Lecture 12: The Dirac Well and Scattering off the Finite Step

[@sonarbangla8711]

Enjoyed this lecture, the most detailed analysis of a wave with classical and quantum effect and also for standing wave. Simply great. I wish I understood it.

[@davidhand9721]

Now that we are done explaining how the wavefunction behaves when bound to defined space, here's why wavefunctions cannot be bound to a defined space.

[@rashidalali6510]

lmao

[@ignaciomartinalliati6293]

3:47 "How surfaces bless you how surfaces reflect"

[@Harry-cs2zr]

I am always concerned by the size of coffee cups in america, literally all lectures i watch some dude has some humungous cup of coffee.

[@charlesnathansmith]

That's the secret to our productivity. Less time spent waiting in line at Starbucks

[@mississippijohnfahey7175]

@@charlesnathansmith the secret to our productivity is almost no one can afford their rent, and a very small portion of the population has nearly all of the money, so most of us spend most of our time working for those rich people in order to earn a tiny chunk of that money, so we can continue to barely pay our rent..... The coffee just makes it slightly bearable.. but that's why we drink such big cups of it :-)

[@SSNewberry]

@@charlesnathansmith We do not use Starbucks.

[@hardikrajpal2410]

Honestly, Mr. Gopal Dixit, I would have very much preferred you recommending this to me yourself.

[@peterpalumbo1963]

RE time mark 30./34 bell curve, you seem to be talking about the big bang and inflation incremental to infinity being a function of probability.

[@anugrahmathewprasad172]

his office must be flooded during office hours

[@K8AG]

No one has commented on the fact this entire lecture was done in socks.

[@sashankkrishna8672]

Thank you for the great lecture! That said, 2:52 Caption corks -> quarks

[@Dr.Sortospino]

Colorado's Group --> 5 Nobel price in 12 years. more than excellent :D

[@tombombadil1868]

It seems to me Colorado is leading in many fields!

[@xinzeng-iq7zv]

i love open courseware, anyone in the world can have an open course degree from MIt

[@atithi8]

(25:18)I have tried to take the norm of the wave function but i am not getting a 2 in the denominator of the time-related factor. Is P(x,t) correctly written. Its not matching with mine

[@andrewandrus3296]

I had the same problem, working on it now

[@vedantkashyap5703]

i think there is a problem with the normalization term, it should be sqrt(2a sqrt(pi))

[@Yash-ML-Sharma]

I don't know if you will read it but I am getting the same answer. He had written the wrong factor in the previous term but was corrected by his students so I think he may have calculated wrong terms using the error in the previous term.

[@prathamlokhande2215]

I don't understand a single thing here bt I can watch him whole day , lol !

[@youcanknowanything8489]

you will learn something anyway...simply through repetition and will build up your vocabulary and capacity to remain optimistically curious

[@ketang7276]

I like this professor

[@physicsperadox7849]

At 17.57 you wrote the solution of that curl psi as. Root a over 4th root pai e raised to something , how u get that pls explain

[@arkyachatterjee718]

11:26 Why is the the minimum uncertainty wave packet for a free particle the gaussian? Also, what uncertainty is being talked about, is it the spatial uncertainty?

[@arkyachatterjee718]

I think the minimum uncertainty he's talking about is delta(x) * delta(p) i.e. the uncertainty product. Incidentally, from signals, we know that for a gaussian, sigma of its fourier transform (also a gaussian) varies proportional to the sigma of the original gaussian. This would be analogous to product of uncertainties in position and momentum being h bar upon 2 (minimum possible value, a la Heisenberg), if correct constants are chosen. Hence minimum uncertainty product is obtained. Also, adding a lot of plane waves can generate any kind of wave, not just gaussians, the final structure of the sum/integral of a large number of plane waves depends on what coefficients you use. Cheers!

[@arkyachatterjee718]

"Also, adding a lot of plane waves can generate any kind of wave" This claim is proved by Fourier convergence theorem.

[@louishamaide53]

Hi Arkya, I was asking myself the same question, so don't worry! There is probably a bit of a divide between us and those who also do recitation etc. at MIT, so things that seem super clear to them, and not yet to us, because there wasn't much emphasis on it during lectures. So the answer to this question can be put a bit clearer than what Chris Metcalf said I believe. Because it can be hard to visualise what the SE solutions are for V=0 easily. Anyway here are two ways of seeing it. Firstly, if you solve the ground state eigenfunction using just the definition of the annihilation operator: a(phi 0) =0 , you will find the solution of the eigenfunction is a gaussian. Secondly, if you look at problem set 2 on MIT's OCW website, you can prove that the Dirac delta function (where there is 0 uncertainty) is the limit when a converges to zero of the following: 1/(a*root(pi)) * e^(-x^2/a^2). So actually Prof. Adams is taking here in his example exactly that, where a is probably considered very small. (The reason a is not equal to 0 is because the Dirac delta function representation of an eigenvalue is never really achieved physically. It is only mathematical, i.e. in the limit.) Finally, yes he is talking about the spatial uncertainty. By combining complex wave functions like Chris Metcalf said, you converge towards 0 uncertainty. Hope this helped!

