Spinors for Beginners 5: The Flagpole and Complex Projective Line (CP1)

0:00 Review of Jones Vectors and Quantum States
5:04 Flagpole Visualization of Spinors
10:12 Real Projective Line
13:45 Point at infinity + Opposite points on circle
17:55 Real Projective Plane
18:54 Complex Projective Line
21:33 Summary

[GeorgeClny]

Man this is the first time I feel like I actually understand why the phase can be ignored. The projective line is such a great visual tool to explain this thank you.

[MichaelBlade]

Shouting to eigenbf for enabling those sick geogebra animations

[karimshariff]

Thank-you. Loved the Geobra animation. I will not anymore be afraid when I see the symbol CP^n.

[StevenG]

Amazing video! Thanks for putting this together!
You’ve probably seen this already, but there’s a nice construction that allows you to take the ratio β/α that uniquely defines a spinor and find the point it corresponds to on the Riemann Sphere. It uses stereographic projection, and it’s illustrated on the “Bloch Sphere” Wikipedia page. You basically take a unit sphere with the complex plane cutting through its equator, pick either the North or South pole to be your “point at infinity, ” and draw a line from that pole to the point β/α on the complex plane. Where that line intersects the sphere is precisely the Riemann Sphere representation of that spinor state. Just wanted to share that in case you haven’t seen it, because I found it very appealing when I first came across it!
Keep up the great work! You’re an inspiration to both students and teachers of physics.

[orthoplex]

Thanks for counting out the number of independent parameters. It had taken me a very long time to make sense of it the other day, when I was staring at the Bloch sphere Wikipedia page.

Also, made me smile imagining your boyfriend showing you how to use Geogebra, so thanks for including that note (and thanks to him).

[carbonised_mutton]

Salutations eigenchris!
I had this epiphany after watching your wonderful exposition of spinors:
If we consider sin(x), cos(x) and tan(x) as functions that return the y-coodinate/distance-to-O, x-coordinate/distance-to-O, and the ratio y/x of a point on the plane, it is apparent the tan(x) has the same effect as mapping the point onto the real projective line RP1; whereas sin(x) and cos(x) merely project points onto a unit circle.
This explains why tan(x) has a period of pi radians - exactly half the period of sin(x) and cos(x).
Conversely, the fundamental reason behind the angle-halving properties of spinors is precisely the reason tan(x) has half the period of sin(x) and cos(x).
Do advise me if this line of reasoning holds water.
Thanks again for your wonderful lesson!

[lourencoentrudo]

This was such a good video. Loved the projective space visualization of spinnors it makes a lot of sense!

[netgrok]

EigenChris just became my new homepage. Great work.

[Duskull]

Finally spinors make some sort of sense :)
This series was immensely helpful (2 days before my Quantum Mechanics II exam)
Thanks!

[Wielorybkek]

It's probably my favourite series of videos on YouTube at the moment. :o

[tracyh]

A lovely explanation, and a lovely shoutout to your boyfriend as well. :)

[tomgraupner]

Wonderful video! Thanks a lot, Chris. Please allow one minor remark from me - as a nitpicker: For Quantum Mechanics the phase factor for a single state can be ignored, as you said correctly. But, when you start combining states as superpositions, these phases of each single state do matter. You'll get additional (local) phases and these describe different physical states.

[ez]

Absolute brilliant! This video answered many important questions for me although my university teachers could not. A HUGE thank you!

[KipIngram]

This is *fascinating* - I've never seen these topics brought into discussions of quantum theory before. It makes wonderful sense.

[giorgosg]

I'm currently learning GR and your tensors for beginners saved me! It is hard to find such good teaching information for inexperienced people in this math language.

[_tgwilson_]

This is brillianlty explained. Wow. I can't wait for the rest of this course.

[BiswajitBhattacharjee-upvv]

In your previous video on spinors you have introduced two concept, in case of two sphere alike , they are positive by Nature and in video of spin as spinor in SG experiments outcome oriented in different axis of space orientation an and probabilistic distribution of states an element " i" to make a sense of different dimension.
A member of complex projective line, those spinor a special transformation unknown to to known.

[mastershooter]

Beautiful! Beautiful math and beautiful video!

[dirichlettt]

This video is really helpful! The visualization of the projective line is super clear!

[joshuazeidner]

eigenchris finally finishes this series