Quantum Theory Made Easy [2]
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- Опубліковано 26 чер 2024
- PART 1: • Quantum Theory Made Ea...
Today we’ll be exploring the evolution of the atom, starting from J.J. Thomson’s Plum Pudding model, on to Rutherford’s Planetary model, and then on to the Bohr model, which is one of the central focuses of this lecture. After going over the strengths and weaknesses of each, we’ll see how de Broglie’s crazy ideas about the duality of matter manifest themselves in the Bohr model, which we’ll use to (partially) derive the Rydberg equation. Finally, after calculating the Balmer series in order to show off how successful Bohr was, I’ll ruin the moment by explaining why Bohr’s model wasn’t complete after all. Some very important questions will be raised, and the answers… well, the answers will blow your mind in the next episode.
This video is less dense than the last one, and certainly less heavy on the maths than the next one. For those of you who paid attention in high school chemistry, at least half of the content here will be review. If this video seemed a little bit too basic, don’t worry- the next video will be the most difficult one in the series. The goal here was to provide a setup for what’s to come- you’ll probably need to revisit this video a few times, since there are things that I mention here that will be explored in much greater depth later on. Now that all the pieces are in place, though, we can finally get to the bread and butter of quantum mechanics in the next video. Stay tuned.
Don’t forget to like, share, comment, favorite, and subscribe!
Soundtrack: selected works from Stellardrone and Nights Amore
Special thanks to NOVA, Best0fScience, and others (to be specifically credited shortly) for several animations used in this and other videos.
Twitter: / kingcrocoduck
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![Quantum Theory Made Easy {3} [April Fools Edition]](http://i.ytimg.com/vi/llz8WtLv2X0/mqdefault.jpg)
![Quantum Theory Made Easy {3} [April Fools Edition]](/img/tr.png)
"Quantum theory stopped being strange and started to become completely insane"
Oh WOW what a cliff hanger and what a story!
KC, thank you for putting in so much effort, this is quite some journey. I can't wait for the next part!
PS
I hope you'll include Richard Feynman, he's my favorite.
I’m finally back! I know it’s been a couple months, but I’m here again. Between classes, research, graduate school applications, and Graduate Record Exams, it would have been crazy enough to make even an ordinary video, but this series requires an extraordinary amount of time for polishing the scripts, narrating (and re-narrating,) producing, and editing, not to mention double checking all of the equations, facts, and derivations! And STILL I’m bound to make a mistake! But don’t worry, after this core series is produced, I’ll make an ERRATA video correcting all of the errors.
To my dear Patrons, I have opened up a new account that will allow you to support me per video rather than per month, if you are so inclined. The average amount of content this year is about the same as with previous years, though it’s more spaced out due to the dramatic increase in quality and runtimes. Just to give you an idea of where we’re at, the next video will be uploaded in a week and a half, and that’ll just be a shoutout video so I won’t add it to the “per video” feed. I’m expecting to make a slightly shorter video than this 30-minute marathon at the end of the month, and then try to release “Quantum Theory Made Easy [3]” within January. Without all of the crazy grad school preparation going on, that’ll definitely be much easier to accomplish!
Thank you all for being so patient and just all around awesome. If you have any questions, ask them here in the comments section or here: ask.fm/KingCrocoduck
Hit me up on Twitter: twitter.com/KingCrocoduck
Patreons:
www.patreon.com/user?u=581251 (per month)
www.patreon.com/KingCrocoduck?ty=h (per video)
If you liked the video, like, comment, share, and subscribe! Spread the knowledge, ask questions, teach others, learn!
+King Crocoduck I subbed on patreon a few months ago and was getting frustrated by your lack of activity. then I remembered you're a student doing real science. take as much time as you need, bro.
+King Crocoduck Aaaah, ended at the PERFECT cliffhanger! I had goosbumps when you said "they were... Half-correct". I'm loving this series KC, I hope to see more. I hope you can crank up the amount of videos you release, because these are AMAZING!
+King Crocoduck
Is it a north American thing to say "shined" rather than shone?
Good stuff KC.
+King Crocoduck
Fine work, KC!
+King Crocoduck Very good!
How can someone dislike this? It's like a whole school year in one 35 minutes video.
"How can someone dislike this? It's like a whole school year in one 35 minutes video."
