Linearity and nonlinear theories. Schrödinger's equation

MIT 8.04 Quantum Physics I, Spring 2016
View the complete course: ocw.mit.edu/8-04S16
Instructor: Barton Zwiebach
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@StaffordGreen]

I appreciate these lectures being online.

[@atharvakanherkar9502]

And free

[@mrsulaman9901]

I would like to compliment the camera man for his fine work. Also whoever is responsible for recording the sound did a great job. It's so important to be able to hear and see these lectures clearly. My thanks also to MIT for making this content available.

[@stephenanastasi748]

I love this form of explanation. It is so complete. And I love that the facts and reasoning are explained in a human-centric simple form, where so many others throw a bunch of fact at the screen. Thank you! I will use this information in the most powerful way. I have tried to wrap my head around the supposedly simple idea of linearity for a long time. Somehow this shifted me over my preconceptions.

[@mjackstewart]

There are some people who are perfect communicators of complex subjects. My calculus teacher, Martha Kasting, was one such person. She would smile the entire time as she described Green’s Theorem, or she would say, “Isn’t that a pretty equation?” Dr. Zwiebach is another.

[@SanDiego_J]

Thank you MIT OCW and all MIT staff!

[@chrisl3987]

As someone who works in (classical) fluid mechanics, I can confirm that it's very very nonlinear.

[@chuuuu1131]

Can you give an example?

[@frun]

Does it look like electromagnetism? I think he meant classical mechanics.

[@LifeForAiur]

@@chuuuu1131 Not him, but in fluid mechanics, to quantify the deformation of a fluid particle in a continuous medium you need something called a stress tensor, which is a 3 by 3 matrix describing the direction of the stress imposed and on which "face" of the fluid particle it is acting on. Check out the Navier Stokes Equation expanded out.

[@MrMathjordan]

Agreed.

[@user-fb4zo8wd5n]

True. Many applied mathematicians research in the field of fluid mechanics.

[@cidorodrigues6087]

I'm Sido Rodrigues Brazil I really like Quantum Physics Classes. Very important to know quantum physics. Teach everything the universe knows and you gain self-knowledge about everything. Great series of really useful lectures on quantum mechanics. I am also very grateful to MIT OpenCourseWare and Barton Zwiebach... etc...

[@axis_8]

I feel illuminated. A clear and concise lecture by a lecturer who comes across having an authentic passion for learning and understanding. Thank you 🙏

[@homerodaniel_007]

This is actually an excellent class. It worth's its length in Gold. Thank you very much

[@anywallsocket]

First time I’ve ever heard the process of QM explained outright, every professor I’ve ever had on the subject just jumps right in and it’s hard to grip without foundational information.

[@retepredlef5212]

Phantastic lecture!

[@MikeDbean420]

Thank you. Great teacher. Easy to follow.

[@Anb-ng2ou]

How is called this teacher pelase?

[@ZapytajRedditPolska]

@@Anb-ng2ou what are you doing here if you have problems with reading description?

[@nichokind5233]

@@Anb-ng2ou Dr. Barton Zwiebach

[@mohammednour1534]

MUCH THANKS MIT

[@yordygarciamalca3487]

Thank you so much!

[@TheFenny]

Nearly 200,000 views on the first video, and the second video only has a quarter of that? Shame so many people gave up already

[@HighestRank]

Brandon Smith maybe it was just a review which they needed, I myself got this only as a recommendation even though I’m not a subscriber to OCW, and wouldn’t have expected anyone who didn’t realize there is a second part because they weren’t told in the video to go looking for it or to recognize it if it had bit them in the nose.

[@MegaFunkysoul]

They were looking for pseudoscience

[@abrarfaiyaz6503]

Maybe they moved on to the ocw site.

[@ramonasosna]

Great teaching ❤

[@java_Marcelo-xx5nw]

Thank you for share!

[@khaledal-homam6482]

You are great.

[@fujiexia2515]

Very excellent open course on QM, thanks MIT professor!

