProfessorMdoesScience]

Thanks for watching!

[@khurshedfitter5695]

When would the other videos (the coming soon ones from description) come out? I absolutely loved this video.

[@ProfessorMdoesScience]

Thanks! "Coming soon" means that we have plans for these videos so they should come out in the next few weeks to months. But if there is particular interest in some videos, I am happy to prioritize them! :)

[@francescodiotallevi1]

@@ProfessorMdoesScience I would really love a video on the mixed states density operators xD

[@ProfessorMdoesScience]

Thanks for the feedback, we are working on this :)

[@ProfessorMdoesScience]

Just published the video on mixed states density operators: kzbin.info/www/bejne/mJiThICPaZWkbrM Hope you like it!

[@khurshedfitter5695]

@@ProfessorMdoesScience thank you so much :)))

[@diegocarrera7520]

Very amazing video, thanks so much from Germany !!!

[@ProfessorMdoesScience]

Glad you like it! :)

[@Seiven2077]

Thank you very nice video

[@ProfessorMdoesScience]

Thanks for watching!

[@iliasp4275]

When I enjoy a video I want to write a comment, but I am very lazy to think about a unique way to express me gratitude, apart from saying "thanks for the video", or "amaizing" so the algorithm wont think I am a bot. tought times we live in,

[@ProfessorMdoesScience]

Tough times indeed; in any case, thanks for watching! :)

[@hassanchhaiba154]

Waaaaaaw, this is awsome. Im working on some mathematical aspect of coherent state but i m struling in physics language.this video give me the motivation to continue the understanding of tgis domaine. Do you have a draft for this?thank you

[@ProfessorMdoesScience]

Thank you very much for the kind words! We don't have written material at the moment, but we'll definitely consider it for the future, thanks for the suggestion!

[@IzzatZubir]

This video is perfectly delivered! Do you have any recommended textbook for this topic? I think I need to do few exercises to be comfortable with the topic.

[@ProfessorMdoesScience]

Thanks for watching! An area where density operators are used thoroughly is in quantum statistical mechanics. A good textbook is "Statistical Mechanics" by Feynman.

[@IzzatZubir]

@@ProfessorMdoesScience will you do more videos on such topics? And have some exercise video too😋. I'm studying Quantum Stats Mech for my masters. Im doing BH information paradox.

[@ProfessorMdoesScience]

@@IzzatZubir we do plan to get to topics such as quantum statistical mechanics, but it will probably take some time before we manage to get there...

[@IzzatZubir]

@@ProfessorMdoesScience Awesome! Are you still teaching at cambridge? I might pursue my doctorate there if I can get the scholarships to do part iii.

[@ProfessorMdoesScience]

@@IzzatZubir Yes, still researching and teaching at Cambridge. Good luck with your scholarship application!

[@adamfattal9602]

Under-rated

[@ProfessorMdoesScience]

Glad you like it!

[@guilhermescorza4412]

Muito bom!

[@saadatsalman4578]

Prof., One Problem in many of the books on advanced Theoretical Physics is that they don't give the detailed step by step derivations of their Results and that gives a hard time for the first time learner. It would be great if could do this neat and clear process of deriving things for the other fields like Condensed Matter Theory, Field Theories, Particle Physics, Statistical mechanics and so on.... P.S -- Just now found that you are a Prof. and ended up reading some of your Research Works and i would be glad to know the name of the other Prof. MC Anyways great work sir!

[@ProfessorMdoesScience]

Thanks for your support! It is our plan to cover at least some of the other topics you mention after quantum mechanics, although it will take some time to get there. As to the other Prof. M, they will hopefully make an appearance soon! :)

[@garciadiegos]

Great video! What is the book that you based your lecture on ?

[@ProfessorMdoesScience]

Thanks! We don't use a specific book for our videos, we instead consult several and then try to come up with what we think may be a useful pedagogical approach. Books we like include Sakurai, Cohen-Tannoudji, and Shankar :)

[@garciadiegos]

@@ProfessorMdoesScience Thanks!

[@priyankasiwach5066]

Thank u so much for this video. I have a question, 12:40 why are we taking degenerate eigenvalues "an" ?What will be the case for non-degenerate eigenvalues?

[@ProfessorMdoesScience]

Glad you like it! We are using degenerate eigenvalues because this is the most general case. If a_n was a non-degenerate eigenvalue, then g_n=1, and the projection operator becomes P_n=|a_n>

[@utkarshbhatnagar9106]

Can you please make a video on quantum fidelity? Really need some easy and simple explanations!!!

[@ProfessorMdoesScience]

Thanks for the suggestion, we'll add it to our list! (although it will probably take us a while to get there, we have a long list at the moment...)

