Unpacking the Schrödinger Equation

We've talked about the Schrödinger equation before, but we really didn't dig into it with any depth at all. Now it's time to really get in there and do the math. What is the Hamiltonian operator? What is the time-independent Schrödinger equation? What we can we do with this equation? Let's find out!

Script by Hèctor Mas

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[Lien]

Think I'll get back to my trigonometry and polynomial division for now. I'll see you guys in about 50 years

[lordspongebobofhousesquare]

Hi professor Dave, just wanna drop in and say thank you. I watched your tutorials to cram for my university joint entrance exam. I don't have the money for tutoring, but I managed to collect enough money to rent a home wifi so I can watch your videos among other things. Your videos are very concise, which I like. I managed to get accepted (to study mathematics) and your videos played a huge role in making that happen.

[higgs_boson]

I love how you are one of the few people on yt to get fully in depth into QM in a way in which we can understand. You deserve a larger following!

[matthewrussell]

Still remember my prof teaching this 15 years ago. The colored fonts are incredibly helpful!

[adleneboulebtateche]

Professor Dave really knows a lot about the science stuff. impressive !

[kazumasatou]

Thanks for this video, professor. Too many people are far too intimidated by the math behind quantum mechanics, though I'd argued it's undoubtedly the most beautiful aspect of this theory. We definitely need more science communicators.

[ultradripstinct]

I've gained 40 brain cells by watching this

[davidpage]

I've been following this series as an amateur. I understand very little but Professor Dave's talent for explanation still allowed me to gain some mild insight into the topic. Thank you

[smlankau]

A very profound explanation. Thank you so much Prof. Dave.

[lowkey_entertaining]

I have never needed to pause a video so much in my life

[brianrasmussen]

I’m in quantum mechanics right now and this is extremely helpful.

[adamdrewko]

This is so... enlightening! Thank you so much for this series and the whole of your work.

[cgw]

A nice way to think about the Schrödinger equation is to take Ψ=exp ( iS(𝐱, t)/ℏ+o(1)) and substitute it into the Hamilton-Jacobi equation: H(𝐱, 𝛁S, t)=-∂S/∂t in the limit as ℏ→0. So for a non-relativistic particle in a potential V(𝐱) we have
H(𝐱, 𝛁S, t)=(𝛁S)²/2m+V(𝐱)
and -iℏ𝛁Ψ = Ψ𝛁S+o(ℏ), so that -ℏ²𝛁²Ψ = [(𝛁S)² -iℏΨ𝛁²S+o(ℏ²)] Ψ, and -iℏ∂Ψ/∂t = Ψ[∂S/∂t+o(ℏ)].
Hence, neglecting terms which are o(ℏ) we have the leading order equation:
-ℏ²𝛁²Ψ/2m+V(𝐱)Ψ=iℏ∂Ψ/∂t,
i.e., the Schrödinger equation. This is related to the path integral approach, where exp ( iS(𝐱, t)/ℏ) occurs in the path integral.

[swipe]

Dude, I just had a class on this today. Didn't make any sense so thanks for this.

[tommasoprocida]

Thank you for this video man. Studying Quantum at the moment but struggling to follow the narrative so to speak of some parts. This was really well explained and helpful!!!

[asdfghjkllkjhgfdsa]

Whoa! Way over my head. Ill be back after watching a few other videos so this makes more sense, i promise!

[akshatpandey]

I recommend ur channel with my friends. They loved it. Ur explanation is superb.

[Ihab.A]

Prof, I started giving thumbs up to your videos before I even see them! Those are invaluable treasures! Grazie!

[cliffordwilliams]

YAY Thank you !! I've always wanted the Schrodinger equation fleshed out for me !!!

[Letsfeelthenaturee]

You are really very brilliant, sir.
Hope you will be more and more