A watcher who dislikes (imagine in a moron, hick, religious zealot, or speech impeded voice): Huh? This sounds like school! Lots of school words! No cats! Nothing I can jerk to! Nothing I can jam! DISLIKE!
1:30 "The outgoing light produces parallel lines" - this is not an explanation of anything that's actually happening! That's why.
EXACTLY! And it's all lies and bullshit. School is not to educate you, but to pump out employees for the billionaires! It's all "memorize the bullshit". No real thinking at all. Just like a year of school is RIGHT!
@@frankdimeglio8216YOU'RE A LEGEND (IN YOUR OWN MIND)!
@@frankdimeglio8216 bullshit. yawn.
I really enjoyed parts (1) and (2). I would love to see part (3) as soon as you are able to create it. So far, very compact, with a lot of information that's easy to understand in a short period of time. My compliments.
I really REALLY love the way you show us the equations as a result of a theory and experiment, and not the other way, I don't know if anyone else agree with me on this but, that's the reason why I'm remembering so many topics that I saw on highschool but I wasn't capable to grasp. I even read books on quantum mechanics (the one you recommend on your "So you want a degree on physics" video description) and I'm capable to understand so many things just by watching your videos. Now everything (not quantum mechanics itself xD) makes sense. You're a really good teacher and put the topics in a manner that makes everything easier. Thank you for this and I will spread the knowledge
Apparently the act of observing part 3 makes its location on UA-cam simultaneously impossible to determine.
I love these videos. They're a breath of fresh air on UA-cam.
OMG, this is the first explanation of Quantum Theory that my poor simple monkey brain can wrap around, please, please, please continue!
The poor kitty in the last scene seems to be saying, "why are you looking at me like that Erwin?" Not aware he is about to become the most famous cat in physics history.
sorry dude but this isn't quantum theory but just physics history with very very early beginnings of quantum physics
It's ALL bullshit for them who are too stupid to think anything besides what your teachers tell you. What a joke!
The "poor kitty" knew ...
The Lord went to human and said: "Go and rule over the world!"
Human: "Wit will be my lever to the world!"
Then Lord went to cat and said: "Go and rule over the human!"
Cat: "Insanity will be my lever to the human mind!"
this is probably the best video series about quantum theory. it's not dumbed down (which at this point in my learning turns me off to pretty much all of the nova media) but not math-y enough that never having taken physics or anything past pre-calc is a hindrance for me (like the susskind stamford videos). it's a concise, thorough, lucid explanation of the most interesting thing in existence. cannot wait for future installments. thank you for making it.
Alex Ruimy it is spectacularly dumbed down. The entire content of these two videos can be summed up in the first few pages of a quantum mechanics book and is taught as standard at AS level physics.
I agree. The historical process has to be understood to even begin to understand. Still, when does a theory finally get to be considered as science? Is it really only when a theory is agreed upon by some community of persons as being the most likely physical proof? And how do mere theories being used to be built upon over time get to become reality or info worthy of being considered premice?
CORRECTION: I just noticed that in the test of the Rydberg equation where I reproduced the Balmer series, I placed 2 in the wrong denominator- it should have gone in the other one. The numerical answer is still correct, but the sign is a little awkward. I'll nevertheless add this to the ERRATA video when this series concludes.
+King Crocoduck
I saw that. Chemistry uses that equation a lot for some reason. Nice video.
+Inorganic Vegan It's used primarily to calculate the first ionization energies for hydrogen-like atoms (e.g. the alkali metals, which can be approximated by considering the lone valence electron to orbit a "nucleus" composed of the actual nucleus and all the lower electrons which screen the positive charge). You can do that by plugging in your ground level to n_L (n_L=2, 3, 4 for Li, Na, K respectively) and infinity for n_U.
StarFox008
Cool.
Why not just modify the original video?
Jeremy Morton
Because by the time it was caught the video was already up, reuploading a minor modification would consume time and bandwidth unnecessarily.
34:10
“... quantum mechanics stopped being strange,...”
Me: Ok, somebody’s off their meds.
“... and started to become completely insane!“
Me: Ah, that’s more like it!
FUCK we need the last part !!!! plz
there'll be several more parts don't worry
+King Crocoduck YES
WHEN? SOON?
These are absolutely the most intelligible and best-made introductory explanations of the development of quantum physics - we SOOOOO need part 3!
