The Quantum Barrier Potential Part 1: Quantum Tunneling
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- เผยแพร่เมื่อ 2 พ.ค. 2021
- Now that we've covered the particle in a box, we are familiar with the concept of a quantum problem. Let's move on to our second quantum problem, that of the quantum barrier potential. With this one, we don't have an infinite square well containing a particle, we have a free particle that encounters a barrier of finite potential. This is a good model for an electron that encounters a defect in a wire. How do we know if the electron will be able to traverse this barrier? As it turns out, even if the energy of the electron is less than the barrier potential, it may still be able to be transmitted due to a phenomenon called quantum tunneling. This is pretty weird, but entirely rooted in sound mathematics, so let's go over the math and learn more about this freaky concept!
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I'm a third-year physics student in college and this is by far the best explanation of QM I've seen. Would love more higher-level physics and math videos from you (Differential Eqns, Schro Eqn in more complex situations (like in an atom), quantum numbers, etc). Thank you so much more producing this stuff, you, Pauls Math Notes, and MIT OpenCourse are the best resources on the internet.
From a struggling physics student, THANK YOU
You have no idea how this material has helped me and my classmates, from the bottom of my cold engineering heart, thank you!
I can't even begin to put into words how much this series has helped me understand quantum mechanics. Don't stop making such videos! People need you!
This is getting deep! No wonder you have to chop this up into so many parts. Thanks for your patience in plowing through the math for us, Professor Dave.
Please...the only thing deep about Dave is, well... everything.
Exactly when I'm thinking and studying QM... thanks for the great work! Keep it up Dave!
Quantam Jesus strikes us again with fax and knowledge
Biggest cliffhanger 2021.
I’m taking theoretical mechanics next semester, this series will definitely be rewatched lmao
Flat earthers be like: I don't see any curvature on that x-axis, you played yourself Dave.
Professor Dave is an exact facsimile of Dave Grohl in the second dimension and in that sublime dimension he impales flat flat-earthers with his drumsticks. It all works out.
@@rogertoaster9385 yay!!! Someone finally realized he is like Dave Grohl :D
Yea..But...?!..😁
Earth is flat!!
@not noot ?
Beautiful video. I COMPLETED MATH FULL PLAYLIST TO UNDERSTAND QUANTUM MECHANICS. PLEASE upload full subject of quantum mechanics so that our species can get a little wiser about our universe. Dave, I am a high school student wanting to know about quantum mechanics. u, thus, came to saviour. proud to be a student of Dave
Greatly explained. Thank you so much, Professor Dave.
2 days to my modern physics exam and in class this seemed an extremely hard topic and I was so afraid that I postponed studying it. Even though it still is not an easy topic, now it is clearer and understandable, thank you, Professor Dave, if it wasn't for you then I would be lost in textbooks and insufficient notes from my professor.
Thanks!
Would it be wrong to assume a spacial symmetry between regions A and B? It would certainly help us out with our calculations.
Another question- Why did you not convert the e^ikx terms to their respective sine and cosine terms? I am slightly confused
Professor Dave, U are FUCKING AWESOME! I absolutely love how calm and cool you are as you debunk the planets dumbest. I envy you, you possess a true talent my friend. Keep keeping on!
Could you explain schrödinger equation for hydrogen atom?
It should be -k_above^2 instead of +k_above^2 at 13:25. You already corrected for the sign by changing (V_ 0 - E) to (E - V_0). This gives the complex exponential solution to the ODE.
Bro Ive been staring at that for the past 10 mins thinking the same thing lol good to see that someone noticed the same thing and I’m not dumb lol
Thank you Prof. Dave !
Thanks sir for explaining this concept 🙏
Sir, could u please tell, what reference books or publications u referred to prepare this video.
Professor Dave u have enlighten me more on dis topic
Thank you Dave :D
Great explanation thank you very much.
