Density operator for mixed quantum states

Operators in quantum mechanics

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Dropping In from the Clouds 🌁

Density operator for pure quantum states

Professor M does Science

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💻 Book a 1:1 session: docs.google.co...
📚 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

Опубликовано:

8 сен 2024

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Комментарии : 120

@shreyaspradhan8546 3 года назад

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 3 года назад

Thanks for your support! :)

@saadatsalman4578 3 года назад

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 3 года назад

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

@rahulmathew4970 2 года назад

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

@attackontitan9834 Год назад

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

@prikarsartam 2 года назад

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 2 года назад

Thanks for your kind words! :)

@armalify 3 года назад

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

@ProfessorMdoesScience 3 года назад

Thanks for your support!

@alexbyrne1437 3 года назад

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

@ProfessorMdoesScience 3 года назад

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

@zairamartinez4274 3 года назад

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

@ProfessorMdoesScience 3 года назад

Glad you found it useful! :)

@ernek89 3 года назад

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

@ProfessorMdoesScience 3 года назад

Thanks for watching, glad you like the video!

@fernandojimenezmotte2024 Год назад

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 Год назад

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... :)

@abejacob566 3 года назад

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

@ProfessorMdoesScience 3 года назад

Thanks!

@hephzibah1048 Год назад

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

@ProfessorMdoesScience Год назад

Glad you like this! :)

@kals2007 3 года назад

Excellent presentation.

@ProfessorMdoesScience 3 года назад

Thanks! :)

@TheFirstCreator90000 3 года назад

You are saving me, thank you very much!

@ProfessorMdoesScience 3 года назад

Glad to hear you find it helpful!

@QUniversity 3 года назад

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

@ProfessorMdoesScience 3 года назад

Thanks! Working on it!

@ProfessorMdoesScience 3 года назад

We have now uploaded the video on mixed states here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-bg0SMY40q8Q.html You can also find the full playlist here: ru-vid.com/group/PL8W2boV7eVflL73N8668N0EQUnID1XaEU I hope you like this!

@syedrumman3920 4 года назад

You guys are great!!

@eatyoualiveinc 3 года назад

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

@mithleshdewangan6787 3 года назад

awesome, you must be the professor of my university.

@ProfessorMdoesScience 3 года назад

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

@attackontitan9834 Год назад

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

@ProfessorMdoesScience Год назад

😄

@AdrienLegendre Год назад

Good video. Simple presentation but also complete and informative.

@ProfessorMdoesScience 11 месяцев назад

Glad you like it!

@ashfakhalikm2444 3 года назад

amazing!!

@ProfessorMdoesScience 3 года назад

Thanks for watching!

@bobdorsett6572 3 года назад

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 3 года назад

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

@bobdorsett6572 3 года назад

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

@ProfessorMdoesScience 3 года назад

We have now published the video on the density operator for mixed states: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-bg0SMY40q8Q.html Hope you like it!

@jaydutta4355 2 года назад

Fantastic for learning and going beyond.

@ProfessorMdoesScience 2 года назад

Glad you like it! :)

@xtnovak 2 года назад

Very nice presented! Thanks

@ProfessorMdoesScience 2 года назад

Thank you too!

@theaceman989 3 года назад

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 3 года назад

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

@saadatsalman4578 3 года назад

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 3 года назад

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! :)

@khurshedfitter5695 3 года назад

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

@ProfessorMdoesScience 3 года назад

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 3 года назад

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

@ProfessorMdoesScience 3 года назад

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

@ProfessorMdoesScience 3 года назад

Just published the video on mixed states density operators: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-bg0SMY40q8Q.html Hope you like it!

@khurshedfitter5695 3 года назад

@@ProfessorMdoesScience thank you so much :)))

@diegocarrera7520 Год назад

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

@ProfessorMdoesScience Год назад

Glad you like it! :)

@aniksingharoy59 3 года назад

Thank You Professor 🤝.

@ProfessorMdoesScience 3 года назад

Thanks for watching!

@LifeIzBeautiful10 11 месяцев назад

Great stuff!

@ProfessorMdoesScience 11 месяцев назад

Glad you like it!

@sinaasadiyan Месяц назад

great explanation

@ProfessorMdoesScience Месяц назад

Glad you think so!

@hassanchhaiba154 3 года назад

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 3 года назад

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!

@shubhamvernekar6547 2 года назад

Again a very helpful video, Thanks a lot!

@ProfessorMdoesScience 2 года назад

Glad you find it helpful!

@Seiven2077 Год назад

Thank you very nice video

@ProfessorMdoesScience Год назад

Thanks for watching!

@IzzatZubir 3 года назад

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 3 года назад

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 3 года назад

@@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 3 года назад

@@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 3 года назад

@@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 3 года назад

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

@VC-dm7jp 2 года назад

Thank you so much, you just made my day.

@ProfessorMdoesScience 2 года назад

Glad you like it! :)

@adamfattal9602 2 года назад

Under-rated

@ProfessorMdoesScience 2 года назад

Glad you like it!

@iliasp4275 2 года назад

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 2 года назад

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

@user-dz7vb2xq4h 6 месяцев назад

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

@ProfessorMdoesScience 6 месяцев назад

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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-odLwUXKY0Js.html I hope this helps!

@utkarshbhatnagar9106 3 года назад

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

@ProfessorMdoesScience 3 года назад

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...)

@priyankasiwach5066 3 года назад

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 3 года назад

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>

@nkm6275 3 года назад

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 3 года назад

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 3 года назад

@@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 3 года назад

well for at least a time-independent case?

@ProfessorMdoesScience 3 года назад

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 3 года назад

@@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.

@garciadiegos Год назад

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

@ProfessorMdoesScience Год назад

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 Год назад

@@ProfessorMdoesScience Thanks!

@guilhermescorza4412 3 года назад

Muito bom!

@srushtipatil9351 3 года назад

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

@ProfessorMdoesScience 3 года назад

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 3 года назад

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

@nitink4076 Год назад

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 Год назад

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 Год назад

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

@ProfessorMdoesScience Год назад

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: ru-vid.com/group/PL8W2boV7eVflL73N8668N0EQUnID1XaEU I hope this helps!

@nitink4076 Год назад

@@ProfessorMdoesScience Thank you

@user-wj1qb3qu1y 8 месяцев назад

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

@ProfessorMdoesScience 8 месяцев назад

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: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-bg0SMY40q8Q.html I hope this helps!

@subhendumo2619 2 года назад

very well explained, but hearing 'Psi' is weird

@ProfessorMdoesScience 2 года назад

Changed pronunciation in the latest videos...

@1991jee 3 года назад

Now I can die in peace!

@ProfessorMdoesScience 3 года назад

Glad you liked it! :)

@_BhagavadGita 2 года назад

4:00

@MessedUpSystem 3 года назад

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

@ProfessorMdoesScience 3 года назад

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 3 года назад

@@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 2 года назад

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

@ProfessorMdoesScience 2 года назад

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...

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