How Schrodinger Came Up With His Famous Equation (But EASIER)

The Schrodinger Equation is one of the most important equations (if not THE most important equation) in the theory of quantum physics. But where does it come from? How did Schrodinger derive it?

Although it's a quantum form of the law of conservation of energy, its derivation isn't discussed anywhere near as much as it should in my opinion. I've even heard some people say that the equation CAN'T be derived, only verified experimentally.

We start with the electromagnetic wave equation - which described electromagnetic waves (as you may have guessed from its name). One of the solutions of this equation is a sinusoidal wave showing the electric field oscillating back and forth between some field value E_0 and its negative, in both space and time.

However this solution is only a solution if a required relationship, or condition, is met. This condition can be boiled down to the idea that any photon corresponding to our wave must have an energy equal to its momentum multiplied by its speed. Luckily, this equation literally defines what a photon is - it's an object whose energy is related to its momentum in that exact way.

Now this relationship (or condition) is actually part of a bigger picture - mass-energy equivalence. For any generic object, we can relate the energy of the object to its momentum and mass through this equivalence relation. For photons, m = 0 so it reduces down to E = pc. However for objects with mass, we get Einstein's famous E = mc^2 equation.

So what if we now go backwards, but starting with the FULL equivalence relation as the condition rather than the reduced one for photons? This way the wave equation we'll end up with SHOULD describe not just photons, but objects with mass (and possibly momentum too). That's exactly what the Schrodinger equation needs to do.

However, when we go backwards like this, the wave equation we end up with is actually the Klein-Gordon equation. This is a relativistic equation, whereas the Schrodinger isn't. So we do need to complete one more step, which is to reduce the Klein-Gordon equation into non-relativistic scenarios. To do this, we see what the equation will look like at low speeds. This is because relativistic effects show up at high speeds, and so if we set the object's speed v to be much less than c, we reduce the Klein-Gordon equation down to the Schrodinger equation!

This derivation is intuitive for those who know the mathematics because we can follow the math through, but it's also intuitive for those who don't know the math because the logical steps can be easily followed.

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Timestamps:
0:00 - The Schrodinger Equation
1:25 - The Electromagnetic Wave Equation and Its Solutions
5:31 - Mass Energy Equivalence - Let's Go Backwards!
7:05 - The Klein-Gordon Equation and Relativity
8:23 - Finally, The Schrodinger Equation (Again)

Videos in Cards:

[iamharsh]

Please make a separate video about how Schrodinger came up with the idea of his equation

[Dr_LK]

Yes, show us the original derivation. Thanks.

[pintuhalder]

Yes I wanna see the Schrodinger's process

[jesusmoralestrejo]

Yes, it would be incredible to see how Schrödinguer derived it. It doesn't matter if the video is 1 hour long👀

[matrix]

Yes! I want to see Schrödinger's way of approaching his equation.

[mathunt]

You missed out the potential term in the explanation. Schrodinger also "derived" a hyperbolic version of the equation but dropped it as it didn't predict good results of something which is why he dropped it. I'd be interested in seeing a video on that.

[FrancisFjordCupola]

Of course, we'd love to see the separate video. This may just help make the other graspable, but even so... :P

[jakeadams]

Brilliant content. Thank you for sharing your knowledge with us 🙌

[patb]

This is really good. Spent months pondering many derivations of SE. This looks the best. Will read that paper with interest.

[ForanoDeBergemarc]

Thanks for this simpler intro. And Yes Yes Yes to Schrödinger's one :) Simply drooling over your own usual incredibly talented coverage of complex matters into layman terms.

[jamesblank]

I'm very impressed with this video. Simple logical steps that get us to Klein-Gordon. Details worked out by the student.

[petergreen]

Beautiful derivation. ❤thank you very much.

[cirdiam]

Another great video from you that has made it to the top of my favourites list. Yes please do the other derivation too.

[carlosg.anguiano]

Would be fantastic to see a video with Schrödingers specific derivation. Amazing content as always!

[satechknowledge]

Please make a separate video how Schrodinger derived his equation

[bramfran]

Yes, please, show us the original paper of Schrodinger.

[giannism]

I would definitely love to see Schrodinger's derivation

[alihan_s_berk]

Please also make a video about Schrödinger's derivation
Also I love your videos, don't stop making them

[Aritra___Roy]

Yess for the original method used by Erwin Schrodinger

[swatichaubey]

Yaa make a vedio on schrodinger's equation 👍