[@beenishmuazzam]

Thanks

[@karthigamanivannan7922]

thank you MIT AND ALL FACULTIES FOR PROVIDING INTERESTING LECTURES ON QUANTUM MECHANICS...

[@abdulbaqui9499]

Good lecture

[@emersonfranzuaaldanagavarr231]

thank you

[@nikhilgoyal7814]

This professor is amazing.. Though my stream is not linked with this subject but then also i have seen the whole video :)

[@prabudeva2547]

I'm from India.... great thank you mit gives the online courses...little tweak are arises...but most of the phenomenal are not predicted..which means my life time scenario is the one of the examples...some atoms are vibrated...but no losses. After some times the illusion are visible... 🤔

[@deepakkumarravi9217]

Thnx to mit n your staff to spread your valuable contribution in enhancing the concept in worldwide.. Respect n love to you all guys.....

[@antoniolewis1016]

@MITOCW Is this the same room where they did the old 2013 QM course, but renovated??

[@mitocw]

Good eyes! Yes, this is the same room where they did the 2013 version of the course. :)

[@tarunpurohit6522]

What a great intro

[@arushaacharyya6376]

Where and how does the non-linearity get introduced in classical mechanics when quantum mechanics is all linear?

[@geoffrygifari3377]

If schrodinger's equation is linear in any case by default, is it not possible to observe nonlinear behavior in quantum system?

[@i.m.Q.2]

Thanks for confirming something I've been wonderimg about for some time now! You've got no idea what you just helped me out with. 👍😁

[@user-fc3wx7bp4i]

Good

[@123string4]

Why is the Schrodinger equation linear when the Hamiltonian depends on V(x), and earlier he said that V(x) can be arbitrary? The quantum harmonic oscillator is a perfect example of a nonlinear potential and as far as I know you need special techniques like Hermite polynomials to solve it.

[@commonlistener87]

Linearity depends on the dynamical variable you’re solving for. In the example the lecturer presents for Newton’s equations, you are solving a second-order differential equation for the “variable” x (which is a function of time). The equation is nonlinear *with respect to x* whenever V’(x) is nonlinear with respect to x. In the case of Schrödinger’s equation, by contrast, you are solving a partial differential equation for psi (the wavefunction), not for x. You’re right that the Hamiltonian has a potential term V that depends on x (often nonlinearly), but V doesn’t depend on psi, and it’s psi that you’re solving for.

[@FreezerBurn.]

I think I am going to treat myself to hotdogs in my mac and cheese tonight.

[@spencersabet8601]

I respect that. Have fun

[@ProgressiveTeen]

How evil. Torturing animals for your tongue's evil delight.

[@FreezerBurn.]

@@ProgressiveTeen ... kind Sir, sadly you are mistaken. Mac and Cheese is not an animal.

[@Mystic0Dreamer]

@ 9:30 he talks about Schrodinger not knowing what the wave function is. How did Schrodinger come up with this equation in the first place. Professor Zwiebach doesn't offer an explanation of how Schrodinger came up with this equation. But Schrodinger must have had reasons.

[@farahsalam1887]

thanks sir for this amazing lectures but can any one give me the notes of the course please?

[@mitocw]

The lecture notes are available on MIT OpenCourseWare at: ttp://ocw.mit.edu/8-04S16. Best wishes on your studies!

[@abubakarejaz5539]

Hey U a physics student too?

[@gustavodeoliveira702]

In what extent can someone assert that classical mechanics or quantum mechanics is linear or not? Is in regarding to the description of fundamental interactions and not merely idealized models? Because a classical harmonic oscillator is a linear system inside classical mechanics and systems that respect Ginzburg-Landau equation are non linear examples in quantum mechanics. Why those aren't consider counter-examples to the thesis defended in the video?

[@debanujchatterjee2768]

The Hamiltonian operator may contain a potential term. So how is the Hamiltonian always linear?

