Bloch's Theorem in Crystals

/ edmundsj
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In this video I sketch out a basic proof of Bloch's theorem in crystals and also talk about where it breaks down and why we might want to use it.
This is part of my graduate series on optoelectronics / photonics, and is based primarily on Coldren's book on Lasers as well as graduate-level coursework I have taken in the EECS department at UC Berkeley.
Hope you found this video helpful, please post in the comments below anything I can do to improve future videos, or suggestions you have for future videos.

[@Cunboss]

"Space is curved anyway, so deal with it" That got real for a second lol.

[@abdulkadirbiffo8937]

Honestly. Let's deal with it!

[@abdulmueenalrashide8236]

it got me laughing for seconds lol

[@wittleliener]

Haha I knew somebody wouldve commented that quote haha :D

[@dheerajkumar2534]

Best explanation of bloch's theorem in YT

[@_The_Alchemist_]

Clear and lucid explanation. Finally understood the Bloch Theorem after so much struggle.

[@husseinh.thebyani508]

It's an amazing explanation, Thanks a lot from Saudi Arabia

[@jaugretler9140]

I have a qm exam tmrw and only learned about this now. Wish me luck

[@dwarapureddijagadeesh5374]

Goooooooood luck You're a lucky person to get wishes from me

[@dddhhj8709]

lmao you learned this is a solid state video 🤣.science is science

[@Shudarsanpoudel]

What was your exam bro? I'm asking after 3 years

[@jaugretler9140]

@@Shudarsanpoudel the subject was called: quantum physics

[@zarintasnim2634]

You explain things quite visually. It's so helpful to understand.. thank you!!

[@PetitCorpsSalade]

Studying for my physical chemistry class, this is actually great! Thanks!!!!

[@taraschutora9166]

The best explanation so far in internet!!!

[@behnammohammadi504]

Very clear explanation. Thank you very much for your time and effort. You helped me a lot and saved my time by this video

[@oliviahoran4525]

Wow, I was struggling so much to understand this bloch theorem but now sir you made me to love 🎉this bloch theorem, thank you so much for your efforts ❤❤❤

[@saifamu1989]

You have no idea how much favor you are doing for explaining these things...

[@amandobhal4264]

Nicely explained. Thank you

[@m.sureshthanjavur9975]

Smart and good explanation about Bloch theorem. I understand the theorem. Thanks a lot

[@lawanyasingh4497]

best explanation of blochs theorem ..very informative

[@saicharanmarrivada5077]

Thank you sir for the wonderful explanation

[@romerofuentero]

Great video! cheers from Spain.

[@alicansaray2940]

Amazing video, thank you very much. Already subscribed by just watching one video.

[@stringsam]

amazing explanation!!!

[@fungi42021]

Great explanation 👍

[@josemariacolingalvez2725]

Bro, thaks for the explanation, it was so cool

[@dexterfang7516]

great video, thanks

[@tcl98387]

The new index at @10:44 should be (2*pi*s)/(N*a)

[@chloe7808]

thanks, it got me confused !

[@Ayah_S_Taihi]

So clear! Thanks :)

Hi Jordan, thank you for your videos, they are very good. Please, I have a question about this one. Just before you introduce u_k(x) function 10:30. There is a step where you rewrite PSI(x) function with index k in the exponent. Where did the " "a" go?

[@JordanEdmundsEECS]

Whoops! k should be equal to 2*pi*s/(N*a), that’s a mistake on my part. k has units of inverse length.

[@biswajeetpatro9990]

Hey Jordan, can you share your mail id, can I contact you for any of my doubts?

[@y3rzhan]

@@JordanEdmundsEECS was looking for the comment, thanks!

[@mausamgupta888]

Thank you so much sir 💞💞

[@emankopeal4210]

Thank you for the amazing explanation (:

[@JordanEdmundsEECS]

You are welcome! I shamelessly steal the best explanations I find from others, of course :)p

[@brotherstech3901]

Enjoyed.

[@delafrog]

Why should the wave function of a particle in a periodic potential be an eigenfunction of the translation operator? (As it was written in 2: 55) How does this derive from the equality of the values of the wave functions for "x "and" x+a"? It is not entirely clear why we should assume namely this relation- with the product, and not with the sum operation, for example? Other issue. Is it really necessary to make a periodic closure of the wave function on the lattice boundaries to prove Bloch's theorem?

[@shanujwilson1204]

A small question. Does this also explain if asked for super lattices? Like those infinite square wells? Thanks in advance. ♥

[@JordanEdmundsEECS]

100% yes sir, you can use it any time you have periodicity.

[@navjotsinghdhiber3454]

Thank you sir.

[@kevinsweeney2809]

Thank you!

[@zacharythatcher7328]

So I understand this, and can see that crystal momentum is just spatial frequency. But I am having a lot of trouble seeing how photons come into the picture. I’m cool with Laue and Bragg, and I’ve seen that there is a classical way to prove that wave number is conserved at the interface with snell’s law. But why and how does all of that relate to band structure? Any good videos covering this?

