Ch 13: Where does the Schrödinger equation come from? | Maths of Quantum Mechanics

Hello!

This is the thirteenth chapter in my series "Maths of Quantum Mechanics." In this episode, we'll finally understand where the Schrödinger equation comes from. We'll first derive that time evolution must be unitary, then use that to get to the form of the equation. We'll then use the fact that energy is the generator of time evolution to take it all the way home (after fixing the units).

If you have any questions or comments, shoot me an email at:
Thanks!

Animations:
All animations created by me within Python, using Manim. To learn more about Manim and to support the community, visit here:

Music:
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♪ Mind Palace by Rosentwig
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[quantumsensechannel]

Hi everyone! A quick note:

In rewatching the video, I don’t think I give enough credit to what Planck’s constant is doing in the Schrodinger equation. It’s true, it really does settle the units, but this is way more important than you might realize.

The Schrodinger equation has energy and time in it, but how big/small are these time energies? Mega Joule seconds? Pico Joule seconds? On its own, we don’t know the energy-time scales that show up in the Schrodinger equation. So Planck’s constant does exactly this. It tells us that the energy-time scale is ~10^-34 Joule seconds, ie really really really small!!! So Planck’s constant sets the scale of Schrodinger’s equation (and hence quantum physics) to be on incredibly small energy and time scales!!! So the hbar being there is the reason we don’t regularly see our dog in a superposition of two states: you need to be in tiny energy and time scales to see the effects of quantum mechanics.

Hopefully this gave Planck’s constant the respect it deserves in the equation! Units are important because they set the scales of physics!

-QuantumSense

[thatdude_]

I think you left out the definig characterisitc of one parameter unitary groups U(t), namely that U(t+s)=U(t)U(s), which also makes a lot of sense intuitively. With this the proof that each U(t) is indeed unitary can be shortened quite a bit, also it automatically implies the existence of the inverse U(t)^-1=U(-t).
On another note, due to Stones theorem, each unitary group can be written as U(t)=exp(itA) for some unique self-adjoint operator A. This is what i find fascinating. Each unitary group U(t) determines an observable A, and each observable A determines a unitary group via U(t)=exp(itA). It's similar to Noethers theorem in classical mechanics, that each symmetry of a system corresponds to some conserved quantity

[ReyhanMehta]

Hey Brandon, your Maths of Quantum Mechanics series is amazing! Keep up the great work, and thanks for creating such beautiful, quality content.

[DKonigsbach]

The best series on quantum mechanics I've seen. If mathematics and physics textbooks were written using your approach, more students would understand and enjoy those subjects. Thank you!!! Please continue your work. With luck, you could become an inspiration for others in how to teach.

[NPCtje]

Bro this shit blew my mind. I was easily able to use schrodinger equation but the connection to classical mechanics is enlightened my perception of the formula.

[Re_Soraa]

Absolutely amazing. I've never thought of proving time evolution operator have to be unitary, that's so pretty and neat derivation, love it!

[rachidbz]

Thank you for making physics as fun and comprehensive as it should be, Huge respect for you Dr Brandon and I hope you continue doing more to humanity!

[pepaxxxsvinka]

Is there going to be a continuation of the series? it is incredible!

[jamesu]

As someone who’s just done an intro to quantum course at uni this series is really brilliant. We didn’t really approach quantum mechanics from the same kind of mathematical perspective of this series. We just started from the schrodinger equation and worked through increasingly complex systems up to hydrogen and helium and the periodic table. But we never really went into the underlying linear algebra with which you do quantum mechanics - we didn’t even learn Dirac notation.. It’s just kinda annoying that all this maths was sort of hidden from view so that we could just trudge through all the usual quantum systems that every undergrad has to learn.

[jaunesmith]

I feel like this series is going to save me a LOT of headache later in my uni course, much like what 3B1B did for me in the past.

[Duskull]

Thank you so very much for this amazing video series!
My Quantum Mechanics I professor literally butchered the foundation of quantum for me with his dry "shut up and calculate approach".
He barley gave us any context, motivation or intuition for any of the derivation he thought us, so when i finished that course i could say i somewhat _knew_ the basics of quantum mechanics but i certainly didn't understand any of it.. but this video series really helped me fill in the gaps!

Please keep up the good work! you are really helping physics students everywhere understand this wonderful and confusing subject!

[cheesecake]

This is the best video series on YouTube! Thank you!

[QuarkyMathChannel]

Your series has been helping me so much in my undergrad QM course. Thank you for all the time you put into making these videos, they are both aesthetically pleasing and easy to understand. Would you consider expanding this series to discuss more topics in QM such as expectation values, transmission probabilities, and different potential profiles (ie. Delta Potential, SHO, QMW, etc.)? I hope that your graduate studies are going well so far, and keep up the excellent work!

[BapiKAR]

You rightly reignited my passion to learn QM, which was lost in the last decade or so. Thank you very much from the bottom of my heart.

[gideonwiersma]

I currently following quantum physics 2 and this helped my a lot in understanding the basics! I am looking forward to the rest of the series.

[fallon]

You are a legend. Please continue the series.

[gvsgaius]

Thank you so much for this series. I remember failing my Quantum Mechanics II class in university more than 10 years ago, and subsequently never really studying the field in any detail. This not only brings back a lot of old memory but also finally things are put into place in my brain. Really helpful and it feels very good to finally start to really understand the meaning of all these bras, kets and operators. Thanks!

[ItsJushJosh]

I have really been struggling while learning linear algebra and feeling like I have any appreciation for how all these different concepts I'm learning work together to do something meaningful. This video series has not only been incredibly helpful to me in crystalizing many linear algebra concepts through a fascinating application, but has provided me with fresh motivation to move forward. Really fantastic work. Best of luck in your doctorate work and I can't wait to see more!

[franciscopereira]

For someone who is just now starting a course on modern physics (basic concepts of quantum mechanics) I am loving this course although I wont need it until the next semester when I actually take Quantum Mechanics I. Beautifull job!

[nerdphysics]

OMG worth every second. And probably then most beautiful seconds of my life. The Beauty of quantum physics can't be explained better than this.