The Schrödinger equation

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📚 The Schrödinger equation governs the time evolution of quantum states, just like Newton's second law of motion governs the time evolution of classical states. The key term in the Schrödinger equation is the Hamiltonian, the operator that describes the total energy of the system. In this video we discuss time evolution in quantum mechanics, and solve the Schrödinger equation for the special case of conservative systems in which the Hamiltonian is time independent.
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⏮️ BACKGROUND
Dirac notation in state space: • Dirac notation: state ...
Representations: • Representations in qua...
Eigenvalues and eigenstates: • Eigenvalues and eigens...
Measurements | Concepts: • Measurements in quantu...
State collapse: • State collapse in quan...
⏭️ WHAT NEXT?
What happens after a quantum measurement?: • What happens after a q...
Density operator for pure states: • Density operator for p...
Time evolution operator: • The time evolution ope...
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Director and writer: BM
Producer and designer: MC

[@nomanahmadkhan7791]

Completed first playlist "postulate of quantum mechanics". From "nowhere in Q.M." to "feeling at home" is what this playlist has done. Thank you so much for your effort!

[@ProfessorMdoesScience]

Glad you liked it! :) A good one to consider next is the one on "angular momentum", and then the one on the "quantum harmonic oscillator" :)

[@nomanahmadkhan7791]

@@ProfessorMdoesScience Sure, I am looking forward to it!

[@TheWingEmpire]

Rigorous as it may be, quantum mechanics is really beautiful, and it's beauty is enhanced by the way you explain it.

[@ProfessorMdoesScience]

Thanks for your kind words! :)

[@iotaphysics909]

this is so interesting that i keep on watching without skipping

[@ProfessorMdoesScience]

Thanks for your support! :)

[@aesopw6324331415926]

your videos are amazing I have watched many youtube lecture playlists on QM in the past, some of which is perhaps at the postgraduate level but I never felt so crystal clear about the concepts, before I discovered your channel

[@ProfessorMdoesScience]

Great to hear, thanks for watching! :)

[@arisfluid]

Great videos, great channel! Keep up the good work! Would be nice to add a video where you show how Schrondiger "constructed" this equation from scratch (like if we were inside Schrondiger's head), so the audience could understand the simplity that governs this equation but unravels so much about the state of a system!

[@ProfessorMdoesScience]

Glad you like the videos, and thanks for the suggestion!

[@hassanchhaiba154]

This is awsome. Thank you so much. I hope that you continue this great videos.

[@ProfessorMdoesScience]

Thanks for your support!

[@dcrespin]

I have seen half a dozen videos from this channel and all are excellent, this one is outstanding. Yours are the very best in KZbin about Quantum Mechanics. Three hurrah!s for rigorous Quantum Mechanics. My respects. However... As you start explaining after 2:00, the linear Schrödinger Time Dependent Equation (STDE) is a postulate non deducible from other principles and is fully deterministic. The STDE: 1.- Has no stationary state (fixed point, singularity) other than the wave function identically zero. 2.- There are trajectories passing through any initial wave function; the eigenfunctions are not preferred in any obvious sense, beyond being periodic, which does not make them fixed points. 3.- The energy is constant along the trajectories in the space of wave functions hence no trajectories join states with different energies. This means that the wave function will never evolve starting at a given eigenfunction to end at an eigenfunction belonging to a different eigenvalue. The above inconsistencies constituted an embarrassing situation for a time dependent equation aspiring to be a model of the hydrogen atom and motivated the creation (already in 1926) of the quantum postulates with the obvious intention of repairing the failures of the STDE, then to measurement theory, collapse of the wave packet, angular momentum where rotations are denied (spin), quantum electrodynamics, etc. The most astonishing step taken by the founding fathers of QM was to keep the failed STDE as a valid physical Law of Nature. Perhaps they felt that such an equation, although dysfunctional and confusing, added an air of credibility to the newborn theory, and it did. But the STDE is so useless that you can eliminate it from the quantum postulates, keep the self adjoint energy operator H and thus obtain a "Purely Probabilistic Quantum Theory" (PPQT) where everything follows from the remaining quantum postulates. All that said, it could well be the case since the STDE is simply postulated and does not predict the physical phenomena it is modelling, that the STDE is definitely wrong. If it actually is such a mistake then the quantum axioms, with the probabilistic interpretation included, could become a historically extraordinary system of fallacies. It turns out that a Deterministic Time Dependent Equation (DTDE) does exist. It is non-linear, is based on Schrödinger self adjoint operator and has all the correct dynamical properties. With the DTDE everything is as causal, continuous and deterministic as any classical wave theory can be. Give it a trial. The big question is how to overcome the instinctive resistance of the powerful and vigilant quantum establishment that needs to defend the theory (for them QM is a truth) and salvage their own scientific reputations. For rigorous and technical details about these matters please google our paper entitled "Deconstruction of Quantum Wave Mechanics". With cordial regards for both Professors M. Daniel Crespin

[@nastyavicodin6229]

Thank you very much. Your way of explaining things is perfect!