[@user-dz7vb2xq4h]

I think in 12:47 P(am) = , not Or the Pm = |am x am|, not Pm = sum(|am x am|)

[@ProfessorMdoesScience]

I think our expression is correct. P(am) is the probability that, given a state |psi>, a measurement of property A will give eigenvalue am as the outcome. This probability in general can be written as the expectation value of the projection operator on the eigenstate associated with the measured eigenvalue am, and when this eigenvalue is degenerate we need the projector to the subspace spanned by the associated eigenstates. We discuss this in some detail in this video: kzbin.info/www/bejne/pZWvqIiOgL5jgNU I hope this helps!

[@srushtipatil9351]

if possible could you make a video on aharonov bohm effect and gauge invarience in quantum mechanics? :)

[@ProfessorMdoesScience]

Thanks for the suggestion! We are hoping to have a series on Berry phases and related science (e.g. Aharonov-Bohm, topological materials, polarization). We would first like to finish the basic quantum mechanics that we are doing now, but we'll add to our list that you woud like us to get to these topics sooner rather than later.

[@srushtipatil9351]

@@ProfessorMdoesScience thank you so much 😊 this means a lot !!

[@1991jee]

Now I can die in peace!

[@ProfessorMdoesScience]

Glad you liked it! :)

[@nkm6275]

I guess it's a partial derivative of the density operator at the last 🤔. isn't the total derivative of the density operator is zero?

[@ProfessorMdoesScience]

Fair point. But just to be clear: the density operator can depend on time in general, although it may be time-independent in some cases (e.g. for an ensemble at thermal equilibrium).

[@nkm6275]

​@@ProfessorMdoesScience and usually, the time derivative of any operator p follows the relation 1/ih [ p̂ , H] + partial derivative. why is the commutator reverse in this case? I thought given relation 1/ih [ H, p] maybe it's partial derivative considering total derivative zero in which case it will satisfy the above condition.

[@nkm6275]

well for at least a time-independent case?

[@ProfessorMdoesScience]

This is a very interesting but somewhat subtle point which has to do with the concept of time evolution "pictures" in quantum mechanics (Schrödinger picture, Heisenberg picture, interaction picture). We hope to publish a few videos on the pictures of quantum mechanics in the next few months, where this should become clear. In the meantime, let me simply say briefly that in this video I am working in the Schrödinger picture, where the time evolution is in the states, not the operators. The reason why we get a time evolution for the density operator is because it is a "special" operator that in reality is simply a different way of writing the state of the system. In this language, you can think about the density operator more as a "state" than as an "operator", and the equation I have in the video shows the evolution of the state in the Schrödinger picture. By contrast, the equation you quote for the time evolution of general operators corresponds to how time evolution is described in the Heisenberg picture. But again, this is not the picture I am using in this video, hence the different sign. I hope this helps!

[@conorpower409]

@@ProfessorMdoesScience Just a small follow on question to this. Using the Schroedinger picture assumes that the operators (not the density operator however) have no time dependence. As a result would it be more technically correct to drop the time dependence in the Hamiltonian operator shown in the Von Neumann equation near the end of the video? The Hamiltonian operator is shown with an explicit dependence on time.

[@nitink4076]

Are you saying that the system exist in any one of many quantum states and we don't know in which state does the system actually exist? Or are you saying that the system exist in a state known as mixed state which is a mixture of all the possible states?

[@nitink4076]

Is the concept of density Matrix used to represent an imaginary quantum state which is a statistical average of the all the possible states?

[@nitink4076]

Is this density Matrix used to calculate average properties in a statistical mixture?

[@ProfessorMdoesScience]

In this particular video we are considering pure states, so no statistical mixture. However, if you follow the entire playlist on the density operator, we do consider all cases, including statistical mixtures of states. You can find it here: kzbin.info/aero/PL8W2boV7eVflL73N8668N0EQUnID1XaEU I hope this helps!

[@nitink4076]

@@ProfessorMdoesScience Thank you

[@subhendumo2619]

very well explained, but hearing 'Psi' is weird

[@ProfessorMdoesScience]

Changed pronunciation in the latest videos...

[@user-wj1qb3qu1y]

What's the difference between projection operator and density operator?

[@ProfessorMdoesScience]

For pure states as discussed in this video, the density operator is a projection operator. However, when we generalize the discussion to mixed states, this is no longer true. We do have a video on mixed states and density operators, so I suggest you take a look: kzbin.info/www/bejne/mJiThICPaZWkbrM I hope this helps!

[@_BhagavadGita]

4:00

[@MessedUpSystem]

So... In the operator formalism you deal with a Lie Algebra, huh?

[@ProfessorMdoesScience]

In this series of videos we are describing these concepts at a more elementary level. However, Lie groups do play a big role in physics.

[@MessedUpSystem]

@@ProfessorMdoesScience I see, it's just that it ringed a bell since I'm studying Lie stuff in the context of Differential Geometry atm

[@mdshaifullah9598]

I love your channel but presentation very very hard to follow and tedious.

[@ProfessorMdoesScience]

Sorry to hear you find the presentation hard to follow and tedious. The aim of our videos is to cover every single step in every derivation, something that most textbooks don't have enough space for. But the price we pay is indeed some tediousness...