Lecture 10: Clicker Bonanza and Dirac Notation

[@zepkid5678]

I encourage anyone who is really interested in this to read the online problem sets and solutions. They are much harder than the clicker questions and provide a lot of insight and ability to develop more intuition about the material. Also, I really wish we had the exam solutions.. but MIT already provides such a great service that I'm splitting hairs.

[@bestfilms2646]

Where we can find them ??

[@pranaychorge6888]

@@bestfilms2646 look in the description.

[@sirewan]

Did you ever find the solutions? I'm trying to follow the class along with the assignments and exams. I took the Exam 1 after this lecture and would like to know how I did...

[@kiiometric]

The start of the video is already so wholesome

[@ANUPAM337]

I have reached to this point in this awesome lecture series and I cannot thank Allan Adams enough for finally helping me to understand and enjoy this subject. I wonder why are there not thousands of comments on these lectures where people share new insights and discuss questions. This is such an amazing resource. Thank you MIT

[@rodrigoloza4263]

I am no physicist formally. But, i think that everyone would fall in love with physics with this teacher.

[@chandus2496]

Rodrigo Loza Lucero True that. If I had a professor half as good as this one, I would have been satisfied for life.

[@aleksidragoev5626]

17:32 Yay! prof. Adams likes Python!

[@zacharythatcher3267]

SO. The definition of an eigenfunction is a function that gives the same function multiplied by a constant. When [x,p] acts on a function you get the value of ih in front of that function(particularly the energy eigenfunction). Therefore the energy eigenfunction is not an eigenfunction of both the x operator and the p operator. It is for the p operator. But the x operator gives a function as a coefficient rather than a constant coefficient. And that fact is the main reason that there is uncertainty when determining momentum and location. Correct?

[@bobbyge3626]

not quite. You are correct, though, that the definition of an eigenfunction is a function that, when operated upon by some operator, returns the same function multiplied by a constant. when [x,p] operates on a function - ANY function in x (or in momentum space, really) - it returns that function scaled by ih-bar. this is, however, completely unrelated to energy eigenfunctions being an eigenfunction of the x and p operators. If you wished to find if a function could be a simultaneous eigenfunction of two different operators, you could simply compute the commutator for them. As the professor explained in the video, if you were to take the commutator of [A,B] and find that the end result is zero, then you can assert that A and B have simultaneous eigenfunctions. You seem to have stated that energy eigenfunctions are generally eigenfunctions for the p operator. However, if you take the commutator [E,p], you do NOT get zero generally, since the potential energy term in E - typically dependent on x - need not be zero. The only case in which energy eigenfunctions are also momentum eigenfunctions are for free particles, in which case the eigenfunctions are plane waves. Similarly, if you take the commutator for [E,x], you also generally get a nonzero value, so you are correct in saying that energy eigenfunctions aren't generally eigenfunctions of the x operator; however, there are actually situations in which the x operator acting upon a wave function returns a constant times that wave function. This is when the wave function is a Dirac delta function. Thus, the only times that [E,x] = 0 are for the times in which energy eigenfunctions are Dirac delta functions. I'm not sure when this would ever physically occur. The main reason that there is uncertainty when determining momentum and position has, in fact, almost nothing to do with energy eigenfunctions. The main reason why they obey an uncertainty relation is because they are conjugate variables, meaning they are Fourier transforms of each other. In other words, if we are in position space, we can represent any wave function as the Fourier series of infinitely many superposed momentum eigenfunctions (and vice versa - in momentum space, we can represent any wave function as the Fourier series of infinitely many superposed position eigenfunctions). Momentum eigenfunctions, however, are plane waves, and have no localized position. So, when you more uniquely specify the momentum - i.e. you construct a wave function out of less momentum eigenfunctions and thereby less different momenta - you delocalize your position. Conversely, since a uniquely specified position corresponds to a Dirac delta function in position space, building that out of a series of momentum eigenfunctions (plane waves) requires in fact an infinite number of plane waves, just by the Fourier definition of Dirac delta functions. So, one of the main reasons the uncertainty principle exists between position and momentum is because they are conjugate variables. The same is then true of, for instance, energy and time, which are also conjugate variables. Hopefully that made a mite of sense :P

[@abhisheknuwan320]

Woow to see the lecture halls of MIT..that may be an another university for US..but a dream for south asians.