OMG WHERE'S PART 3.?? noooooooo. this was the best series I've found so far. thank you so much for this.
Odd that its not obvious that time and its constituents are our source of Gods attention.
ua-cam.com/video/llz8WtLv2X0/v-deo.html
Part 3 is here now
@@mathnerd97 it only been 3 years. Think I'll pass bruh.
This is the best series of videos about QM I've ever seen, anywhere. These videos are absolutely priceless, they cover every nook and cranny, every problem and explain the "why", all with a dash of very important history and math. Keep up the best work!
I taught electron optics for 35 years. I find the light and electron similarities fascinating.
Well written and produced explanation.
What can a hobbyist study to get a fundamentals? Especially how things are measured.
34:14 Hi Schrödinger! Keep it up man, I'm loving this series. The saddest part about choosing to become an engineer is that I only get to take a relatively few physics courses.
+MrPhilsterable Trust me working with Schrodinger's equation is not fun at all. One of the 'simple' problems I had to solve took 3 pages.
The series is awesome indeed, in this video KC condensed a semester's worth of physics classes into 30 minutes.
+MrPhilsterable hhahahahahahaha I think exactly the same thing, that's why I'm currently pursuing a physics degree
+MrPhilsterable He picked an appropriate animal. Cats are insane - cuddly one minute, fierce furballs of rage the next.
God damn it. Why do I know understand so much about what you are talking about? Either my high school education was awesome, you can learn a lot by watching documentaries or both.
Well done again. From a neophyte's perspective, you've taken what is often given in lectures as a series of facts and theories, but here you've employed an interesting narrative format, which takes the viewer from one complex point to another, where one does not need to fully understand any given point, but intuits as one progresses. Alternatively, the complexity remains sufficient to create a range of understanding-levels.
This video makes complex scientific subjects look simple.
Fantastic. I understand everything you said in terms of comprehension, I only wish I had your dedication to get to understand this in terms of knowledge. Can't wait for part 3.
+Daryl Soar
Well KC summarized a semester's worth of classes and if the people watching understand the concepts I'd say that's job well done.
Really the only thing you should need to understand how the equations are implemented is solving some problems.
it's fun to know what makes it tick, not any use for me to calculate and verify or apply it. Same as learning how mechanical engine works... doesn't mean I need or know how to build working one :_)
When do we get part 3? this series is awesome! I want to hear more!
I thought I knew a thing or two about quantum mechanics but this guy glued all that knowledge to really help me understand what it's talking about. Hurry up with part 3 please.
Man, please make the part 3. How many times can I watch part 1 and 2 on and on?
At least a dozen more times. That's what I do.
Amazing video. I am a layperson interested in science and many things have been clarified for me here.
Some points have confused me and will require further thinking by me.
Thank you for a very informative video series.
i neeeed a third video this is so interesting
georgie I agree
Most original content I have seen on this. Honestly, the inclusion of the credit role is so original i might influence others to adopt it. VERY impressed with everything here. Consistent voice no studder, Words flow with the music and the animation, perfectly... you.. ARE.. AWESOME!!
15 years back I struggled hard to understand them. Now all of sudden it brings tears to my eyes. Thanks a lot King Crocoduck.
2019 has started and no part 3 yet, come'on man!
Dammit! I really want to see the rest of this series...
I want to say a huge thank you to you for putting these so far two series's on. Even as non total non- scientist, I find I can understand what's being shown, I find quantum physics very interesting indeed; thanks again.
One of the best educational videos I've seen, and definitely the best about quantum physics. Looking forward to number 3. Thank you!
1 person's brain exploded at the amazing information and pressed the wrong thumb :-p
hahahaha it seems
+cr0nosphere
There are two thumbs or one equation describing two thumbs
but then the equation must collapse and there is one thumb
and the thumb is dead
and spca is asking inconvenient questions
Their knocking at my door
+cr0nosphere the mere act of observing the like to dislike ratio forced there to be a certain outcome, before observing, there were both upvotes and downvotes together, and upvotes without downvotes, and downvotes without upvotes, and neither downvotes nor upvotes
any cat that down votes this is either alive our is dead. I won't open the box and find out which is true, I'll hope his energy state has achieved zero ;)
233 trolls and-or exploded brains now.
WHERE'S PART III DAMMIT
yes where the f*** is iiiit!!! need my dose!
