9. Dirac's Bra and Ket Notation

MIT 8.05 Quantum Physics II, Fall 2013
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

[iwonakozlowska]

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.

[raamannair]

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

[coltonboxell]

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.

[ahmedhegazy]

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

[mhlanganofreedomnkalanga]

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

[ΜαρίαΜπρέντα-βκ]

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

[rv]

*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 <ψ| in fact denotes the hermitian dual of ψ under the hermitian product <_|_> of H.
But the most awkward thing is perhaps that the ket can be used as a placeholder for _anything_ whatsoever, not just elements of H: we can have |n> 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 <φ|A|ψ> pushes you to believe that it is always possible to "move the operator on the other side" e.g. that <φ|A|ψ> is the scalar product of <φ|A with |ψ> (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.

[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?

[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?

[giannisniper]

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

[henrique]

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

[pamlarauvu]

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

[randomguy]

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

[압둘하미드이드리스]

How do I understand the difference between bra and ket?

[leiladousti]

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

[progra_kun]

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

[deiselia]

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

[igf]

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

[aeroscience]

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