Ch 6: What are bras and bra-ket notation? | Maths of Quantum Mechanics

Hello!

This is the sixth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll intuitively understand what the bra is in quantum mechanics, and why we need it. We'll also finally justify the power of bra-ket notation, and its relation to the Riesz representation theorem.

If you have any questions or comments, shoot me an email at:
Thanks!

Animations:
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Music:
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♪ Overworld by Patricia Taxxon
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[quantumsensechannel]

One quick cute tale: if you read Dirac's "Principles of Quantum Mechanics, " in which he widely introduced his notation, he actually makes no mention of the Riesz Representation Theorem. In fact, he says to assume the correspondence between a ket and the corresponding bra. We don't know if he had such mathematical prowess that he felt the Riesz Representation Theorem was obvious enough to assume, or if he simply was not aware of the theorem's importance.

That being said, this is the same man who remained completely silent after a student said, "I don't understand the second equation, " during a lecture. After being asked why Dirac didn't answer the student's question, Dirac said, "that was not a question, that was a statement."

So who knows. He dances all around the Riesz Representation Theorem in his book, so he likely knew about it on some level.

Anyway, hope to see you next episode where we learn how we represent observables as operators in quantum mechanics!

-QuantumSense

[evilotis]

PSA: if anyone wants/needs more material about dual spaces etc, eigenchris's videos are also really good

[HeyKevinYT]

I wish I took Intro to QM a semester later so I could watch your videos side-to-side with my lessons. But better late than never: your series will literally save my career now

[johnwickfromfortnite]

This really reminds me of how little i "got" QM before i took a functional analysis math course. QM is basically just that applied to the Schrödinger Equation :D

[narfwhals]

The Dirac notation makes a lot of connections look so elegant its insane. It took me a while to understand its usefulness. Why do we need a new symbol for a vector? But being coordinate free and able to basically just forget about the basis is so powerful!

For me it's basically tensor algebra without the index juggling.

[alexpapas]

I'm having my QM I exams in a month.... This channel is a miracle before my eyes🔥

[Semispecula]

In love with this series, it is lovely. Well made!

[it]

0:00-Recap
0:38-Introduction
1:35-Linear Functional Definition
2:48-Dual space
4:15-Necessity of linear functionals and bra vector
5:23-Riesz Representation Theorem
7:58-Example for bra-ket notation

[Joel-fszh]

thank you for your handwork and inuitive explanations. have first QM test tomorrow

[AnkitSingh-emit]

Hey, i just want to let you know, thank you for this amazing series. Making it easy and simple to understand . Please keep posting more videos like this !

[thomaspetz]

I am very happy this series exists now. It will help many people with quantum physics, Thank you!

[jarlhamm]

Thank you from a graduate student going from an engineering undergraduate degree to a physics degree 🙏

[gauravprabhu]

Bro this is simply awesome! Thank you so much for this! This video just cleared so many things for me.

[putin_navsegda]

im a math major and enjoying your videos ! who knows probably i will do master studies in physics

[MrCrypto-pvkc]

Not the video I thought I'd get when searching what are bras. and now i know why so few people understand them... You would have to know quantum mechanics.

[zaktoid]

the equation at 9:06 makes me think about the "bi-dual" space, i.e: the space of linear form which takes a linear form as an input.
When we think about it, this space would look exactly like the "original space" (more rigorously, they would be isomorphic)
One example of an element of the bidual space would be Ev_x(f) : V*-> R | f-> f(x), i.e, it takes a linear form and return it's evaluation at input x.
if V=R², and we "indentify" Ev_(e_1) (f):= e_1 and Ev_(e_2) (f):= e_2, we can se that 9:02 as a "go and return" :
The first "bra" goes from V to its dual, and the second does the opposite !
My point is that the "ket" notation makes a vector look likes an "operation" on "something", and it's reminiscent of that bidual space stuff.
PS: thanks for the videos !

[googoogaga_]

THANK YOUUU LEGEND makes so much sense <333

[GuilleEseA]

I have a question from the end of the video. I see that a bra followed by a ket is the inner product braket. But what is a ket followed by a bra? (What is a "ketbra"?)

[LarghettoCantabile]

Multiplying a vector by its transpose yields a matrix. The identity operator is the identity matrix. Each term of the sum has a distinct diagonal element equal to 1.

[davidwright]

I'm a bit confused. At one point is seemed as if the output of a linear functional is a scalar; but then its output was expressed as a row vector - which is not a scalar. It also seemed to me that a linear functional operated on a column vector and produced its transpose, but this isn't true - it selects one component of a column vector and produces a row with the same number of elements, but all the non-selected elements set to zero? Those replacements-by-zero are what prevents the output row vector as being a transpose of the original column, am I right?