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Bloch's Theorem in Crystals

16 768 views | 29 May. 2019


If you want to see more of these videos, or would like to say thanks for this one, the best way you can do that is by becoming a patron - see the link above :). And a huge thank you to all my existing patrons - you make these videos possible.

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.

Zarin Tasnim

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


amazing explanation!!!

Ayah Taihi

So clear! Thanks :)

Josemaría Colín Gálvez

Bro, thaks for the explanation, it was so cool

omansh dhawan

Study bloch theorem in hindi at https://youtu.be/-wiVyKdhwhE

Rye Fungus

Great explanation ?

Jaú Gretler

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

Zachary Thatcher

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?

Atul Bisht

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

Dheeraj Kumar

Best explanation of bloch's theorem in YT

Matyáš Zetek

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?

Prathamesh Deshmukh

Thank you sir

nomin munkhtuya

lot of thanks from mongolia ;)))

Beat The Kraken

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).

dexter fang

great video, thanks

Taras Chutora

The best explanation so far in internet!!!

Randy Dandy

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


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

Aman Dobhal

Nicely explained.
Thank you

Tim Tok

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


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?

Veechi Brothers Tech


Arun B

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?

Lawanya Singh

best explanation of blochs theorem ..very informative


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

Sai Charan Marrivada

Thank you sir for the wonderful explanation

Bhat Saqib

Which book u have been used

Shanu J Wilson

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

Hussein H.thebyani

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

Mind Agony

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

Waves block explorer

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Understanding Waves Blockchain

135 views | 7 Feb. 2020

Thank you for

Thank you for watching.

In this video, you will learn:

1. What Waves is

2. History of Waves Blockchain

3. Waves Development Progress

4. Technological Foundation of Waves Protocol

5. Approaches to the development of the Waves protocol

6. Distinctive Features of the Waves Blockchain

PDF version of video: https://github.com/Wavesafrica/Waves-education-Series/blob/master/Understanding%20Waves.pdf

Connect with us:

Telegram: https://t.me/wavesafrica

Twitter: https://twitter.com/waves_africa

Facebook: https://facebook.com/wavesafrica

Crypto Phrenik

Great summary. I recently discovered waves and am thoroughly impressed. Subbed

Waves block explorer

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Overtones, harmonics and Additive synthesis

384 065 views | 30 Mar. 2010

Overtones are the basic

Overtones are the basic building block of sound. Come and learn how to construct sounds at http://www.synthschool.com

This video is about additive synthesis, and how sounds can be constructed by pure sine waves. If you are a musician or sound engineer, this video is a must for you!

On our site there is also a download for the harmonic explorer, which is the device used to create this video. so you can explore for yourself!

C M33

Very good work, thank you.

Gotex 27

Awesome tutorial, what's the melody in the beginning? Sounds like something by Beethoven. Name eludes me

Lee Bee

ευχαριστώ :)

Berry the Technoman

So basi caly it's fucking awesome

Julian Edwards

i have an essay due in on this tomorrow, ur my fucking hero

Lily Arciniega

I remember watching this video when I was like 14 and barely learning what audio engineering or synthesis was and it helped so much. Thank you for kickstarting my interest in the one thing I'm good at. :p

Justin Martin

sound exists in two dimensions...time and amplitude


Seriously! What a great video! Well explained, demonstrated and easy to follow! Thank you so much!


Another Israeli synth school.

Ferenczi Róbert

Thank You!


Great video.
One bugbear for me: I don't believe there's a good reason to think that the overtone series is "the building block of all music"


If the vertical plane represents amplitude, and the middle line represents zero, how can you hear it when the wave is BELOW the middle line!? You cant hear a sound below amplitude zero.

Hooman Talakian

Great Tutorial. Big Thumb !


hey great video! it sounds like when you hit the 16th harmonic thats an octave? tks!



Teology Music

Thanks for real, another question closed now, a 10 minutes synthesis course.

Simon Kormendy

Also notice how a sound that's rich in harmonics will sound louder to the human ear than a sound that has relatively few harmonics in it.


What is the name of the music?


I watched this when I was 4 or 5, I’m eleven now

Ling talfi

best representation of harmonics I've seen so far. Thanks.


This changed my life

James Kemp

Thanks - that's a really good intro to harmonics - the basis of music - way more useful place to start than whatever they did in school that was just confusing and ungrounded in how the phenomenon works. Thank you thank you.

Josh Nesbitt


Nikhil Paul

I think Sound frequecies have effects on mental health



Rishi Sen

Now I know what Jacob Collier was talking about in music



Riccardo Dell'Orto

what software can analyze the waveform of a sound like the one you are using here?

Mr Dan Master

Why is the frequency not a dimension in sound?

Noah Rosser

THIS is a good video, exactly what I've been looking for. Everything else I've found is riddled with irrelevant information, doesn't tell me anything new, and doesn't answer my questions. Very well done!

Jesse Johnstone

Lovely lesson! Thanks for this

pavanvirat 78

good explanation sir

Noah Barnes

this video makes me want to die



ארז פרביאש

what software you used?

Iggy Wake

This was amazing! I just have one question: were overtones invented by humans? (deciding to add tones with double frequency, and playing them at 1/2, 1/3, 1/4 of volume...) If not, how did we find/discover them?
This is such an amazing topic, when you start to research math and geometry, nature seems so deliberate.

Scott Carey

Wow! I had an epiphanic moment when you compared the osciloscope's positive and negative values to that position of the speaker's membrane! Thank you so much!

