Ch 11: What are unitary operators? | Maths of Quantum Mechanics
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- Опубліковано 26 січ 2023
- Hello!
This is the eleventh chapter in my series "Maths of Quantum Mechanics." In this episode, we'll define unitary operators and understand how they preserve the inner product. We'll then show how unitary transformations preserve probability in quantum mechanics, and why that makes it an incredibly important class of operators.
If you have any questions or comments, shoot me an email at:
quantumsensechannel@gmail.com
Thanks!
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If you make a PDF I will buy it, and probably I will not be the only one. This really associates the formal math and the intuition in a striking way. Well designed, brilliant accomplishment.
6 minutes of joy.
That's what she said
I agree juliangomez8162. This is truly 6 minutes of pure joy.
Analogous to joeltheperv6292's comment, I too experienced 6 minutes of pure ecstacy.
Loving the series, excited for the upcoming episodes!
This is amazing content! You have reminded me back of the time when 3b1b used to post his series on calculus and linear algebra. This is just awesome. What books would recommend me to follow along with this and self study quantum mechanics ?
Leonard Susskind Theoretical minium :)
@@megiddo67 I actually started my journey into quantum mechanics from that book, but it was quite confusing to me starting off from it. However, I can see that after this video series that book would probably be a very good resource to get more comfortable with the notation, because that book was not quite this good at giving intuition but did explain the theorems more rigorously and of course reading gives more time for pondering.
@@SynaTek240I have exactly the same expirienfe with this book, it covers some topics with less focus than others, it’s title is “theoretical minimum” but it depends on knowledge that reader have to have before reading
Your videos are the ones which I will be coming back to see again and again, in present and future.
Amazing series! Very good job. Please continue doing this.
amazing series, keep it up and hope you continue with the amazing content!
I recommend the people who are all beginners for quantum mechanics before watching the video try to read QCQI by Nielsen & Chuang. It might improve your understanding. Thanks for the series Sir.
amazing. easy to understand
3:00 ÛωÛ, what's this?
Da unitawy opewataw ÒwÓ
-QuantumSense^_^
😂
@@quantumsensechannel lmao
Thank you for making this video and saving me from the hell of math midterm!!!!!! This video is amazing, so clear, so understandable.
you're criminally underrated
Great video!
Love it!
The presentations have been quite educational, admittedly. Yet the Unitary Operators concept found immediate applications in several projects currently engaged upon.
Have you covered why we need complex numbers?
Two dims in a scalar adds extra degrees of freedom with convenient ways to turn into real numbers. Plus eulers formula creates a really nice way to multiply complex numbers and decompose those into fourier series, both of which make them super nice to work with. But idk why theyre specifically necessary to represent these ideas physically, i think it was just the convenient convention that held the necessary properties.
Super !!!
Hello, I like your videos for they're reasonable and convincing. I have problem understanding why the general unitary gate in quantum computing has 3 angles to write into. I don't understand what it means. Could you explain it? Thank you.
Tenho uma duvida: A matriz do operador unitário somente possui valores diferente de zero em sua diagonal principal e estes valores serão sempre iguais a 1? Como construir este operador a partir de um Ket?
Just waaw ☺️
The content is really good. Can I use parts of these clips in my educational videos in my regional language?
Hello, thank you so much for the kind words.
And yes absolutely! As long as you give me credit for the clip, I am very happy to have the content distributed in other languages. Thanks for asking.
-QuantumSense
@@quantumsensechannel Thanks a lot, it will be very helpful 😊
I still finds it weird that there could be orthogonal eigenbasis. Like isnt that mean the operator is just scaling the probability of their definite states?
Not quite, if you look you'll see that the value of λ has the property that |λ|^2 =1 that is to say λ = e^iθ where θ is some real number. Thus in its eigenbasis, unitary operators in QM have the property that they can change the complex phase of the wave function. This doesn't have a real effect as the complex phase is not a physical observable and so there is a Guage invariance in this way.
@@justynpryce well i dont mean just the unitary, i mean all observable
Hello! Thank you for watching.
It seems there may be some confusion about observables and probability. You’re correct, an observable would “scale the probability” of its eigenstate, since they aren’t in general unitary. But this isn’t how we calculate probabilities in quantum mechanics. We only use the eigenstates, not the observable itself, to calculate the probability (ie in the form ||^2). Note that the observable doesn’t show up here.
Now I think that begs the question: what do observables actually do then? This will be the topic of next episode, where we review generators in classical physics. We’ll find that each observable has a corresponding transformation associated with it. So for energy this is a time transformation, for momentum it’s a position transformation, for position it’s a momentum transformation, and for angular momentum it’s a rotational transformation. We’ll then use this to find that the action of the energy operator in QM is a time derivative (schrodinger equation), the action of the momentum operator is a position derivative, the action of the position operator is a momentum derivative, etc. These actions in general do not conserve probability, as you say (ie the derivative of the wavefunction doesn’t necessarily have the same total probability as the wavefunction itself).
Hopefully this cleared it up a bit!
-QuantumSense
Which software you're using to make that videos..??
Infamous equation... Yeah, true.
why is left slot of the inner product antilinear?
It's the property of inner products. It's non linear in first term and linear in the second term.
I think there’s an interesting question about whether unitarity is _always_ preserved. It’s considered bedrock in standard QM, but I wonder if when we finally include gravity we’ll find there are exceptions. (And, if so, perhaps that solves the BH information paradox.)
It all makes sense now, that is " Quantum Sense" HA😆.
I think your definition of a unitary operator at 1:46 is incorrect, this is the formula for a Hermitian operator, a unitary operator is where U*=U^{-1}. If I am wrong please correct me I am here to learn.
U^† = U^{-1} = U* The Hermitian conjugate (†) and conjugate transpose (*) refer to the same operation, you might also see it called transjugate or adjoint matrix
@@cillians7407 Thank you so much for this clarification, so is there no way to denote a complex conjugate without a transpose of a matrix. E.g.
[ a+bi, c+di *
e+fi, g+hi ]
=
[ a-bi, c-di
e-fi, g-hi ]
I believe * can refer to complex conjugate without transpose. But for clarity in Quantum mechanics the hermitian conjugate would be used most often & Conjugate transpose (*) is more common in a linear algebra setting
And yes, six hours ago, i probably failed my quantum mechanics exam🙂
Looking on the bright side, in the many worlds interpretation at least some versions of you will have passed.
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