[@JordanEdmundsEECS]

Yeah so I have a video on "optical band structure" and a bunch of videos on optoelectronics/photonics (you might want to just see the introduction) I made studying for my prelim exam :)p. Photons come into the picture as they are the mediator that lets electrons go from states in the valence band to states in the conduction band (and vice-versa via emission).

[@physicsdiva1871]

Well explained

[@JordanEdmundsEECS]

Thankyou Diva!

[@akashcallofdutymobile8888]

Thank you sir

[@md.rayidhasanmojumder5039]

At 9:28, where did Shi(x) go in the equation of Shi(x+a)? why did you wrote (x+a)/a over the exponential omitting the Shi(x)?

[@jatinkashyap1491]

"Space is curved anyway, SO DEAL WITH THAT" some serious gangsta shades 😎! JK, love it.

[@sherinsaraphilip1869]

Thank you

[@ghezzalimohamed2483]

you are the best

[@maryfucci3131]

At 9:25, when you divide the (x+a) term by a, I don’t see why you only get and x/a left and not (x/a+1). Is it because (x+a) is equal to x? So you’re not actually diving out your just substituting x for the numerator x+a ?

[@coursecczu152]

at 9:22 why we add (x+a) to e ^i 2piS/N, where does it come from? could some one give me a hint?

[@atulbisht5774]

Helo sr nice explanation..I want to ask how u made such video....means device or software..plz reply

[@JordanEdmundsEECS]

iPad Pro/duet Pro on Windows 10 PC with Autodesk Sketchbook for drawing and OBS Studio for recording.

[@beatthekraken7088]

4.46 it is not clearly nonsense, it is the definiton of infinity in complex analysis, where the straight lines equal to circles. Why? Because 2 straights cut themselves in the same amount of points like circles. 1,2 or infinity (The breaking fact is that infitiy is always a point where they cut themselves) This is because e^ikx is non ambigious. Imagine a map on polar coordinates (stereographic projection from the north pole) then every radial path you take to infinity leads you to the north pole. It will help you if you draw the mapping of the square Rx[0,2pi) via exp(x+iy).

[@JordanEdmundsEECS]

Yup, I'm familiar. It's a beautiful definition, and the world of pure math is a bizarre one indeed.

[@randydandy8196]

So k is the crystal momentum and not the wave vector?

[@PresCalvinCoolidge]

I'm confused because we still don't know u_k(x). It seems like we have not gained anything, we just traded not knowing psi(x) for u_k(x).

[@shivammishra6624]

Great

[@TheIndolence]

What is happening when you say at around 9:05 "the e^(i2pi s/N) term has to be a function of x. Say, 'times x, and then we add an a to it". This seems completely arbitrary, and yet it's an essential step in the whole development to arrive at the final Bloch theorem. Why not a function of x in a way more complex way? Does it not work if it does not depend on x?

[@arunbrvce]

Very interesting explanation. I followed you very well up until the point where you established C^N=1 (i.e. C is the nth root of unity). Why do we need a fancy function to satisfy this condition? C=1 itself is a solution isn't it?

[@EspirituDeLaSombra]

trivial solution

[@shayanmoosavi9139]

Because 1 isn't the only solution.

[@pseudolullus]

@@shayanmoosavi9139 That's it

[@mariamhasany5250]

space is curved, is that referring to the Einstein's theory of general relativity?

[@FernandoMartinez-wf4jd]

Thank you for the headache 😂

[@JordanEdmundsEECS]

Ahahahaha 😂😂😂 This made me laugh.

[@nomingirl]

lot of thanks from mongolia ;)))

[@JordanEdmundsEECS]

Thanks! I’ve always wanted to visit Mongolia! XD

[@nomingirl]

Always welcome :))

[@LL-mq7gj]

where did the a term go at 10:34?

[@LL-mq7gj]

using your definition of k, i cannot reproduce your calculation of psi(x+a) at 11:30

[@user-tz3hp6jo9d]

he tried to explain about a point in circle refer first 1min again

[@LL-mq7gj]

@@user-tz3hp6jo9d thanks

[@caleb7799]

just gotta watch them uhms/ahs

[@santalos5]

I dont see the benefit of assuming its a circle? 04:00

[@GBY13]

The explanation from 9:00 is bit chaotic...

[@miro.s]

If we use symmetry, wave function should be identical after shifting the position of some periods, not rotated in complex space. Wave function from principle of equivalence between atoms can not number them. From its perspective all of them are same, there is no other information. This is against what you are showing. You are describing circulant matrix where order matters.

[@miro.s]

The same for electromagnetic potential functions, they are same after shifting some periods. The same in optics with periodic entangled coherent sources of light. These examples are totally against what you claim.

[@miro.s]

You are showing not standing wave that is equivalent to the Mexican wave and present at all cycles, because atoms in that constellation can never reach synchronization all together.

[@black1blade74]

Only the probability density is actually observable in this system so the phase can vary.

[@pseudolullus]

@@black1blade74 Everything is fine unless there is some topological funny crap going on, of course :P

[@pseudolullus]

@miros There is an extra degree of freedom in condensed matter physics which is both important and denied by your treatment. The phase indeed can vary and this is crucial for some effects.