John Doe

In the spectrum analyzer, the sine wave is shown to have a peak that tapers off. Since there's only one frequency, shouldn't the sine be represented by a single line? Why would there be activity present in the neighboring frequencies?


Wtf this shit actually makes fucking sense to me???? Thank you so much for this lol. Very thorough and clear

Ivan Alekseev

Great video.

David Edelman

Who are the 18 people who don't like this?!?!? Beautiful tutorial!

Toge Inumaki

1:09 I'm sorry, do You mean to say that the speaker literally moves to and fro within its casing in the Audio System with relation to the Y-Axis?

Aaron P.

Finally, I got a good explanation of sine waves and frequencies. Great video.


But every harmonic have the same or different volume? or can be modified?

Eugene Bogira

Wow! The best explanation ever! Thanks a lot!

Eric Keinrath

Amazing and didactic!

Tom Shea

most excellent --- more more



yes, but time is actually 3 dimensional. See Dewey Larson

NikiWin — Music

i love it

Mars attacks

Excellent and thank you for that great demo !!

sanjai mishra

Thank you sir

Suraj Deshkar

Thanks ,even my english is not good but I got my answers

Berry the Technoman

That was pretty tasty information. I knew about all waves are made from sines but now I finally have seen and understood this. Thanks a lot for the visual information linking to sound.

Simon Kormendy

And they are both present to varying degrees in the sounds we can hear.


What software is this?  Thanks

gil regev

a mathematical point of view: Fourier Series
האם זיהיתי מבטא ישראלי?

Abra Cadavra

It's like I'm preparing for graduation from Singularity School where God teaches it's first Elemental Soul Group how to construct the Cosmos.

David M. Johnston

Question : does phase matter between the principal and harmonic frequencies? What does it change?

Simon Kormendy

Any sound that humankind can hear has three basic qualities, it's pitch or frequency, it's loudness, and it's tone or timbre, the timbre of a sound, is determined by it's fundamental frequency and all harmonics that are present, a note played on a brass instrument, such as a Trombone will sound different when compared to the same exact note played on a woodwind instrument, such as a Clarinet, both instruments may be playing an A 440Hz note, but notice how much brighter and louder the Trombone sounds in comparison to the Clarinet, and also notice how a Sawtooth wave has a similar bright and brassy timbre like the Trombone, or how a Square wave has a hollow timbre like the Clarinet, Sawtooth wave has a mixture of both odd and even harmonics, whereas the Square wave has mainly odd harmonics in it.

Helio Lago Junior

If you put the video velocity on x0.5, looks like you are drunk.


Thank you.

Teacher Kenny Callow

Soooo goooood


Wow. Great Video. I can only mimick what others had said :-)

Rishabh Bhatia

Excellent video! Now I understand why same tone on guitar and piano would sound different despite the same frequency (fundamental frequency) . Thanks.

Howard Anderson

Excellent! Thanks for the solid tutorial, You totally nailed it, many thumbs up

Gabriel Elias

so you're telling me even when you make a square wave... the speakers send out sound waves in the shape of the square ??

Sreenikethan I

I love the explanations!! Finally I know now how sawtooth waves and square waves are generated from sine waves!

Simon Kormendy

This is a great example of Fourier's theorem, and how it can be put into practice.

Celestial Elixir

Hella informative. Tyvm

H z

this is the best explanation ever ! thank you very much sir !

Noah Stolee

This was an amazing video, sums up synthesis concisely, and in a way that makes perfect sense! Made me think about how this is applied in organs, which I never really thought about before, but now it makes sense - they're altering what harmonic frequencies are used in the organ! This will really help with my music, thank you so much!

Official Khastik

1:59 What about triangle waves?

Vasia Patov

This is one of my FAVORITE videos EVER.


Great video! I'm not even a musician, I just fell down the internet vortex of curiosity, and your explanation was very clear!

Space vs Pitch

Such a great video. I would recommend it to all musicians period!

Michał Adamczyk

Human music. I like it


And what's about subharmonics?

Rick Ellis

This is how all tutorials should be done.


So easy to understand, great tuto, thank you so much !


thank you

Dan M

Don't forget the exponentials. Any mathematical function (i.e. audio signal) can be built using combinations of sinusoids and exponentials. Good video, thanks for posting.

Toge Inumaki

4:35 I sincerely thank You for helping Me unlock my Sharingan, I'm serious. Thank You. ?????

Simon Kormendy

Some harmonics will have a "harmonious" sound to them while others will have more of a harsher sound to them, as it turns out the more harmonious sounding ones will be the even harmonics, whereas the harsher ones will be the odd harmonics.


i just hope i haven't been brainwashed :p
nice video!

Rumman Muhammad

the best demo i have ever watched ( relating to the complex nature of the experiment, made dynamically simple) ..best regards to u Sir..

Guido A secas

I don't know if somebody will answer this question, but i'll make it anyway because I couldn't find it anywhere:
Haw a triangle wave is made in additive synthesis?
I hope somebody can help me with this!

Machina Music

Excellent, thanks for this!

Angry Vaginas From Space

7:48 https://youtu.be/-3JCESdFNyw


Hello where can I Download the software for Mac. Thanks great explanation dude!



Its Me 679

I like this so much!!
I wish there was a second like button


Bruuuuuuh wtf. So much makes sense now

Dimity Magnus

Please open the synth school! Keen!


Please analyze Jackie Evancho's voice with this


i still dont understand why those shapes make those sounds scientifically i need that and like how does a speaker vibrate in a square wave surely it goes somewhere in between it cant go instantly up or down like a computer


Very good and easy-to-understand tutorial




What does a wave with only even harmonics look like?


Brilliant tutorial, thanks!