[@youcanknowanything8489]

tis an awesome privilege for us too🇨🇦

[@Vercongent]

@1:03:41 eigenfunctions must be non-zero

[@elliottpittam9564]

Walter Lewin vs Allan Adams.... Who wins???

[@pmcate2]

@29:50 How is the answer C) when we are literally leaving out the information about u_n. @35:27 To me this says that the answer should have been 'less information'.

[@anandapatmanabhansu]

For determining cn we need psi(x), so I think that's the reason why they contain same information. If we know basis psi(ns), then Psi(x) and Psi(x) expressed as a sum have same information.

[@ardarico]

Where does he touch upon Dirac notation in this video?

[@jasonplayer9047]

He mentions at the end of the lecture that he ran out of time and that he would cover it in the next lecture (but doesn't). Here's another video on the Dirac Notation: kzbin.info/www/bejne/qGOxfop7qLOWitUsi=bteIIImDrCzqa69k&t=1640. Best wishes on your studies!

[@LydellAaron]

25:55 mind blown. The "Hamiltonian" is just the energy operator.

[@gammakay521]

at 39:41, I think INAUDIBLE was supposed to be E_n

[@mitocw]

Thanks for your note! The caption has been updated.

[@gammakay521]

at 25:14, I think the word is "pain", French for bread.

[@mitocw]

Thanks for another caption suggestion! The caption has been updated.

[@gh-by9qj]

23:45 Couldn't we answer this question using the definition of a dagger operator? x/x0+ip/p0, when we apply this to the u(n) we see that there is no proportionality, or am I wrong?

[@fawzyhegab]

Such a fantastic explanation at the end!

[@hakuni123]

In the first clicker question, why is answer C wrong?

[@qianhe3420]

The eigenvalue in the first choice has a wrong sign

[@Amit1994-g9i]

The question is how many, not which one, so II and III are correct which is 2

[@NewYouTubeHandle1]

@@Amit1994-g9i Wow, okay that makes sense b/c both B and C are solutions. Thanks!

[@ShashiBala-u5s]

I don't understand how 2 and 3 are right could you please explain it to me ​@@Amit1994-g9i

[@zindoalbert]

I could find the exam, but couldn't find its solutions. Does someone know if they are available? Thanks

[@mitocw]

Sorry, the solutions to the exams are not available.

[@zindoalbert]

Thank you very much for your attention, it is a great service you provide :)

[@arkyachatterjee718]

Third clicker question, on finite and infinite well.. How does the curvature of the wave function have anything to do with ground state energy? Prof Adams says that the wave function for the infinite potential case has to curve more inside (perfectly reasonable), hence higher energy..

[@consecuencias.imprevistas]

Look at the TISE, there's a second derivative with respect to position for the kinetic energy. The first derivative with respect to position is the slope, the second derivative is the curvature, approximately

[@sirewan]

I took Exam 1 but the solutions are not available on the ocw website. If anyone wants to compare results, let me know.

[@bhagyalakshmi1053]

Craft waterfall, and pink inverter ,definsn

[@sumitparida7993]

Why the first answer is not "c"

[@Amit1994-g9i]

The question is how many, not which one, so II and III are correct which is 2

[@ShashiBala-u5s]

​@@Amit1994-g9ihow are 2and 3 correct and why not no. 4 option is correct???

[@mrpotatohed4]

0 is not an eigenfuction of any operator by definition.

[@anandapatmanabhansu]

Sir at 28:19, for option B, As x tends to infinity, wave function tends to infinity, then how it is possible. Is A the correct answer. There is similar question in griffith

[@alvaropernas6275]

I think the answer is that not all functions with positive second derivative need to go to infty as x goes to infty. Think for instance about a decaying exponential (as is the case for the quantum harmonic oscillator). This family of functions tend to zero as x goes to infty but their curvature is also positive as x goes to infty

[@anandapatmanabhansu]

Thankyou ​@@alvaropernas6275

[@hurtcolor]

32:00 continue!