[@ProfessorMdoesScience]

Glad you like it! :)

[@fernandojimenezmotte2024]

Thank You professor M does Science for your wonderful channel in Quantum Mechanics

[@ProfessorMdoesScience]

Thanks for your kind words!

[@davidroux7987]

I'm really enjoying and learning from your videos. I teach undergraduate QM, and wish I could explain this clearly! What a lot of hard work to make these videos, thank you for sharing.

[@ProfessorMdoesScience]

This is great to hear! May we ask where you teach?

[@bene1443]

Great job!!!

[@ProfessorMdoesScience]

Glad you like it! :)

[@porit1023]

Now I understand conservation of the norm! Thanks a lot!😊

[@ProfessorMdoesScience]

Glad you found this helpful!

[@redaabakhti768]

among the countless ressources on Qm your lectures along with f.schuller's are among the the best of the best god bless you

[@ProfessorMdoesScience]

Thanks for your support! :)

[@fatemehmaroufkhani7104]

amazing video! thank you so much.

[@ProfessorMdoesScience]

Glad you like it! :)

[@ashrafabdelrahman1358]

Thanks for the videos. I would just suggest a big picture approach to study quantum mechanics and subsequently quantum field theory. As student, I have the motivation to understand the subject and approach it directly but I think students are overwhelmed by the information available so I think it would be very helpful if you can provide something like map. Thank you once again

[@ProfessorMdoesScience]

This is a great suggestion, and we are thinking about doing something along these lines, perhaps organising an online course giving a high level view of the topic that builds on the videos. We'll share more information when we have it!

[@ashrafabdelrahman1358]

@@ProfessorMdoesScience I hope this comes soon

[@rubiramirez5617]

Wow, excellent channel!! One of the best explained, Have you ever considered talking about special relativity? It would be awesome ;D

[@ProfessorMdoesScience]

Thanks for your support! :) Our plan is to continue with quantum mechanics for a while (there is still plenty to cover), but hopefully we'll move to other topics in the future, thanks for the suggestion!

[@workerpowernow]

very good video :) thanks

[@ProfessorMdoesScience]

Glad you like it!

[@AlexAlex-fs8bk]

Have you considered doing a video on the Dirac equation? Great channel!

[@ProfessorMdoesScience]

Thanks! We do plan to cover more advanced topics, including the Dirac equation, once we finish with elementary quantum mechanics. Thanks for the suggestion!

[@iotaphysics909]

superb job sir..if you can then please make a video on symmetry

[@ProfessorMdoesScience]

Thanks for the suggestion! We are hoping to look into group theory and its use in quantum mechanics in the future, although it may take us some time to get there...

[@BruinChang]

It is so unimaginable that Schrodinger equation is a postulate.

[@ProfessorMdoesScience]

Why do you think that? If you compare this to classical physics, the dynamics of the system is also given by a postulate, in that case Newton's second law :)

[@BruinChang]

@@ProfessorMdoesScience I think so because of the combination of a constant, the imaginary number and a derivative of a wave function.

[@ProfessorMdoesScience]

@@BruinChang Oh, I see, you mean that the specific form looks strange. It is indeed something that is well beyond our everyday experience, and I guess every student of quantum mechanics is surprised by this theory :)

[@quantum4everyone]

It doesn’t really have to be, but in the end, we just push things back to other postulates. So, for time evolution, if you instead prefer to use the fact that probability is conserved, so the norm of the state cannot change in time, then we learn U(t) is a unitary operator. If you want the semi group law, which says evolve from t1 to t2 and from t2 to t3 is the same as evolving from t1 to t3, then it must be an exponential function with an exponent linear in t. If you then say the Hamiltonian must be the operator governing time evolution, so it gives the same behavior as classical systems, then you end up with U(t)=exp(+/- iH t) for a Hamiltonian independent of time. Convention has us pick the minus sign. Next, You can break this up into tiny time steps Delta t so the U operator is a product of many U(Delta t) operators. Then if H depends on time, we replace the constant H in those products of factor by H(t) corresponding to the time in the given factor. This is called the Trotter formula. Now all of the latest time dependence is on the far left of the operator. Take its time derivative and voila, you get the time-dependent Schroedinger equation. In this way, the Schroedinger equation arises from the postulates of (i) probability conservation, (ii) the semi group rule, and (iii) the fact that the Hamiltonian operator governs time evolution. Is this better? Depends on your personal tastes. For me, I think it is, instead of postulating the equation.