[@rezokobaidze8501]

hamiltonian has potential energy inside and why it is linear?

[@AbhishekSachans]

Because potential energy is not a function of the 'wave function'- the independent variable in schrodinger's wave equation (or its derivatives); unlike in Newton's equation of motion in which P.E. WAS a function the independent variable(s) e.g. x in general.

[@FredBakker]

Mister Zwiebag, you absolutely rock! Explaining complex stuff simple is a trait of true genius!

[@sagarwadhwani1610]

Can't we use linear qm to solve 3 body problem

[@nayemabdullah7627]

I am from Bangladesh Love Quantum mechanic

[@cedriccappelle2036]

For some reason I keep watching this guy even though I don't understand ßhit of this

[@pmcate2]

Aren't maxwell's equations only linear for some materials?

[@kaushaljain5999]

5:13 how is Hamiltonian operator linear? Since it also contains potential energy term which need not to be linear.

[@zacharythatcher7328]

Kaushal Jain the potential in the Hamiltonian can be thought of as a set of values that span relevant space (a normal line or surface over space) that the wave function will be multiplied by at every single one of those points individually. So the wave equation (the input) will be transformed in essentially a multiplication style operation. Multiplication is linear, and so is the “potential operator”. If the potential was somehow squaring or logging the wave equation, that would be nonlinear, but that is impossible. The potential isn’t that weird. It just multiplies the wave equation by its own predetermined values, which you could do before or after multiplying by a constant and get the same result.

[@aryasingh8173]

@@zacharythatcher7328 wow

[@pranjalsharma3370]

Amazing👍 Can anyone say whether these are graduation or postgraduation classes? Or anything else?

[@pranjalsharma3370]

@pippo Thanks!

[@nsudhir_here]

Can someone explain what is potential V of x? I'm noob in quantum physics. Does it mean a kind of potential which is providing force?

[@bencegabor9228]

Potential V(x) is a classical quantity, whose negative derivative is force. For example: en.wikipedia.org/wiki/Gravitational_potential or en.wikipedia.org/wiki/Electric_potential#Electric_potential_due_to_a_point_charge

[@nsudhir_here]

@@bencegabor9228 thank you sir

[@geoffrygifari3377]

hmmm i guess quantum mechanics is linear because the potential operator is applied ("multiplied") to the wavefunction, instead of the potential being an arbitrary function *of* the wavefunction, as in classical mechanics

[@eternapesadilla2355]

Arent you the dean of the university of architecture in copenhagen?

[@AlexBlade27]

I have a question, isn't Hamiltonian operator also a non linear operator, because it also contains Potential term which may be quadratic or cubic depending on the condition. Thus, isn't then Quantum mechanics also, non linear in nature. Please, explain if I am wrong.

[@sylvenara]

While the potential energy term in the Hamiltonian operator of quantum mechanics can be nonlinear, the dynamics of quantum mechanics are fundamentally described by a linear equation, the Schrödinger equation. Therefore, quantum mechanics is considered a linear theory.

[@AlexBlade27]

@@sylvenara ok understood. Thanks for the help😊

[@gamalf123]

Why can we assume the Hamiltonian is a linear operator? Isn't it another measure of potential, and theoretically could be made some non-linear result?

[@LusidDreaming]

I don't know enough about the Hamiltonian to directly answer this, but in general an operator is linear if it satisfies the following two conditions (O is an operator): 1.) O(f + g) = O(f) + O(g) 2.) O(c*f) = c*O(f)

[@frun]

@@LusidDreaming yes. I think that's the definition.

[@fredrikj8491]

The difference is previously your solution was in terms of x(t) and the potential depends explicitly on x. Now your solution is in terms of the wave function, of which the potential is not a function. The Hamiltonian is a linear operator on the space where the wave function lives. The potential is not a function of your wave function.

[@ericsmith1801]

@@LusidDreaming So there cannot be time compression to satisfy linearity... experiments seem to suggest that in addition to spatial nonlocality there is temporal nonlocality involved in entanglement. I doubt that changing the inertial frame of reference will get rid of such nonlinearity.