Part III has not been discovered yet lol
It's coming this Christmas
The very act of observing or measuring part III changes it and so it can't exist when you are trying to see it.
would we wait part 3 like we've waited for Half Life 3? (justgamingthings)
4:28 "This was known as the Plum pudding Model. This model had tremendous appeal because it sounds delicious, served warm, with a scoop of vanilla icecream."
I like the metaphors and comparisons with every-day like situations or objects: it really facilitates understanding. Make your videos and its dense content accessible to people with no background in Physics, nice!!
A few more pauses at times would help getting things in but it is probably because I am a foreigner
I hope that You will find time and money to continue to make this amazing series, instead of criticizing and bashing Creationists. They're stupid, and I doubt You'd convince anyone. Unfortunately it seems that some bashing is required in USA, where so many Christians are in some extreme sects. I'm Catholic, and I don't see a problem with Universe being approx. 14 billion years old.
Also, You do a great job at explaining, not only giving us foundations, but later also equations, and real life examples of calculations done on this equations - it is just the right amount. Perfect - better than Leonard Susskind, even better than more famous science UA-camrs, like minutephyics, Veritasium, Vsauce etc. And that is not small feat. You should be proud. And I'm not saying that L. Susskind, or those UA-camrs are bad. They're great. But THIS series is PERFECT. History, pacing, everything.
"Easy" being a _relative_ term ;)
Thanks immensely for this series of videos. You're making quantum theory - something many people are fascinated by but have difficulty grasping - much more accessible to those of us less mathematically inclined folk. I very eagerly, yet patiently, await the third video. Take care and best wishes to you and your educational endeavors.
are you ever going to complete this series? i would love to see it!.
yes
Part 3 ... 😒
Great videos! Is there a part 3?
there will be soon
Looking forward to it!
Yay!
Indeed this series is fantastic! Totally got me hooked tonight, but when I saw the third part hasn't been released almost a year after the second one, I couldn't hide my disappointment. Then I considered helping you a bit on Patreon, but as you haven't posted any news about that series there, I didn't know if you were still considering finishing it.
Indeed although I don't disagree with you on your favorite topic that seems to be debunking creationism, I am only interested in that marvel of educative science and entertainment you've started about quantum physics. I will gladly support that series if you make it a priority! Seriously you should do, you are very talented for it!
Can't wait!
Brilliantly produced and narrated.
Really enjoy how you root your narrative in a historical context. It is much easier to retain the concepts this way. Plus you really know what you're talking about. Brilliant. Thanks for these gem videos worthy of being included in any time capsule.
I absolutely loved these two videos, I watched them back to back and I found that you not only refreshed my knowledge of physics but you've actually helped me bridge the gaps in my relatively basic knowledge of quantum mechanics. Thanks, looking forward to the next video :)
dat ending
+Gego/XAREN ゲゴザレン It produced in me a mix of laugh, tears and goosebumps.
+Gego/XAREN ゲゴザレン
A cat in a box - lolz!
Damn you Schrodinger!!!!
+rataflechera Felt the same.
+Gego/XAREN ゲゴザレン I think it schrodinger's cat
Hi Crockduck. A small correction .. at 03:55 you say "most of the mass of the electron". supposed to be "... of the atom".
Other than that, As always - great video!
goddammit I thought I fixed that -____- will annotate.
These are some fantastic videos, probably a more accessible take on this subject than I've ever seen before. Really looking forward to the next in the series!
This was great! The amount of detail was perfect for my level of understanding, so I really feel that I learnt a lot.
Subscribed! Where is part 3?
Coming this Christmas
I can't wait. You're videos are awesome! Thank you so much!
Tom Stanway-Mayers *Your. It's a possesive pronoun.
King Crocoduck And thank you for the christmas present.
The Internet Quite right -> typo
TThis is much simpler than I expected. The media really makes this stuff look like magic.
I don't think that a regular joe would understand the first part, let alone the second part that wasn't easy to follow even for little me who is interested at this science as a hobby, let alone the third part which probably will melt my brain within the first five minutes.
I really don't think that even as simplified as it is here every single person can understand the science.
The observations themselves are simple enough. It's when you think deeply about them that you get confused.
After several years in grad school, I've managed to distill my lack of understanding of quantum mechanics to one aspect: what is called "the measurement problem". But that is a very hard one. It's not magic, but it's hard.
