Spinors for Beginners 4: Quantum Spin States (Stern-Gerlach Experiment)

0:00 Introduction + Stern-Gerlach Experiment
3:38 Internal Angular Momentum
5:34 Bra-Ket notation
7:55 State Collapse, Born's Rule
10:00 Z-oriented S.G. Experiment
12:34 X-oriented S.G. Experiment
16:01 Y-oriented S.G. Experiment
18:37 Bloch Sphere, U(2) Matrices
20:44 Global Phase Shifts with Born's Rule, SU(2)
24:10 Conclusion

[kikivoorburg]

Just these four videos have already given me a much better understanding of spinors, quantum spin states, polarization, and even O(n), SO(n), U(n), and SU(n) matrices/groups. You are doing genuinely amazing work!!

[charbinger]

The part where you added the "also called: dual vector, covector ect.." was incredibly helpful. Using different terms in different contexts for what are basically the same general object fucks me up.

[KipIngram]

I do have to say that this is the best treatment of this material I've seen. I mean, it's the same material, of course, but you're just giving a concise clarity that more rare out in the world than one might hope. You clearly put a lot of work into all this - thanks very much for your efforts!

[alejrandom]

Never had I seen such a simple breakdown on the SG experiment

[taibilimunduan]

Incredibly clear explanation, as usual!

[KipIngram]

8:11 - Ok, this is something I feel pretty strongly about. We should stop saying things like "a particle can be in several states." A superposition is not a particle being in multiple states at once. The particle is in ONE state - the one we write |v>. We can *express* that state as a linear combination of basis vectors using any basis we wish - but the particle is not IN any of those states.

Usually this basis is defined by a measurement we're just *thinking about* making at some point in the future, so those basis vectors cannot possibly have any physical significance to the particle *before* we make the measurement. Their physical importance is that they define the set of *possible* states that the particle *may* go into post-measurement. Before the measurement the particle's state is |v> - a single quantum state. After the measurement the particle's state is *one of* those measurement eigenvectors.

This may seem nit-picky, but I think statements like "the electron is in multiple states at the same time" and "the electron is in multiple positions at the same time" are just very confusing to a lot of people and paint an incorrect picture of what's going on.

Ok, I'm getting down off of my soapbox now.

[locusf]

Quantum Sense channel is also making a series on Quantum Mechanics in general, this is an excellent supplement to it and its really nice to see independent work from two really great creators!

[alegian]

great video :) I always skip these sections where the results are super similar for the X, Y, Z axes, but I think its really important to include them in the video, as you do. It's reassuring and clears up misunderstandings 😄

[bobnob]

thank you for these 5 videos. They've really helped clarify some of the math my textbook has thrown at me and will help very much with my quantum computing assignment. All i have to do now is figure out the bell inequality. Can't wait for the rest of this series!

[nthumara]

this vidio is really good it is also gives idea of quntum formalisem

[CarlosRodriguez-mxxy]

Thank you for giving us this chance !!
"You're the top!"

[raulsimon]

Very clear, and at the right pace. Thanks a lot.

[parreiraleonardo]

9:34 Just a quick correction: in linear algebra, the euclidian inner product is not exactly the shadow.
The euclidian inner product (or dot product)
v⃗ • w⃗ is the length of v⃗ times the shadow of w⃗ in v⃗, giving the same result as the product of the length of w⃗ times the shadow of v⃗ in w⃗ .

The shadow itself can be of different sizes depending on v⃗ and w⃗, it is what we call the “component” of a vector along another. The shadow will only be the same size if v⃗ and w⃗ have the same length.

We denote the “component of w⃗ along v⃗” as
comp w⃗ (v⃗) = (v⃗ • w⃗)/ ||w⃗||.

[ytpah]

🎯 Key Takeaways for quick navigation:

00:13 🧲 Quantum spin states are described using the same mathematics as the polarizations of classical light waves.
00:40 🌀 The Stern-Gerlach (SG) experiment involving neutral atoms in an inhomogeneous magnetic field reveals quantized spin angular momentum.
03:18 🪙 The magnetic dipole in silver atoms is due to the electron's internal angular momentum, called spin angular momentum.
05:00 icon Internal angular momentum (Intrinsic magnetic moment)
07:00 icon Bra-ket notation
09:00 icon State "collapse" aka measurement, or projection onto real values via fourier transform
11:00 icon Z-oriented S.G. example
13:00 icon X-oriented S.G. example
14:53 🔀 Quantum superposition is represented as a linear combination of states with probability amplitudes.
17:00 icon Y-oriented S.G. example
18:51 🌐 Quantum spin states can be arranged on a block sphere, similar to polarizations of light on a Poincaré sphere.
21:00 icon
23:52 🔄 SU(2) matrices are used to rotate quantum spin states on the block sphere, taking into account the determinant's phase factor.

[theronsosachavez]

You should give a course, it doesn't matter if you ask some money for it. I certainly pay it. Furthermore you can give a constance.
Your videos and way to explain are amazing 👏

[shaguftanaseem]

Thank you for providing such an awesome understanding.

[manabranjanghosh]

i think before studying quantum mechanics one should really know what even complex no can do inside matrices, if that picture is somewhat clear then i think it really oils up the path towards spookiness,

i am an undergrad student taking q.m. at univ, and boi its really sad that they really dont know how to teach or they just love to hang in the level they they have been taught.

whatever, ur vids(tensor, relativity and this) and many more generous ppl and the effort u guys put in make me feel math and physics are not worthless things to do. much much love to u guys all.

[greenguo]

"just remember quantum superposition is a fancy way of saying linear combination" 🤣

[cmilkau]

26:10 Would've preferred "we CAN set the coefficients", as this choice is convention. You can swap the coefficients of |+x⟩ and |-x⟩ without violating the constraints, and you can multiply all the coefficients with any complex number on the unit circle as well.

[nicolasPi_]

How do we know the states |x> and |y> have to be distinguished? So far the experiments on axis X and Y are symmetrical. And also, why not going further by introducing more states |d0>, |d1>, etc... for any random axis D0, D1, ...? As far as I understand the Stern-Gerlach experiment, the axis of measurement is orthogonal to the trajectory of the particle. How do we know the spin is not related to the particle's trajectory in the first place? Can we measure the spin on an axis parallel to the trajectory or on a non moving particle?