[@forheuristiclifeksh7836]

Dynamic quantum variable, wave function

[@hadlevick]

Can you catch the sensation of simultaneity, can you do 1+1...

[@infinity-and-regards]

9:35 How did Schrodinger come up with his equation before there was any physical interpretation for the wave function? What did he try to derive? What was his starting point?

[@durgeshgaikwad741]

When de Broglie proposed the idea of matter waves, Schrödinger tried to find an equation which could describe these matter waves and hence came up with the famous Schrödinger equation

[@lambda2693]

it is quite easy. you just have to prove that what the classical operators become in qm. like p=-ihbar d/dx or E=ihbar del/del t

[@infinity-and-regards]

@Lambda that doesn't sound easy at all, could you elaborate?

[@lambda2693]

@@infinity-and-regards look finding the operators is tough but the derivation of the equation is very easy if you know the operators. okay look i will derive it for you but i will assume the operators if you want the proof for why the operators are equal to what i am assuming you will have to look it up as the proof is very long. E=KINETIC ENERGY + POTENTIAL ENERGY KE= P^2/2M. PE=V(X,) LET US QUANTIZE THIS EΨ=P^2/2M Ψ +VΨ NOW EΨ=Ih/2π dΨ/dt. and p=-ih/2πd/dx Ih/2π dΨ/dt=(-ih/2πd/dx)^2/2m Ψ+VΨ Ih(ΒΑR)dΨ/dt=-h(BAR)^2/2m d^Ψ/dx^2+VΨ AND YOU HAVE DERIVED THE SE. YOU CAN DERIVE ITIN OTHER FORMS BUT THE PROCESS IS SAME. THE REAL CHALLENGE COMES IN PROVING THE ASSUMPTIONS AND YOU TO USE BRA'S AND KET'S FOR THAT. ALTHOUGH THE PROOFS ARE GIVEN IN SOME TEXTBOOKS BUT ARE VERY COMPLEX. EVEN GRIFFITHS DOES NOT GIVE THE PROOF

[@kaushaljain5999]

9:22. Why is one wave function unable to explain both spin up and down state of e-?

[@benwincelberg9684]

Initial conditions aren’t given

[@SarojKumar-lt8qy]

Sir can a wavefuntion determine the dynamics of a macrobody?????or it is just applicable in cases of microbodies

[@farooq8897]

It can.. But Classical Mechanics is a good approximation and easy to use..

[@hadlevick]

Hamilton?

[@ahmedafifkhan]

Can anyone elaborate a bit from @2:20 to @2:33. What did he mean? Where did that graph came from?

[@NoName-vq6cg]

Graph of potential energy over time. (Potential energy meaning the work that force has to do. Force × distance) The derivative is the force acting on it at a specific time. Like if a ball is rolling down a hill, hes basically just saying that because there's mass, gravity would be pulling it down, and it loses potential energy as it gets closer to its destination and force is used. So the force is the negative of the derivative of potential energy.(someone correct me if I'm wrong)

[@surendrakverma555]

Excellent lecture Sir. Thanks 🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏🙏

[@kaushaljain5999]

4:39 Is time is dynamical variable? what is definition of dynamical variable?

[@friendlyone2706]

@@damariscalleros4631 but time changes, sometimes

[@kostasargiris748]

Very good teacher!! I have a question : 3:34 Why x squared is not a linear function?

[@mateusmarinho72]

f(x) = x² f(a + b) = a² + 2ab + b² f (a) + f (b) = a² + b² So f(a+b) is not equal to f(a) + f(b), therefore it's not additive. So it's not linear.

[@kostasargiris748]

@@mateusmarinho72 yes, okey, thank you very much!

[@urpisarmiento5385]

​@@mateusmarinho72muchas gracias, entendí la explicación.