[@ignaciomartinalliati6293]

Problem set 2, problem 6, WF 8: Δx = a/2 and Δp = hbar/a.This means Δx ·Δp = hbar/2, so the Gaussian wavefunction saturates the Uncertainty Principle i.e. it is a minimum uncertainty wavepacket.

[@renelfcr]

Good day! What thats program used for simulation plot to behavior the wave function and the energy and the probabilities in this class? I'm brasilian guy, so sorry for the english errors within sentences. Thanks!

[@Yash-ML-Sharma]

I am not sure if you are still looking for an answer but that is PhET simulations and you can easily find that on the internet.

[@HT-rq5pi]

At 24:40, I am not getting the time dependent wave function that he is. I think he continued using that extra factor of 2 in the denominator of the power of the exponential which was a mistake that was pointed out earlier. I'm probably wrong though, so can anyone let me know where I could've went wrong when taking the square of the modulus of the wavefunction such that I would end up without that factor of 2 next to the 'm' everywhere in the probability density function?

[@telelight]

When I found the probability density for that wavefunction I had absolutely no factors of 2 anywhere. I think he just got confused with his notes and made an algebra mistake.

[@tedsheridan8725]

I think he made a mistake - I didn't get those factors of 2 either.

[@volcanic3104]

How can we solve the system at 1:10:00 if there are 3 coefficients and 2 equations?

[@entonenton7712]

For Normalization we use A=1 if we send a particle from the left side to the barriere, so D= B+1 and set both expressions equal to each other so we can solve for B and than doing the same with B= D-1.

[@pavlooliynyk3738]

Does anybody know and could explain why A=1 at 01:10:52? Thanks. UPD. A typo was made, there should be A.(look up 13 lecture)

[@pranaychorge6888]

What are they laughing at 56:00?

[@ikarienator]

When was the minimum uncertainty wave function at 11:59 mentioned before?

[@iandsouza86]

+ZHANG Bei Yeah, don't think he mentioned it in these videos. Perhaps they're on the problem sets or may be in the recitation classes that have not been video recorded.

[@ignaciomartinalliati6293]

a little late, but just in case it helps someone else, Problem set 2, problem 6, WF 8: Δx = a/2 and Δp = hbar/a.This means Δx ·Δp = hbar/2, so the Gaussian wavefunction saturates the Uncertainty Principle i.e. it is a minimum uncertainty wavepacket.

[@zuzhichen6400]

@@ignaciomartinalliati6293 Many thanks~

[@not_amanullah]

Thanks 🤍❤️

[@xinzeng-iq7zv]

i prefer a in person lecture and a music minor

[@dannycrofts8138]

Ever since I was young I've been interested in the potential of spheres 😉👍🏽 xxx

[@mintontable]

Especially the anal vibrating type.

[@JohnVKaravitis]

Shit, it's true. He picks up his drink yet does not drink. Schroedingers' Coffee Cup, perhaps? You do and don't drink from it?

[@TheViscweaver]

@ 42:50 phase velocity= ω/k. How can group velocity be= ẟω/ẟk? Logically group velocity should be= ẟ(expected value of x)/ẟt. I tried calculating with Matlab but got an intractable expression. Please comment.

[@Regularsshorts]

If you try the expression you provided the result will be the same taking it as v and p=mv & p=ħk

[@data7traveller]

What was the joke at 55:53, about the tunneling barrier? I don't seem to get it... :/

[@aeroscience9834]

Shreya Ray I think someone said "what" in the audience

[@santiagoarce5672]

Yeah. I know this is like 5 years late but there is no joke. The students were all just bewildered at the idea of quantum tunneling. One of them even said "What!?", which made the others laugh because they could relate to his surprise.

[@data7traveller]

@@santiagoarce5672 Thanks for the explanation! Never too late 😉

[@rashidalali6510]

@@data7traveller the return of the king

[@peterpine15246]

Anyone has solution for challenge at 29:25?

[@aleksidragoev5626]

Why does he leave the exponentials separate, when the base is the same, so he could just add the exponents?

[@ryanyates6761]

it's often clearer to leave certain exponential terms separate

[@rotem1201]

16:09 my guy is not wearing no shoes LOL

[@CHistrue]

I actually have something serious to ask. No jokes today. The question of levels of countability infinities came up in the speech. Is it the case that only the countable infinity can be renormalized? Is the infinity corresponding to the measure of the Set of the Reals non-normalizable? If so, that would seem to make intuitive sense in my mind.

[@CHistrue]

***** Is that mathematically honest? I mean, it obviously works in Physics but I wonder if it plays some havoc with paradoxes.

[@CHistrue]

***** I am familiar with the banach space in a general sense, so I know just enough to be dangerous. Is not a banach space of the same countability as Hilbert space? If I am correct on this, then in what sense do infinities in banach space "cancel"?