Somebody once said anyone who claims to understand quantum theory doesn't understand quantum theory. The math is pretty enigmatic to a layman, but I feel like it really boils down to the famous double slit experiment. I think if you understand that, you're not doing too bad, unless you happen to, in fact, be in grad school for the subject. Cause really, this is just in depth explanation of that. The next step is to actually try to learn the math
Having only gone through the engineering physics required courses, I've always had difficulty with quantum physics. This was so much clearer than most videos. It actually clarified a point I never realized. The concept that the electron wavelength was in integer steps in order to wrap to the same point in its orbit. Did I say that correctly?
That was the original motivation for some quantum ideas, bu what turns out to happen is a little more complicated than that. An electron wave is three-dimensional, whereas the original argument by De Broglie only works in 2 dimensions, right? You have a circle and imagine superimposing a sine wave on it. That was his argument. Well, take the lowest energy state of the hydrogen atom, for example: De Broglie's model predicts that the electron would move on a flat plane. But experiment shows that the ground state of hydrogen is spherically symmetric.
De Broglie's argument wasn't really a "first principles" kind of thing. It was a guess that meant to explain the (then mysterious) Bohr model. Nobody believed the Bohr model, you know: not even Bohr. It was just a guess that seemed to kindasorta work and hint and some more fundamental physics, and that it did.
States of the hydrogen atom can be characterized in terms of three numbers: the energy of the state, the total angular momentum, and the component of the angular momentum along a specific direction in space (you choose which). You can't say anything more, though (for example, you can't specify _another_ component of the angular momentum).
The two angular momentum numbers describe how the wave function behaves "around" the atom. A spherically symmetric state always has zero angular momentum, for example. De Broglie's idea was intended for the total energy, but actually it's closest to a description of how the component of angular momentum in a certain direction works. That one is described by a wave that has to "fit" around the atom. But it turns out that the total energy doesn't depend on the angular momentum, to a good approximation, so De Broglie's idea was a more or less correct argument for an entirely different bit of physics.
Hey, thanks for the hard work doing these videos. You should continue this series. They have a good balance of history, explanation and rhythm.
25:48: "... This is why you don't diffract when you walk through a doorway". Classic!
At 4.00, "electron" needs to be changed to "atom"
Very attentive
4:00 * why do intelligent people insist on injecting their own national idiomatic expressions into other tribes where it’s not just useless but has different meaning? It’s idiotic.
Great! Waiting for the last (?) part...
ehhhhh not quite, we've still got a few more videos ahead of us:)
+King Crocoduck can't wait!! how soon can we expect part 3?
Alex Ruimy I'm producing it right now; unfortunately, this will be the hardest one to make since it has the most advanced material, and breaking it down is challenging. I'm definitely on it though
+King Crocoduck really looking forward, man. thanks for your great work. i watch this series at least once a week.
+King Crocoduck Nicely done. Eager for part 3 and hoping there will be more after that. :D
This is just an amazing series, thank you so much for making this.
I can't begin to explain how wonderful it feels to understand this
You are amazing, keep it up!
Always wait so long for a KC video, but the quality makes it so worth it! Good job KC, please keep them coming.
Part 3?
Still no part [3]?
You have a talent for this, sir! Your explanations are concise, the imagery fits perfectly, your narration is excellent! You're clearly on the path to being the next Sagan, Kaku, or Greene. Benevolently, gently sharing the wonders of physics with those that just don't know.
These videos was my introduction to quantum theory. Simplification of a very complex subject with clarity; amazing. For me (average man) to be able to understand up until Rydberg Equation is remarkable. Will need to rewatch. This has even helped me understand in depth how Spectrometer works. Thanks for creating these videos.
Where is the Part 3, I am waiting for the holly cat of Schrodinger
there was a part one? shit better go watch that...
These videos are awesome! Thanks so much! But, what the hell, where is Part 3?! You left us up in the air at the point where it gets the most interesting!!
"this is why you dont diffract when you walk through a doorway and thank goodness that you dont"
lost my shit at this
The more I study quantum theory the more I am humbled by The Great Designer! How arrogant are we, his creatures....
Who’s watching this in 2019?
I am
Me
Same here.
1st 1 was gud 2 . . . .
