Ch 7: How are observables operators? | Maths of Quantum Mechanics

Hello!

This is the seventh chapter in my series "Maths of Quantum Mechanics." In this episode, we'll go over how we represent physical quantities as operators in quantum mechanics. We'll also use our physicist's intuition to derive that observables must have real eigenvalues with an orthonormal eigenbasis.

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

Animations:
All animations created by me within Python, using Manim. To learn more about Manim and to support the community, visit here:

Music:
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♪ Astral 6 by Patricia Taxxon
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[sarveshpadav]

As you mentioned in the video, any physical quantlty (called observables) associated with a particle is represented as a linear operators in QM which implies that 'mass' being a physical quantity should also have an operator representation right?

[vicar]

This is by far the best video series in YouTube for 2023! Thank you!

[Self-Duality]

I’m absolutely loving this series! This is a perfect precursor to tackling the older treatises (e.g., Dirac’s writings) and newer textbooks.

[stephanecouvreur]

I find that “foundations” help me a lot to understand a subject, by explaining how concepts and laws come about from physical intuitions or observations, and how they relate to one another to ensure internal consistency. It doesn’t need to be historically accurate, it is still of great pedagogical value. QM textbooks often skip this part by slamming the reader on the head with 7 or 8 postulates which seem to fall out of the sky (do you know any exception, is your approach inspired by a particular textbook?) Congratulations for taking a different route, I’m impatient to see the rest of the series! 👏👏👏

[divyaanshu]

Best playlist on QM for students jumping to graduate level QM. I highly appreciated the content of this channel because it took me a complete reading of QM book to learn this.

[physicsbutawesome]

I am really intrigued by this channel. Either there are a lot of physics students watching youtube or there is a general audience that is not scared off by maths. Either way, fascinating!

[henrycgs]

I think the biggest lesson I'm learning from this series is how physics is built.

We observe things that don't follow our current understanding of nature.
We try to come up with a model that better explains what we observed.
We then extrapolate the model based on its mathematical properties.
And then we check if the extrapolations also hold true.
If they do, that's a good model. Otherwise, back to step 1.

And sometimes, these models can get pretty unexpected and unintuitive. But if it works, it works.

[stevenschilizzi]

This was one of the key questions I had been asking myself for a long time (and that no textbook ever really explains: “This is the formalism, mate; now shut up and calculate”… and become a donkey-physicist!). This video is quite illuminating. And physics isn’t even my field. So, well done and million thanks. Looking forward to more.

[pixlark]

Something that confuses me greatly is the nature of these quantum states. I think I'm misunderstanding something.

If we represent a particle with a quantum state |ψ>, that state can be constructed from a basis of eigenvectors for some particular property like energy, momentum, etc. So if |ψ> = c_1 * E_1 + c_2 * E_2 + ..., where E_i are possible *energies* that the ψ particle could be measured at, then we've completely encapsulated ψ's quantum state just in terms of its possible energy values.

But what I don't understand is that |ψ> can be expanded into *any* property, not just energy. Does this not imply that if two particles have the same exact quantum state in regards to one property, say energy, that they must then be identical with every other property? But surely that can't be the case - if two particles have the same exact energy, for example, that doesn't mean they have the same exact position, or the same exact momentum. Two electrons for example could be in the same energy state around two different hydrogen atoms.

Obviously I'm confusing something, but I can't figure out what that is...

[SeattleMarc]

These are great videos. I'm already familiar with a lot of the material but I am still learning things seeing the concepts from a unique perspective. Can I request that you make a playlist with all the videos in this series? It makes it easier to play them all end-to-end that way. (And I can also add the playlist to my library, share it out, etc.) Thanks!

[shiroufubuki]

This explanation for the quantum formalism is the best I've seen teacher thank you so much.

[it]

0:00-Recap
0:49-Observables and linear operator definition
3:55-Eigenvalues, eigenstates, and superposition
5:22-Properties of physical observables, proof for eigenstates forming a basis

[amitshoval]

I love this series, thank you
I have a few questions (sorry if they are basic)
I understand that an observable are a linear operator which takes a ket and returns another ket
First, is the observable a function from the space to itself?
Second, what does the returned ket represent, and how do you use it to get the value this observable needs to meassure, like position, momentum, etc.?

[kesleta]

I am confused on how the eigenvectors of an observables associated operator can form a basis for our Hilbert space. For example, I do not understand how we could add several possible measurements for energy and end up with a vector (or ket) that includes information on a specific position. Additionally and relatedly, what do you mean by the notation at 3:00, "| 2.44 N*m*s >"? This seems like it is constructing a ket that contains information only about angular momentum, which seems like is incomplete. Am I incorrect that the kets in the Hilbert space contain _all_ the information about a given particle? Thanks for the really great series!

Additionally, what do these operators actually "do?". If I have a particle a represented by the ket |a>, what does E^ |a> actually represent? Is it a different particle?

[aramsarkisyan]

Great video and amazing series.

I don't yet get why eigenvectors and eigenvalues are used though. If I undestand correctly, we need a linear operaor that takes the quantum state vector and gives us the particular number for one of the observable properties of the particle (say, angular momentum). If we use eigenvectors, then using them as operators to quantum states will give us the same eigenvector times the eigenvalue. What I don't undestand is why those correspond state of the observable and values of it? I feel the intuition that the set of eigenvectors gives some kind of building blocks of the quantum state, that are least affected by the changes and thus can be considered as defining for the state, but it still seems vague.

[timuralmabetov]

This is amazing! Keep doing what you do and you will get your million followers very soon!

[andywong]

@quantumsensechannel Great video series! One question, does the following mentioned 'vector space' actually mean 'Hilbert space' ?
5:50 Observables' eigenstates must span the entire _vector space_
8:44 span the entire _vector space_

[matusmoro]

I am currently in undergrad and even though I haven't had QM yet, your videos are still a joy to watch!

[pjotrstraver]

Amazing series, looking forward to the next chapters!

[JonasBroeBendtsen]

Thank you for this amazing series. I love the style of the videos, and you do such a good job explaining fairly complex subjects!
I hope the rest of the series will come out in time for my quantum mechanics exam in two weeks, as they really help me get som of the concepts i didn't get during lectures or while reading in Griffiths Introduction to Quantum Mechanics.