[@CHistrue]

***** My thought is that it is workable but not absolutely given. If it works then use it, but any absolute view of physical reality will have to wait a while. In other words, I believe in the Copenhagen Model until something more broad comes along I can jump on board with. It might be String Theory but I have a feeling it will be a bigger theory even than that which includes non-normalized infinities and Set Theory as part of the actual theory itself. What that might be, I do not know and will not speculate. Until then, I accept what is given and taught in Quantum Physics because I have nothing to replace it with.

[@CHistrue]

***** Oh, I am not against it. I just remain open until the full evidence comes one way or the other.

[@dranorter]

1:05:33 Particle/wave duality gets very awkward to talk about sometimes. It seems to me that a "measurable" is a way of thinking about, ie, calculating or conceiving of, a quantum system. Quite literally, a "measurable" is a basis, at least in the sense that the operator corresponding to the measurable provides us with eigenfunctions which serve as a basis. Any wavefunction also has a "native" measurable that it's already expressed in terms of, specifically the position. The intuition that the system is "really a particle" is backed up by the act of measuring that position and finding the particle to be located at a specific place; or, we can measure broader things like "which path did the particle take?" which are still measurables, and fit within the particle-like intuitions we have. And even when we don't measure one of these "particle-like observables", we can think in terms of superpositions of particle-like states, and compute the interference terms from there. Viewing the system as a wave is somewhat broader. In some systems it refers to a specifically very wave-like basis such as the energy eigenfunctions. But at other times it just serves as a catch-all intuition for the fact that interference is happening at all. I guess what I'm trying to say is that right at this point in the class, it begins to show that the philosophical approach was abandoned after the first or second lecture. Once there was enough background to make it clear on empirical grounds why momentum might equal wavelength, the course focused on mathematical tools without worrying so much about epistemological clarity. Right at 1:05:33 the relevant question for me is "why not suppose there's really a rope?" If particles seem to behave roughly like wave packets along a rope, that seems like good evidence that there's a rope they're traveling on. The waves aren't in the "probability" - he misspeaks. The waves are in an underlying complex-valued field. We could think of that field as the "rope". Admittedly the underlying field is expressed in terms of an observable, namely "x", ie, position. So we could say that the wave is traveling through the positional eigenfunctions. The "rope" is the position eigenfunctions. At the same time, though, there are other sets of eigenfunctions which we could use or consider fundamental.

[@Pklrs]

The rope parallelism is very interesting indeed

[@UnforsakenXII]

Spring Break~

[@matsstokes]

Can someone please explain 30:18? I don't understand what he did.

[@KramnikDaniel]

You asked your question a while ago, but in case this is useful to someone else in the future... At x=0. the Gaussian contribution to P(x,t) is always e^0 = 1 (because x^2=0), regardless of time. However, the value of P(0,t) changes because the normalization constant in front of the exponential decreases with time because its denominator increases when t moves away from 0. This is consistent with the probability density moving away from the origin with both forward and reversed time evolution relative to the starting time (t=0).

[@MuhammadShoaib-ng5zq]

where i can find the problem set. any body can help me plz

[@xinwen6926]

at ocw.mit.edu/courses/physics/8-04-quantum-physics-i-spring-2013/

[@saraswatmukherjee7312]

Which software is that..?

[@mitocw]

phet.colorado.edu/en/simulation/quantum-tunneling

[@M_Lopez_3D_Artist]

lol the cell phone alert, its funny if u have on vibrate the professor would never know it, it could ring 10 times and he never no, and yet if you have on volume he would totally be mad about it, its so funny how that works lol

[@dkkim6582]

What is the program used in this lecture? It seems very useful :)

[@mitocw]

+Dongkwon Kim The program used in this lecture is called Mathematica.

[@telelight]

I believe he has in fact referring to the program found here: phet.colorado.edu/en/simulation/bound-states

[@farseer90718]

It's this one phet.colorado.edu/en/simulation/legacy/quantum-tunneling

[@abhirupbhoumik6547]

yo that simulation tho

[@alpteknbaser7773]

💐💐

[@consecuencias.imprevistas]

It's a shame that the probability current isn't explained in this video

[@fredericosevergnini9348]

You can find more information about it reading the problem set (5, I guess) and its solution. There, the probability current is derived and explained in more detail

[@epsilonxyzt]

No knowledge without his notes, boasting and playing theater to recover the lack. Tired to lisen. Pity for the money of MIT, the result will be more playing theater and more boast.

[@jozefkimak3010]

Why do you think he knows nothing without his notes?

[@stevedujmovic6787]

Most lecturers use pre-prepared notes projected onto a screen and READ from these. Allan uses a blackboard and uses his notes on the fly. You're showing ignorant prejudice or you're a Troll.

[@not_amanullah]

This is helpful ❤️🤍

[@youcanknowanything8489]

higher education....on my time...at no charge.....simply marvelous👩‍🦱