Thank you so much professor Dave for the simple explanation, it helped me understand the Schrödinger equation in depths but something is unclear to me and it's the process of calculating the A, B, C,D... "for example if we want to find A. We get a result which is a function of C and D but our teacher said that these values are always a function of the incident wave and I am a little lost in this case because D is not an incident wave " (i am sorry for my bad explanation, I study quantum mechanics in French)
Thank you so much for the video, Im just confused about a few things if anyone is able to answer. First of all, (14:00) the 2nd order differential equation for V>E is almost identical to the 2nd order differential equation for when VE solution complex exponentials?
Hello Dave,
How does a time dependent schrodinger equation change what is stated here? Is it simply treated as an extra dimension?
the time dependent Schrödinger-Equation looks almost like the tie dependent one, but instead of Eψ you have iħ d/dt ψ. This means the particles energy is somehow not steady, but lose energy over time. For example if you particle interact with its surroundings it transfers energy and thus lowers its total energy.
@@Satori_kun oh I see, thanks!
So, a particle with Energy (E) < Potential Barrier (V) can penetrate the barrier. Quantum Mechanics is really spooky!
don't worry though, it doesn't happen super often
“Probably”
@@ferociousfeind8538 It does. Quantum tunneling literally drives the nuclear fusion inside stars. It may soon be a barrier to Moore's law and stagnate the advancement of computers in the near future. Quantum tunneling is everywhere.
@@cinemaclips4497 for a minute I read that as "nuclear fusion inside cars" and was _very_ concerned you misunderstood something.
But... yeah, there are lots of quantum phenomena occurring. I meant, specifically, like, a tennis ball isn't gonna quantum tunnel through your tennis racket, for example
I’ve been waiting for more videos on quantum physics for agessssss
Great video Dave. There were a number of small inaccuracies in the video. Scattering solutions are *not* normalizable, and because of this, it is useful to set the overall scale by taking A=1. By constructing the probability current (which is constant from the Schrödinger equation (it's proportional to the Wronskian of the time independent Schrödinger equation)) we can interpret |A|² as a number flux of incoming particles (and go on to derive |R|²+|T|²=1 for the transmission and reflection probabilities). Also, the coefficients A and B are (relative) amplitudes, and not relative probabilities. The numbers A,B,C,D,F and G are complex numbers, so you get |A|² rather than A² etc. if you need to compute |ψ|² (which you don't really need to do) in your calculations.
A nice way to interpret the solution is to notice that p(exp(ikx))= ℏk exp(ikx) (where p is the momentum operator p=-i ℏ d/dx), so that exp(ikx) is a momentum eigenstate with momentum ℏk, and thus is right-moving if k>0 and left-moving if k
Dunno what that means.. but nicely said 😆
@@gentlyschannel4193 lol
Hey, so ..he said that wave can travel through, just like the WiFi connection. I have a question, that is this a proper analogy? Because WiFi is mainly radio waves and that can travel through walls , because of the large wavelength . Tunneling isn't same as that, I think
@@smitagrangerk4846 correct, tunneling is a different mechanism. though radio waves can also tunnel so… maybe the analogy is more apt than we thought
14:22 Could someone explain to me, how we come to the conclusion that the A term describes movement to the right and the B term movement to the left?
Especially, since we are operating with time independent solutions.
Yes finally more quantum!
Thank you very much for a clear video. Connecting the ideas @9:34 and @10:27 , my question is: can a freely travelling wave be stationary...?
thanks 👏
Thanks for your detail explanation. But can you explain the math a little in section B? Im confused about the sign of the terms (V0-E). Because when u change the terms (V0-E) to (E-V0), the sign before the term should be changed too (i.e. (V0-E) = -(E-V0)). But in the Schrodinger Equation in section B, the sign before terms of the equations inside and outside the barrier are the same (+ve in this case), can you explain a bit about this?
Interesting video but there is something I don't understand. At 11:36 for E < V_0 when writing the Schrödinger equation one can factor both psi(x) so as to have a term in ( V_0 - E ). At 13:03 when E > V_0 the starting Schrödinger equation should be the same and therefore yield the same expression as previously but now we have ( E - V_0 ) in front of psi(x) why is that?
Study about particle in free space and infinite square well and harmonic oscillator you'll get answer
The sign before the 2nd term will simply be negative. Small mistake but it creates great confusion.