[@Davide-bl2wb]

Hi your lectures are really great. Thank you very much, I have "fixed" a lot of things I got wrong in QM. Is there a preferential order to watch the videos? Do you make a printed version (slides/book) available?

[@ProfessorMdoesScience]

Glad you find them helpful! We organize the topics in playlists, so if you go to the Playlists page, you can follow the videos in the order they feature in each playlist. A good starting point is the one on "the postulates of quantum mechanics". In the future we hope to expand the type of material we share, but it may take some time to get there.

[@Bloodwork-ig6dg]

Your videos are great! Is there any literature on rigorous qm that you could recommend?

[@ProfessorMdoesScience]

There are a few books at the level we approach the topic that we like, including Merzbacher, Sakurai, Cohen-Tannoudji, and Shankar. I hope this helps!

[@zhongfenglin4225]

Hi Professor M, thanks for your great video. I just have one question. At 14:48, I wonder if you forget to add a minus sign when |psi(t)> = exp[iEn(t-t0)/hbar] |un> , it should be |psi(t)> = exp[-iEn(t-t0)/hbar] |un> in my opinion. Am I correct? Or do I miss something?

[@ProfessorMdoesScience]

You are absolutely correct, there is a typo in that equation!

[@ajilbabu13]

@0:37 If the Hamiltonian of the system depends on energy, then how does that H corresponds to the total energy?

[@ProfessorMdoesScience]

The Hamiltonian is the operator that represents the physical quantity "total energy". In quantum mechanics, physical properties such as energy, position, momentum, etc., are represented by operators. So rather than saying that the Hamiltonian "depends" on energy, one could say that the Hamiltonian "is" the energy of the system.

[@hasnaingillani2587]

If you guys can please do a video on parity. I don' t know about others but I find it confusing. thanks!

[@ProfessorMdoesScience]

Thanks for the suggestion, we actually have the theory ready for the parity operator, so hopefully in the not too distant future....

[@ProfessorMdoesScience]

We have now prepared two videos that cover the parity operator in some detail. You can find them here: kzbin.info/www/bejne/haTZiZiKerarr8k kzbin.info/www/bejne/pWnPZ4ywbqp7ptU I hope you like them!

[@ajilbabu13]

What are the different representations of SE? You mentioned the energy representations, are there any other?

[@ProfessorMdoesScience]

Good question! In principle you can write states in many representations. The general theory behind this is explained here: kzbin.info/www/bejne/qKGVnGWlh5qNh5o In practice, you want to choose a "convenient" representation, and for the SE this happens to be the energy representation.

[@nomanahmadkhan7791]

@@ProfessorMdoesScience Usually Schrodinger equation is introduced in Quantum Mechanics books in the following form; (-hbar^2/2m) (d2psi/dx2) + V(x, t) psi(x, t) = i hbar(d psi/dt). Is it written in position representation?

[@ProfessorMdoesScience]

@@nomanahmadkhan7791 Yes, what you've written is the position representation of the Schrödinger equation.

[@sayanjitb]

It was a really helpful and excellent lecture as well. At the timestamp, 14:24 shouldn't the exponential contain a minus sign before E_n? [|\psi(t) > = exp( - E(t - t_0)i/h) * |u_m> ]

[@ProfessorMdoesScience]

You are absolutely correct, thanks for finding the typo!

[@TheSADDAM11000]

I have a question. If the state of a system is unknown at first. Then we make a measurement and found \labmda_n, the system will immediately collapse into state un, which is an eigenstate. So, according to time evolution, the system will then stay in this eigenstate forever with global phase factor, right? So, any time we measure again, we should get the same value \lambda_n?

[@ProfessorMdoesScience]

This will be true if the quantity you measure is the energy, in which the eigenvalues and eigenstates are associated with the Hamiltonian. But if the quantity you measure is something else, and in particular the corresponding operator does not commute with the Hamiltonian, then time evolution will again mix in other eigenstates. I hope this helps!