[@forheuristiclifeksh7836]

3:49

[@ZEROCARTOO]

I didnt know i had spedup the videos (to 1.5x) until I read the comments. Oh man, he speaks too slow to the point that his class could be boring. Thankfully it is an online course where you can set the speed, and also for some people that are not familiar with the language and might want to slow it down ;) Greetings from Peru

[@kaushaljain5999]

Explain 0:56 to 1:05 by example

[@hadlevick]

(Reproduction/Feed/Reasoning) decanted selfover hexagon...

[@kaushaljain5999]

explain 0:53 to 1:07 by example

[@brandomiranda6703]

Why is 2nd law for newton not linear wrt Potential? Partial derivatives are linear and so are the normal derivatives...

[@AbhishekSachans]

Because, say for one dimension, x is an independent variable of which potential energy is generally a function so that gives you a non-linear differential equation. That's it!

[@mattmurdock2259]

free knowledge hooray

[@Adam-cn5ib]

why pay when you can not pay? right?

[@ryogakaydc7017]

Ojala algun dia regrese Barton a la Fiee para para una clase de estado solido 🙌

[@RonPaulOrDie]

Whatever it is it's non-linear, and it is the easier explanation. Maybe he says this later Im 30 seconds in.

[@timmy18135]

It is linear iff a linear relationship exists

[@LRahmanGrandUnifiedModelLRahma]

L. Rahman Grand Unified Model

[@riturajanand7133]

sir how force is equal to the derivative of potential...Sir as I know the force is equal to -du/dx (rate of change of potential ENERGY W.R.T X) not potential.....

[@mysteriouspandey3450]

Pehle basics clear karo Bhai baadme quantum ki lectures samjhoge

[@riturajanand7133]

@@mysteriouspandey3450 Thanks sir for your advice, please answer to bta dete doubt ka

[@leonidasloquendero]

Orgullo peruano

[@CharlesSmith-vk8co]

You can aquire all thie knowledge for free.You may even sit down in the lecture und visit all classes and pass the exam.But if you want that piece of paper which says that you did all of that you have to pay thousands of dollars.

[@hadlevick]

The size of simultaneity...

[@hassanbaqer9280]

👍👍👍👍👍👍👌👌👌👌

[@suteguma0]

Can anyone help me understand what the T-like symbol really means in the derivative equation?

[@timetostudy6443]

Yes professor, I found the tutorial irrelevant since I’m no where near being a physician.

[@chandrusekar9575]

Hi

[@ejoe7938]

Where is the teacher from?

[@carloveable1]

He studied at my University in Peru (well known in Peru as UNI) at the same Faculty than me and he finished (Electrical Engineer career) I believe in 1977, then he came to America to follow Master and PhD. degrees.

[@p0lv0jack_]

👁️

[@wassupari2294]

benedict cumberbatch in disguise

[@Shooo117]

Bruh😂

[@sharptongue2972]

Han Solo is now a physicist? Damn...

[@achintyajai2934]

alright he reminds me of dr. peyam

[@SarojKumar-lt8qy]

Sir . I wanted to ask .............we know that a theory has numerous equations in it working all together to state one point .Now if we say that a theory is linear then does it state all the equations of that theory to be linear or there is a possibility for only a few to be linear ???????

[@friendlyone2706]

The moment a non linear factor is introduced, everything affected by that nonlinear factor becomes nonlinear.

[@MS_PrithwirajMaity]

CLASSICAL MECHANICS IS NONLINEAR AND QUANTUM MECHANICS IS LINEAR THEN HOW CLASSICAL MECHANICS IS APPROXIMATON OF QUANTUM MECHANICS.

[@abhijeetbhattacharjee6185]

3:31 What is V(x) ?

[@Inserthandle-ff6jw]

Velocity along the x axis

[@dougiev9287]

Newton's is non-linear because potential could be non-linear function; ok! But Maxwell's is linear...couldn't potentials A and V be non-linear?

[@HighestRank]

dougie v yes, Aa is easy to see, but Vv will always be made using straight lines.