Heyoooo
Excellent movie. Great graphics, the pace was just right for me. Perfect sound and pronunciation and of course great content! Thank you very much!
This is really nicely done. I loved every minute of it even though I've already learned beginner QM. I can't wait to see what comes next. Congratulations on putting together two of the best videos on this topic I have ever seen.
I heard that link to the part 3 will be on the last page of Trump's tax returns.
That is a good THANKS very much for the laugh
And just what dose a person like you know about tax returns....Get a real job first then tell us all about tax returns. After that you will become a Republican.
@@twhelwig5062 Nope, wasn't there, just like the taxes...
I didn't understand a thing, f this, I'm going back to religion.
+Gabriel Incognito 9gag.com/gag/127175/too-stupid-to-understand-science-try-religion
+Gabriel Incognito You could also try feminism. Much like with religion it involves starting with an answer and then finding or inventing "evidence" that matches. Much easier than that science stuff.
lol
Please do number 3!
Your narrative is the best I've read, in terms of communicating the concept. This is the closest I've come to getting a full picture of what is known about quantum physics in my head. Great summarisation.
Brilliant explanations... We all certainly wish you would continue this series. Any chance of that?
Please consider making a part 3. And thank you for this amazing work.
So much information, but yet it's so easily digestible. You're awesome dude.
Even though I know these topics by heart, it was wonderful to watch your video. The way you put everything together was so good. If I were a teacher, I would make this video a part of the Syllabus.
The first 2 were great and I hope you get the time to finish the series some day. Thank you for making these.
Excellent, a pleasure to watch, informative and I am looking forward to part three.
And starting to become "completely insane". I am looking forward to the next video!
Part 3. I am waiting for it. Tons of thanks for the first 2 of the series!
This series seamlessly integrates video and speech at a speed that I can absorb. Some other excellent presenters go so fast that I experience a mental blur. Thank you very much.
So awesome KC. I was just re-watching episode 1 last night and wondering when the next one would be released. Cheers!
Waiting for part 3. This the best explanation of Quantum Theory on the UA-cam.
Love it!
Great work KC! That is what *real* physics looks like.
Your videos are absolutely brilliant. The work you’ve done for these is a great service to the public.
I don't mind waiting for part 3, I just hope that it exists one day xx
Love love love this series. Clarifies what I half knew. Keep them up.
Please finish the series!!! I'm taking the second semester of physical chemistry now and your videos helped me understand the concepts better, now if only the series was complete.
Amazing as always. I love both your debunking videos and the physics lessons. Thank you!!
Really well done! I was reading "A Brief History of Time" and turned to these videos to help better understand the concepts. These are by far the best explainer videos I've found. Thanks!
Excited to see the 3rd video.
Hi! I just wanted to tell you I enjoyed this video so much. It's in my favorite list. I hope you continue posting such great material.
King thank you so much. You have enlightened me no end. These videos are by far the best that I have seen
explaining the Quantum world. Your knowledge seems to have no bounds.
Where is part 3?? Bring it up fast , please. Can't wait now!!!
I just barely understand the "simple" descriptions but this is so beautyful! It must have been such a wonderful feeling when the discoveries were made, when everything lined up.
I have been waiting for this :3
Thank you King Crocoduck!
Yeah, finally. Although not understanding every bit, especially the formula's and stuff, this kind of video's really make me understand QM a little bit more each time. Keep up the good work, I'm eager of part III
Simply brilliant! Everything is crystal clear and straight to the point in an entertaining way
I can only repeat what I've posted you under part 1: really, really great work! Very helpful animations, going back to the basics and origins in the history. Also saying, that some things are important but won't be covert till one of the last videos is very important.
Funny thing is we are now an hour further in trying to grasp quantum theory, but rather than talking about about the implications and how it (could) work you're teaching us where it comes from. Which in turn lets me understand for the frist what is really going on and how all those different experiments and theories are connected and have evolved.
I'm pretty sure you're an excellent professor, as you're able to explain quite complex matter to a non-physicist like me.
I'll defenitly wait for the insane part of the series and take advantage of the time and do one or two re-watches ;)
Keep it up!
That was so good that even my secondary school brain understood that pls make more!
thanks uploader..it was awsome and much better..pls upload 3rd part..im subscribing
These 2 lectures were what i needed...thanks for clarifying so clearly