[@ajilbabu13]

If the state Psi is an explicit function of t, then how will it become functions of x(in position basis) and p (in momentum basis)? Is there any transformation for this change

[@ProfessorMdoesScience]

To see how to write down the state of a quantum system in the position (or momentum) basis, you can check out our video here: kzbin.info/www/bejne/aJ3VZJR3aduUeNU In these bases, the state of the system is a function of x or p. If you have a time-dependent state, then this function *also* depends on t, so you would get something like psi(x,t). We have an example of this here: kzbin.info/www/bejne/eXjFp2yZi9x9gpI

[@ajilbabu13]

@@ProfessorMdoesScience thanks for the reply

[@afaq7564]

Which book do u use for these lectures? i guess Quantum Mechanics by Mcintyre🤔

[@ProfessorMdoesScience]

Interesting guess, but we don't use Mcintyre (we are not too familiar with it). We mostly consult Sakurai, Cohen-Tannoudji, and Shankar to decide how to present things.

[@iotaphysics909]

@@ProfessorMdoesScience yupp...it is quite helpful for me as i am following all these four books in detail

[@arjyadebsengupta8159]

Sir can you recommend some reading material on QM for under graduates?

[@ProfessorMdoesScience]

There are many books, and what I recommend to my students is that they start with a few until they find the one that works for them. But a few that I think are generally good include: "Quantum Mechanics" by Merzbacher, "Quantum Mechanics" by Cohen-Tannoudji (two volumes), "Principles of Quantum Mechanics" by Shankar, and "Modern Quantum Mechanics" by Sakurai. I hope this helps!

[@arjyadebsengupta8159]

@@ProfessorMdoesScience thanks sir

[@rodrigoappendino]

@@arjyadebsengupta8159 Thy Feynman's lecture on physics, for free on the internet. The third book.

[@arjyadebsengupta8159]

@@rodrigoappendino thanks bro I read the first volume as bedtime stories

[@ajilbabu13]

As we are so much inclined to the Hamiltonian of the system, does the Lagrangian play any role in quantum mechanics ?

[@ProfessorMdoesScience]

Yes, one can also do quantum mechanics using the Lagrangian. In that case, the resulting formalism is the so-called path integral formalism. This approach is typically introduced in more advanced courses, perhaps we'll get there ourselves further down the line :)

[@ajilbabu13]

@@ProfessorMdoesScience wow, thanks for the reply❤❤

[@tomgraupner171]

I'm struggeling with the "stationary states". The electron in a Hydrogen-Atom will leave an excited state and return to the ground state. Why is that? Isn't this a "stationary state? Or is this not covered by Quantum Mechanics and needs QFT due to emmitting and absorbing photons?

[@ProfessorMdoesScience]

A stationary state is an eigenstate of the Hamiltonian of the system. If you had a completely isolated hydrogen atom, whose Hamiltonian was only the kinetic energy of the electron and proton, and their Coulomb interaction, then an electron in an excited state would stay there forever. As you correctly say, to describe emission and absorption of photons, we need to go beyond this simple picture: our Hamiltonian now includes additional terms that allow for these transitions. I hope this helps!

[@tomgraupner171]

@@ProfessorMdoesScience Thank you, Professor. Yes this helps me a lot. Wonderful to get not just a fast - but moreover: an understandable answer! Wonderful

[@garvitmakkar]

please clarify that statement. initial state in which the system is stationary state(you written states) ? or the eigenstates of eigenvalue equation of Hamiltonian (which are infinite states corresponding to each energy level) are stationary states. if both statement are not valid then please tell what are stationary states. thanku so much.

[@ProfessorMdoesScience]

A stationary state is an eigenstate of the Hamiltonian, which corresponds to a particular energy eigenvalue. They are called stationary states because, if you have a stationary state at a given time, then you will have the same state at any later time. Does this help clarify your doubt?

[@garvitmakkar]

@@ProfessorMdoesScience The arbitrary state psi(t) that we build from basis as these eigenstates of Hamiltonian is also stationary state?

[@ProfessorMdoesScience]

No, such a state will not be stationary in general, there will be interference because the phases of individual energy eigenstates in the expansion are different. I hope this helps!

[@subhajitsadhukhan8521]

Professor why global phase factor doesn't matter?

[@ProfessorMdoesScience]

A global phase factor doesn't matter because it does not change the predictions of quantum mechanics. For example, we discuss this in the context of eigenvalues and eigenstates here: kzbin.info/www/bejne/pmLdmGCZZtOprbM And in the context of measurements here: kzbin.info/www/bejne/pZWvqIiOgL5jgNU I hope this helps!

[@IsomerSoma]

0:55 but it can be derived out of a path integral and this integral can be related much more inuitively to the physical ideas. I really didnt enjoy QM before learning about the path integral formalism. Without it it seems so unmotivated. What i learned in classical mechanics is the derivation of a theory from simple principles. The S-equation just didnt seem to be such a simple principle and indeed there are much better pathways to QM. So i think you are doing students a disservice to tell them the S-Equation cant be derived. You can quantize the action functional from classical mechanics and arive at the schrödinger equation. This should be at least informally mentioned.