Nice Video but i have got a question: how is the wave traveling to the right or left? In quantum mechanics its just a probability wave that is spread out. It coud only "travel" by evolving in time. But we are considering the time independent case so?
Hey, so I think, time dependence or independence doesn't tell anything about how the particle is traveling. Because, wave function of a particle describe the particle physically, which may remain same even whole traveling, like take a ball is traveling at some speed, it remains the same ball. So the wave function is not changing.
But, for time dependent wave function, it means , the particle is changing physically, wave function is changing with time. This can happen even if the particle is not traveling...just changing with time. Hope it explains
k thanks! I guess that makes sense.
Can you do spherical harmonics? Before the end of next week please
Particles will tend to reside in low potential regions. beautiful way of saying
Wow, I don’t understand about wave. could you explain about equation?
Best explaination ever made man❤.you earned my respect from 🇧🇩🇧🇩🇧🇩🇧🇩🇧🇩
Why were you able to just place E-V in the shrodinger equation for a particle above the barrier?
I'm... Early... For the second time in a row! Yes!
Also, I love you Dave and your videos. Good job, well done, keep going! 😊
Edit: your* videos.
Sorry 🙏😅
Isn't A×e^(ik0x) travelling left? And the ones with negative complex coefficients travelling to the right? In the particle in box, you used them in this order, but here the order is reversed. Is there a reason for this or just a slip of the pen?
I'm here early for a change!
I love you Dave. When I graduate and start earning steadily, I will remember to repay my debt to you. Thank you so much
ah, yes, the reason we cant have faster chips
Carry-on..
Bless You...
part 2, please!
At 18:26 and around you've actually flipped si right and si left because right one would be the negative complex exponential and left would be the positive complex exponential.
Well explain...
thank youuuu sir
18:34 Sir how is it mandatory for a continuous function to have a continuous derivative? mod x is not derivable at x=0 but it is continuous at x=0.
i was thinking the same thing. you can take |x| for example and that would be continuous but have a non-continuous differentiation. i think we do know the first derivatives of the psi have to be equal because we know psi is twice-differentiable, and all differentiable functions must be continuous, so psi' is continuous
Great sir
:D i'm gonna go back to my IT studies now...
Just a few hundred hours more and I will say: AHAAA! Thanks! :-D
Thanks
Better taught and explained than profs in IIT
When are you going to do gaming videos
How is it possible for you to be so knowledgeable?
Dave is a national treasure. I'm not even joking or exaggerating.
bless you omg you helped me pass
Faszinierend
Why do we have the second boundary condition say "continuous functions must have continuous derivatives"?
I just got the covid vaccine and I'm very exhausted at 11:30 pm. Not the best time to watch this, but its so interesting to me🤣
I know! Everything that isn't part of your syllabus seems interesting. But as soon as you do have it in your syllabus, the same thing becomes mind numbing
In the region B above the barrier v is zero so k above should be same as k°
great video. thank you. i have to watch it some more. my ADHD is in the way 😕
me right now: I'm watching so my future self won't have to
future me: here we go again
great
18:31 "because continuous functions must have continuous derivatives"
_Weierstrass function has entered the chat_
HOW DO YOU LEARN IT ALL BY YOURSELF ?
Studying, probably
👁👄👁
I need the second half now! I have an exam tomorrow on this, I didn't know I needed this until now!!
Nice
its 2am.. i graduated highschool a year ago.. i am not in university.. what am i doing here-
same i probably wont even study this lmao
You are demonstrating two things - you are naturally curious, and you are brave.
Hope you're studying science in university by now 😁
Aree bhaisahab yeh kis line me aa gaye aap
0:55 I don’t think charges always flow to lower potential.
In the figure shown, if you place a negative test charge, it’ll flow to the positive charged particles, or higher potential, not to the lower potential.
And in an inductor with an alternating voltage and alternating current, for two quarter cycles the charges are flowing from lower potential to higher potential
for a negative test particle, flowing to the positive charges is the lower potential. just because the positive charges are higher up in the diagram doesn’t mean they have more energy.
side note: in quantum mechanics we will often say potential when we mean potential energy (which is what dave was talking about in the video), which a horrible convention as it will inevitably get mixed up with voltage.