[@czitels1856]

Interesting thing. First video has 2x more views than second :D

[@posthocprior]

Somewhat vague definition of the difference between linear and non linear.

[@schrodingerscat3912]

(steepled hands)

[@uTubeNoITube]

You don't need any of this. Just go to Vegas on weekends and play black jack.

[@Greato_]

Wtf

[@diegofernando4277]

I don't get it, he says that classical mechanics ain't linear because of the potential energy, but the Hamiltonian it's the sum of the kinetic and potential energy, so, how can the classical mechanics be non linear, but the wave function that also depends of V(x, t) be linear?

[@ogoshi]

Yeah, I'm also a little confused by this argument

[@andrewstallard6927]

Notice in the classical equation, m x'''(t)=-V(x'(t)), the potential is a function of the derivative of the position. While the derivative itself is linear, we don't know what the unknown potential function "V" is so we can not say with certainty that V(x'(k*t))=k*V(x'(t)). By contrast, in the Schrodinger equation the potential V is multiplied by the wavefunction psi, so V*k*psi=k*V*psi

[@UnknownBeast41]

The potential function is arbitrary, in most scenarios we approximate it to be harmonic (proportional to x^2) but it can be generally non linear. Alternatively H-hat is the Hamiltonian operator which is basically a constant time the 2nd derivative with respect to position i.e its linear. Its not the Hamiltonian itself, its an operator named after the Hamiltonian.

[@mike4ty4]

The quantum Hamiltonian operator acts on the state variable differently than the potential/kinetic energy (classical Hamiltonian) does in Newtonian mechanics. In particular, "x" in the potential energy function is just a parameter, not the present particle state being plugged into the function like it is in the Newtonian case with Newton's second law or Hamilton's equations, because in quantum mechanics position, momentum, etc. are not actually numbers, but "fuzzy" quantities described by probability distributions (which corresponds to reduced information, as per Shannon), and they are all wrapped up in the linear state vector, |psi>. That state vector is not a real number, but (effectively) an infinite number of them, and hence could not be inserted into the potential energy function anyways, which is expecting only one real number as input. Instead, the potential energy function _becomes_ an operator on the state vector by first considering it in the form of the positional wave function psi_x(x), which is a "basis expansion" (effectively the same thing as vector components of ordinary vectors, but with infinitely many components) and then you multiply this wave function by the potential energy function to get another wave function (i.e. form psi'_x(x) = U(x) psi_x(x)), which then represents, by going backwards through that expansion, the resulting new state vector. Since multiplication is distributive, hence linear, you get a linear action of this operator.

[@mohammadaminmasoomi3597]

I'm from Iran.I love quantum physics and the other parts of physics and absolutely I go to the MIT university.

[@abcdef2069]

QM(quantum mechanics) is linear? where is the laws of physics that says it is, we dont truly know if QM is linear, but the probability representation of psi or ANYTHING is ALWAYS linear, because we MADE it linear so that it could be solvable, this is the ONLY way we could even TRY to solve for the slightest bit. "it is linear" and "we MADE it linear" are two different things. the initial value problems of non-linear classical mech is the SAME as the boundary problems of linear probability representation in quantum mechanics.

[@atmonotes]

for a second there i thought he was Harrison Ford

[@ahmedessam1426]

this continuous montages and cuts through the video made my upset because i want to know everything he says like the real lecture :(

[@justinji431]

2:14 FUNNY

[@ryanyi8900]

I have some doubt about the view that professor mentioned in the lecture about the relation between the many body equation solving difficulty and the linearity of equation.And I think the linearity of shrodinger equation provide us a possible way to solve the superposition problem, which is a one body problem! So, I do not agree the view that the nonlinearity of Newton mechanic equation make three body problem hard to solve.I have not tried to solve many body problem by shrodinger equation or newton mechanic equation, so my point might be wrong.I hope somebody could help me to figure this out.Thank you:)!