[@ProfessorMdoesScience]

Thanks for the suggestion! We have indeed taken the traditional approach to quantum mechanics, and I would argue with a set of well-defined postulates. Some of these postulates can be replaced by alternative approaches, and one could argue relative merits. But we think our approach works for most people as it requires mathematics that are typically known by starting physics students. But we'll certainly keep re-evaluating how we explain things!

[@IsomerSoma]

​@@ProfessorMdoesScience I have no issue with how you explain things. Your presentation is excellent. No complaints. My QM Professor woold feel inadequate, if we would assume he cares about teaching (he doesnt), while watching this. Problem for me is that this standard approach, which you can find in almost any undergradute QM book and lecture, feels terribly unmotivated to me (and i am sure not the only one). The quantization of the action functional works as a great motivation for people just having learned about classical mechanics. You can avoid the mathemtical difficulties for the most part and derive the Schrödinger equation more or less returning to the standard path. Using this approach for undergradutes is an idea by Frederic Schuller. You can find his lectures on youtube, but the undergraduate QM lecture i am referencing is in german. He also has a pretty unique way of teaching undergraduate classical mechanics, where he formulates it in the langauge of differential geometry right from the start.

[@ProfessorMdoesScience]

@@IsomerSoma Thanks for the reference! We are constantly looking for ways to improve, and we could certainly prepare a future series in which we re-evaluate the basics from a different point of view! :)

[@dcrespin]

@Isomer: I disagree about deriving the Schrödinger Time Dependent Equation (STDE) more intuitively from path integrals and Classical Mechanics. Think about this: 1.- Assume that you start from Classical Mechanics, then how do you intuitively introduce into the STDE the imaginary unit, i^2=-1 ? 2.- In Classical Mechanics the state variables are pairs $(q,p)$ with $q$=configuration and $p$=momentum. In the QM of hydrogen the state is the wave function $\psi$ which plays the role of $q$, and the quantum mechanical analogue of $p$ does not even exist. Hence deriving the STDE from QM in a rational, motivated manner is impossible. At some point you make an act of faith and that exactly is an axiom. 3.- If you do accept the STDE, be it axiomatically or otherwise, it does not predict any preference of the wave functions for the eigenfunctions of the energy operator H (quantization of states axiom), or of the energy observable preferring differences of eigenvalues (quantization of energy axiom), nor any trajectories of the STDE exists joining eigenfunctions belonging to different energy levels (quantum jumps axiom). All these properties should be consequence of a correct time dependent equation for the wave functions. The STDE is not such a correct equation and the quantum axioms were invented as an imitation of the older Bohr planetary atom model, where the quantization of circular orbits and of the jumps were perhaps axioms or simply unexplained. Thus, the quantum axioms were an attempt to repair the defects of the STDE. 4.- Still worse, acceptance of the unjustifiable and dysfunctional STDE obliges you to accept the accompanying number of equally unjustifiable quantum axioms. The axioms are so bizarre that they created a still unresolved crisis in the Philosophy of Science, where terms like Quantum Ontology (according to Google 3,960,000 results) and Quantum Epistemology (4,980,000 results) have emerged. So, whatever intuition was used invoking Classical Mechanics, path integrals or otherwise, something very fundamental is failing there. 5.- Once you start the path of the STDE you are free to propose basically anything, for if something does not work just make it work under the duress of a convenient axiom. Please look at my comments only two jumps above this answer in the thread. All these remarks, critical as they are, have the intention of adding rational, constructive and eventually fruitful arguments to the discussion that this excellent video about orthodox Quantum Mechanics has motivated. Finally, mistaken as the STDE may be, Oxford has once more proven with this video its top class ratings. I hope for them one day to explain the correct Deterministic Time Dependent Equation (DTDE) of hydrogen in a manner as clear, concise, rigorous and pedagogical as here done with the STDE. With the best of regards, Daniel Crespin

[@garvitmakkar]

The global phase having time t zero is just zero can we remove that?

[@ProfessorMdoesScience]

You can always multiply your state by a global phase, so if you have a global phase at any point, for example at t=0, then you can "remove" it by setting it to zero (such that the exponential becomes 1). I hope this helps!

[@garvitmakkar]

@@ProfessorMdoesScience in video at 11.36 the exponential having (t_1 - t_0) here t_0 is just time at t=0 can we simply write (t_1 - 0) ?

[@ProfessorMdoesScience]

@@garvitmakkar Yes, if t0=0 you can write it as you do. Keeping it with t0 just makes the argument more general for other initial times. I hope this helps!