2x2 matrices in terms of Pauli matrices

📚 2x2 matrices are used to represent operators acting on 2-dimensional state spaces such as that of the spin angular momentum of spin 1/2 particles like the electron. In this video, we show that we can write any 2x2 matrix in terms of the identity and Pauli matrices. This provides a convenient language for the description of 2-state quantum systems.

0:00 Introduction
1:27 Pauli matrices
2:09 2x2 matrices in terms of Pauli matrices
10:29 Hermitian 2x2 matrices in terms of Pauli matrices
14:58 Wrap-up

⏮️ BACKGROUND

⏭️ WHAT NEXT?
Spin 1/2: [COMING SOON]
Rabi oscillations: [COMING SOON]

~
Director and writer: BM
Producer and designer: MC

[victorquantum]

I'm currently studying advanced quantum mechanics as part of my master's degree, and your videos have been incredibly helpful to understand and clarify fundamental concepts and mathematical procedures. Seriously, keep on doing this excellent job, greetings from Venezuela! 🔥

[linshuaizhang]

I am a PhD student from China, and my research direction is quantum information. I really like listening to your program, which is very clear. I hope you can stick to it, thank you!

[richardthomas]

Thank you -- even for a basic topic, your clarity and precision is very helpful!

[YossiSirote]

I have now watched all 84 of your videos from beginning to end. They are one of the best set of videos on QM on YouTube. Please keep going. There is so much more to teach and you guys are so good and clear. Please produce many many more. ❤❤❤

[yiheyao]

Thank you Professor M! I've watched ALL your videos and they are super helpful! The reason I love your videos the most is that every step in the equation derivation is super clearly stated, no jump of item cancellation or substitution. Looking forward to the Rabi Oscillation as mentioned at the beginning of this video!

[paulbk]

Please give us more videos in your wonderful interpretations! Eagerly looking for a series on statistical mechanics!

[TopConductor]

Very good and simple intro to QM. Please continue your work!

[cosmotheorist]

I like your videos. I find them useful and overall content applicable in my advanced quantum mechanics course, which I'm working on

[paulbk]

After a long as usual liked the video first and then watched it learned from it

[ruifenghuang]

Hi, do you guys plan on doing a video on 3-D scattering/Born approximation? The math is getting to me, I liked the spherical harmonics video because it helped me understand the mathematics behind it.

[DanielStJohn-hcmh]

Hello, I cherish your contribution. The proof the Pauli Matrices forms a bases on M2(C) maybe a more direct and cleaner approach would be using the inner product < O(i), O(j) > = d(i, j) ? Respectfully.

[paulbk]

Wish to see more from your brilliant channel

[someguy]

Thanks this helped a lot I am in quantum now

[Handelsbilanzdefizit]

As a layman, there's one thing I would like to know. Maybe it's a bit off-topic, but in short:
In geometry, If you shift a vector along x direction and then y, you get the same result as, shift y fist and then x.
So, xy = yx --> xy-yx = 0 --> Flat space
If xy ≠ yx --> xy-yx ≠ 0 --> Curved Space.
Has the commutator of quantummechanical Operators [X, Y]=XY-YX something to do with curved spaces?

[TheHoodedMan]

So if we were to derive that density matrix ρ can be written in terms of the Pauli matrices in the form ρ = (1/2) (σ0 +a·σ), with a : a vector of real coefficients a = (ax, ay, az ) were we to start with the general expression you showed?

[Jagann]

Great video. Any textbooks you would suggest for reference?

[felipevidalekelund]

and what about a generic 2x2 matric that can be written only in a linear combination of the Pauli matrices? Im stuck in an excercise where they give you that and then ask for a basis of orthonormal eigenvectors for that generic one (in relation to the standard hermitian product)

[achalvinod]

Hello profs, it'd be great if you could do a video on the famous Bell Inequalities as well I was surfing through your playlist but couldn't find one :) Best Wishes!

[datho]

Why do the expansion coefficients have to be real?