[@xinzeng-iq7zv]

i am preparing for mid term in community college open course ware on my way to work

[@dnsuperstar]

Are these clicker questions representative for the exam?

[@enisten]

dnsuperstar The prof said at the beginning that the exam questions will be very similar but slightly harder, involving "short calculations". He also said he will probably not prepare a practice exam, so this is it.

[@MHamzaAli]

I can't find lecture notes after notes 8

[@mitocw]

Sorry, those were all the notes we had. They could be unavailable for a number of reasons: instructor didn't supply more, the student hired to do the notes stopped, the files were lost, etc. No note was given in the course as to why there are only the first eight.

[@HT-rq5pi]

i don't get how they always ask heaps of good questions in class and the get split 50 50 on the easy clicker questions

[@lokahithagasthya5936]

Or maybe the questions are what are usually understood as "dumb questions" but Prof. Adams never fails to take every little confusion and connect it to something deeper that brings out the underlying physics and maths.

[@hassan.javaid]

Haha.. I think the same.

[@conoroneill8067]

Maybe the students who ask questions are those who are ahead of the curve in class, and therefore ask really good questions?

[@abhishekkawade6494]

What does Delta mn mean at 15:41

[@quentinmarecat2213]

It is the kronecker symbol. It equal to 1 if n=m, 0 else. Physically, it mean that differents eigenfunctions (corresponding of different eigenvalues), are orthogonales. It is true because these eigenfunctions corresponding to discrete set of eigenvalues. You find almost the same result with a continue set (eigenvalues of p,x) but with Dirac operator.

[@AfterThisShutUp]

There was a great sexual joke to be made @48:21 lol

[@JamesAlanMagician]

Why is the instructor not wearing shoes?

[@shinglau9323]

um....lecture 13 at 4:34 ?

[@mitocw]

+Shing Lau There are a number of reasons that this could happen: the instructor could show slides from later in the course earlier, the number could be a typo (instructors are human after all), some courses have different non-lecture sessions mixed in the numbering (like exams, labs, recitations, etc.) that would make the lecture number seem 'off'. See the course on MIT OpenCourseWare to see the lectures in context: ocw.mit.edu/8-04S13

[@ravikola576]

hey, can you help getting notes of this lectures after lecture 8? as they are not the main site

[@sankarshanharidasan6751]

The summation of eigen functions to represent a wave function contains more information. For example if wave function has zero expectation value of momentum, the summation will show individual eigen values adding upto zero.

[@xephyr417]

oh god i see myself sleeping... thats why i failed this easy test. for shame.

[@dranorter]

Hopefully you have somehow acquired better sleep habits by now :)

[@MinecraftworksRUS]

51:27

[@ikarienator]

Can you ever have a question that is not good?

[@AlexHandle355]

It seems like the answer Adams would give you is No hahaha Adams is a Great Teacher

[@raghulsankar1153]

is math related to science

[@xinzeng-iq7zv]

is he having a baby or he is having his baby taken away

[@bhagyalakshmi1053]

C

[@CHistrue]

Thumb this up if you did not do so well on the quiz. (I got some right but not enough!) There is re-assurance in numbers!

[@alpteknbaser7773]

👍👍👍

[@smrt-e]

Again - opening with an utterly backwards approach to reinforcing, assessing, and fairly assessing the understanding of your students.

[@bhagyalakshmi1053]

Grin full fi ,red fulle definsn diioyra

[@atithi8]

No Indian Student will even dare sit like that in front of a teacher. At 49:06 in the front row

[@АлександрБагмутов]

It is not intrinsicly bad in any way. It is just about unnecessary values communities assign to things in addition to logical respect for emotional states of conscious beings. So all you`ve stated is that appendage of unnecessary values is differnt for India and US. How surprising

[@ashismodak6481]

you are a chutia guy

[@ashismodak6481]

you are right the student should not do like that

[@dranorter]

I'm in the US as well, and from my experience some teachers would allow this and some would not.