@@bobross5716 But my example of an inductor still holds. There are periods of time where the charges are flowing from lower to higher electric potential.
@@altuber99_athlete just to be really clear, when you say potential, are you referring to voltage or potential energy?
@@bobross5716 Neither. I’m referring to electric potential, which is electric potential energy per unit charge.
@@altuber99_athlete electric potential energy per unit charge IS voltage. needless to say, Dave is referring to potential energy in this video.
so we may be talking past each other in some sense.
but i will point out in your example, there is a lag between voltage and current because there is a back emf involved (e.g. a magnetic vector potential) in the inductor. if you incorporate this potential as well, you will find that the charges still flow from high to low total potential.
I hope I have the math knowledge to understand this.
Came here for math to understand how mosfets store data with tunneling effect, now i have more questions than i had before. WHAT!?
Never been this early
James Liar Tour be like "Huh???". This was a long time ago I managed this course. To my memory the video is correct. Note that this is a transistor.
18:28 why must continuous functions have continuous derivatives? Continuous functions need not be differentiable and differentiable functions need not have continuous derivatives
I would be interested in an argument why the derivatives need to be equal too. Please let me know if you have found one.
If the Schrödinger equation applies to every quantum wave function. Then we should be able to take the second derivative of the wave function, which actually implies the continuity of the first-order derivative. Especially in 1-D. But I don't know if it's a valid argument.
I agree that the wave functions should have continuous first derivatives because being twice differentiable implies continuous first derivatives (except at the boundary). My issue is with the argument. I think he does an overall great job explaining all of these technical topics. Having a math background, this argument stuck out to me as wrong even if all the resulting calculations are correct and are justified with a different argument
It's a postulate for the Schrodinger equation. You're right, continuous functions need not be differentiable, but the postulate requires the first order derivative of the equation to be continuous too. That's all
i think you made a mistake in the region above the barrier. if you look at your equation and the value you choose for k above, you would get a real exponential solution. i think the mistake is somewhere in the signs of your differential equation for the region above.
Yep, when he changes (Vo-E) to (E-Vo) he should put a minus sign, therefore its [-y'' - k^2y = 0] so it's a complex exponential.
❤
Wish I knew about this as a kid. Woulda gotten me outta soooo many a$$ whoopins!
thank you physics jesus
as I am getting more familiar with QM, I am not able to understand why the F* ball ain't going through the wall, and not going to believe it without solving the schrodinger's eqn
XD
Wow- I just tuned in to find out if the quantum particle is FLAT.-? Lol
Man if you ever need a kidney, I'll give you mine happily.
it might be better to show to particle moving slower in area B; as it loses kinetic energy to potential energy... its more like an instantineous hill. If the ball rolls fast enough, it rolls over the hill, but on top of the hill the ball rolls slower then at the bottom in A and C. The potential barrier as shown here is just an extremum where the slope of the hill becomes infinite, creating a stepwize energy jump. The ball reflecting with lower energy is then analogue to the ball rolling on the hill, not reaching the top and rolling back. This metaphore is better and does not make any incorrect analogies like you do.
then you also better show how the particle has to borrow energy temporarely. If the barrier is small enough; delta t is small allowing for a larger delta E to tunnel with this borrowed energy. Thats it
Find the resonant
frequency of B.
Planck relics.
Hey guys! ⚡️We just released an interview with Professor Dave on our channel. ⚡️Check it out!! It is super interesting!
👍🏻‼️
What?
Fun fact: Quantum tunneling is the reason behind why the Sun can produce so much energy from nuclear fusion
Thanks Physics Jesus🙂
My quantum chemistry days
Como cuando tienes que aprender ingles si o si, gracias maestro
I just hit a lick with the box
Hah! Doing this in first sem!
Did anyone ever tell you that you look like the Mandarin from Iron Man?
Just say it Dave, the infinite square is ____ as the Earth.