9. Dirac's Bra and Ket Notation

MIT 8.05 Quantum Physics II, Fall 2013
View the complete course: ocw.mit.edu/8-0...
Instructor: Barton Zwiebach
In this lecture, the professor talked from inner products to bra-kets, projection operators, adjoint of a linear operator, Hermitian and unitary operators, uncertainty of the Hermitian operator, etc.
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@iwonakozlowska6134]

Thank you Mr Zwiebach for the geometrical interpretation of uncertainty. This drawing explains a lot and could be a leitmotif not only for quantum mechanics.

[@frenandin]

Yeah, as an economist myself, that same geometrical interpretation is frequently used to illustrate the decomposition between predictions and residuals in linear regression models (which is obviously a lot simpler than quantum mechanics, but it retains a similar intuition). Obviously there's a lot more to it, but it is always useful coming back to the geometrical interpretation to try and piece together what properties your data gives you.

[@ΜαρίαΜπρέντα-β6κ]

31:18 How can we calculate |x1+x2> where |x1> = 2|y1> + 5|y2> and |x2>= 7|y1> -2|y2> ?

[@coltonboxell1960]

He's like a laser beam @52:40, the way he directed the room lol. The kid was speaking in waving transverse modes, I thought like Bessel Functions, and the prof pointed his like a delta function.

[@raamannair4272]

A more linear algebraic method of approach to Dirac's notation, instead of talking more about wave functions and quantum state

[@ahmedhegazy699]

1:19:00 Hmm, I think that (Cauchy-Schwarz inequality) is the same as (Schwarz inequality). There is no distinction between them.

[@greywall6726]

It's just the naming, there is also Cauchy-schwartz-bunakowski equation

[@zphuo]

@9:56, eigenvalues which have a higher dimension subspace is degeneracy! Good. @31:00, ket |ax> is vector, a notion about states. But "ax" isn't, it means a state that particles at position ax. So a|x> is just amplitude a times state at position x. Is that right?

[@jimmyb998]

yes

[@giannisniper96]

20:00 except for spin 1/2, the hilbert space of qm system is always infinite dimensional, so why not saying them?

[@vavrespaddr7290]

Except for all the higher spin particles you're forgetting about. And spin-0.

[@giannisniper96]

Vavrespa [DDR] higher spin can still be treated with a finite dimensional hilbert space (just bigger than 2 complex dimensions)

[@randomguy7658]

26:19 i visualize U as rotating theta in xy plane, changing basis to another orthonormal basis doesnt change rotation matrix appearance, other ideas?

[@michaelkrusche1760]

Excellent lecture

[@mhlanganofreedomnkalanga6788]

Is there a lecture, for Kets, Bras and Operators? If there's please share

[@not_amanullah]

This is helpful ❤️🤍

[@not_amanullah]

Thanks ❤️🤍

[@압둘하미드이드리스]

How do I understand the difference between bra and ket?

[@MyCreativeGamer]

A bra is the hermitian conjugate of a ket. Meaning just transpose the ket vector then change the sign of the all complex values within the vec

[@omerkzlkanat2103]

@ 1:12:00 good intution

[@rv706]

*Dirac's notation is the most dis-functional notation ever invented by physicists.* Don't believe me? Well, consider that for ψ an element of the Hilbert space H, we have just |ψ> = ψ, which renders the ket of a state vector useless as a notation. The only use is the bra- notation, where where n is an index, or worse | λ> where λ is an eigenvalue (hence a complex number, not an element of H), so now we don't have | λ> = λ. Also |λ> may lead people to think that every eigenvalue necessarily corresponds to a unique eigenvector. All this induces confusion in physicists (or people that don't know linear algebra), making them believe that there's anything deep or mystical about surrounding a symbol with a weird-looking parenthesis |_> . The wrongness of the Dirac notation is not limited to ugliness. It also induces conceptual confusion: when dealing with hermitian/symmetric unbounded operators A that are _not self-adjoint_ (there's plenty of them in quantum mechanics, e.g. the momentum), the bra-ket notation pushes you to believe that it is always possible to "move the operator on the other side" e.g. that is the scalar product of (A being its own adjoint A*), but this is true only if φ is in the domain of A*. It's not just a bureaucratic detail: it can produce wrong results if the computation if done carelessly.

[@MinecraftworksRUS]

The wavefunction ψ(x) is a represention of a vector in particular basis x_1..x_n. Meanwhile |ψ> is basis independent vector.

[@rv706]

​@@MinecraftworksRUS: That's the type of thing a person who doesn't know linear algebra would say.

[@HilbertXVI]

The momentum operator does have a unique self adjoint extension in most situations encountered in QM. What are you talking about? In fact we refer to self-adjoint operators as observables, not merely hermitian/symmetric ones.

[@przygocki]

For easy acess: ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/MIT8_05F13_Chap_04.pdf

[@leiladousti3769]

Hi , Thanks for your videos. Does anyone have Cohen Quantum mechanics solutions ?

[@NazriB]

Lies again? DK Bra Nipple

[@outroutono4937]

ad joint

[@henrique7612]

I have watched almost every class from this professor, only for fun.

[@justinji8634]

you are learning, not having fun.

[@henrique7612]

@@justinji8634 No, I'm doing both at the same time. Can't you?

[@jackbradley4737]

bet you skipped lecture 7 tho lmao

[@psharmacgk]

I'm rewatching this after spending a fair amount of time in the world of tensors, I see a lot of structure that I didn't see before and I'm curious, is there a complete tensor based description of Quantum Mechanical state spaces anywhere that I can jump into?

[@eyepatch1808]

Where can I find more about tensors from the basics

[@lambda2693]

@@eyepatch1808 you can always go with andrew dotson if you want the basics

[@josephwrinn]

You can’t do a complete description of quantum mechanics with just tensors. Tensors come back to themselves when rotated 360 degrees. QM requires spinors that only come back to themselves when rotated 720 degrees.

[@TheMorhaGroup]

@@josephwrinn would that not be specific to a partical of 1/2 spin?

[@josephwrinn]

@@TheMorhaGroup yes. Spinors are used in the description of fermions, which have half integer spin. They are the particles that make up ordinary matter (electrons protons neutrons etc)

[@pamlarauvu]

Is there anyway that I could have access to the notes the professor is referring about in minute ~26?

[@pamlarauvu]

never mind! I found them :)

[@manmathbagali3859]

@@pamlarauvu you still got those notes?

[@pamlarauvu]

@@manmathbagali3859 I was requesting the notes but I never had a reply. Good luck

[@domagojperkovic3829]

@@manmathbagali3859 ocw.mit.edu/courses/physics/8-05-quantum-physics-ii-fall-2013/lecture-notes/ I think they are here.

[@manmathbagali3859]

@@physicalanish Thanks.

[@m_c_8656]

dang, he good

[@i6g7f]

It's not Vishy Anan at the board, so there will be not chess today

[@progra_kun4331]

hi, I know this is related but do you have something explicitly related to quantum computing?

[@deiselia6918]

Será que eu sou a única brasileira aqui desesperada sem entender nada

[@Bravo_Alpha]

Don't worry. I speak English and still don't understand anything.

[@aeroscience9834]

You can barely do anything in quantum mechanics with finite dimensional vector spaces, so why are you ignoring infinite dimensional space?

[@rv706]

There are a lot of things one can do with finite dimensional Hilbert spaces: e.g. spin; and all of quantum information theory.