Operators in quantum mechanics 

Yeah these are some of the best explanations on the mathematics of quantum mechanics I have ever seen. Although Professor M is a little fast, I can always just rewind if I miss anything. I would have trouble keeping up in a live class of his though, good thing these are recordings.

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Yes, Best for career building.

Thanks for the praise! :) But Feynman was a truly inspiring teacher and we cannot hope to compare ;)

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Really glad you find them helpful, thanks for your kind comment! :)

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Thanks for your kind words! :)

Thanks for your support! And thanks for the suggestion, we're trying to slow down in the newer videos. However, I also agree with you that RU-vid already provides the tools to cater to all needs: I too slow down/pause videos when I am learning something for the first time, but then go the other way and actually speed them up when I am simply revisiting something I already learned for a quick refresher. Overall, I agree with you that the video format provides a more useful flexibility than a normal lecture.

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You are absolutely right

This is great to hear, and welcome! :)

Glad you find them useful! Do spread the word ;)

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The outer product we discuss here is different from the tensor product needed to describe entangled states. We do actually have a few videos covering tensor products, you can check them out here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-kz3206S2B6Q.html ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-T3ynwXrE0Xw.html I hope this helps!

@@ProfessorMdoesScience Thanks! :D

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Thanks for watching! :)

The quantization rule does indeed simply say that to go from classical to quantum mechanics, we need to promote the classical position and momentum to operators. With this, we can then also deal with derived quantities, like orbital angular momentum. There are some quantities in quantum mechanics, like the spin angular momentum, that do not have a classical analogue, and in that case we have to design the operator from scratch. Electric charge is not an operator in quantum mechanics, it is an intrinsic property of a particle. I hope this helps!

As a simple rule, operators act on kets, and their adjoints (i.e. the dual space equivalent of the operator) act on bras.

@@ProfessorMdoesScience thank you professor for the crisp answer! Love your channel from India

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The term Hilbert space technically refers to an infinite dimensional complex vector space with an inner product. In quantum mechanics, we are interested in both finite dimensional and infinite dimensional vector spaces. While you will often hear the term "Hilbert space" applied to both, in our vides we've decided to use the term "state space" to describe the most general case of both finite and inifinite dimensional spaces. You can find our video on state space here: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-hJoWM9jf0gU.html Regarding eigenvectors (also called eigenstates or eigenkets), they are special vectors in state space associated with a particular operator, such that the action of that operator on them does not change their direction. We cover eigenstates in this video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-p1zg-c1nvwQ.html Overall, we go over the mathematical basis of quantum mechanics in our series on the "postulates of quantum mechanics", which you can follow in order in this playlist: ru-vid.com/group/PL8W2boV7eVfmMcKF-ljTvAJQ2z-vILSxb I hope this helps!

@@ProfessorMdoesScience really helpful!

You are correct. I think in most videos we do say inner product between two kets, but we were clearly not careful enough here, thanks for pointing it out!

The main challenge when working with functions of operators is that some of the usual rules we are used to when working with functions of scalars do not always work because operators don't always commute. For this reason it is typically most useful to work with the power series directly, as then all we have are products of operators, where all the commutation rules become clearer. We have a few videos going over some of these topics, so I encourage you to check them out: ru-vid.com/group/PL8W2boV7eVfnb10T_COKPozxEYzEKDwns In particular, the last video of the playlist covers functions of operators in some detail. I hope this helps!

@@ProfessorMdoesScience Thanks, I got it 😊, Your videos are really very useful, 😊

We are hoping to prepare a video reviewing some of our favourite books, but good ones to get started with include: "Quantum Mechanics" by Cohen-Tannoudji, "Quantum Mechanics" by Merzbacher, "Modern Quantum Mechanics" by Sakurai, or "Principles of Quantum Mechanics" by Shankar. I hope this helps!

@@ProfessorMdoesScience Thanks a lot sir.

Operators act on states by, in general, changing them. If the states are eigenstates of the operator, then acting with the operator does not change those states, it simply multiplies them by the corresponding eigenvalue. If you are interested specifically on the Schrödinger equation, you can find more detail in our video: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-CKpx9hkQ3HM.html

Thank you

If I understand your question correctly, then both expressions should give the same final answer. I hope this helps!

@@ProfessorMdoesScience thank you professor for your sincere effort And what about ,

@@pritamroy3766 You can always calculate this expression, and perhaps a good way to understand what it would look like is to consider the matrix formulation: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-wIwnb1ldYTI.html And we do often deal with non-Hermitian operators in quantum mechanics, for example unitary operators play a very important role: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-baIT6HaaYuQ.html But at the end of the day, what you have to ask yourself is: of the many combinations of operators and states that one can build, which ones are the relevant ones for understanding the behavior of our system? And the postulates of quantum mechanics answer this question, for example telling us that physical observables are associated with Hermitian operators. I hope this helps!

Should have clarified that we do work with linear operators.

What step are you referring to?

We are hoping to do a more introductory series in the future. But for now you are correct, these videos are roughly at the level of second/third year undergraduates who have already had an introductory course in quantum mechanics.

Glad to be helpful!