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Schrödinger Equation

Anonymous Sat 23 Jan 20:26:32 2021 No.12619320 View Reply Original Report

Quoted By: >>12619398 >>12619591 >>12619709 >>12619717 >>12620218 >>12620872

What's so specially about it?

$i\hbar\frac{\partial}{\partial t} \Psi(x,t) = \left [ - \frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ] \Psi(x,t) \,$

It's literally "Energy = Kinetic Energy + Potential Energy" expressed as operators acting on a wave function.

What did Schrödinger add that wasn't already obvious in Hamiltonian mechanics?

Anonymous

Anonymous Sat 23 Jan 2021 20:44:02 No.12619398 Report

Quoted By:

>>12619320
>What's so specially about it?
> wave function
/thread

Anonymous

Anonymous Sat 23 Jan 2021 20:51:06 No.12619439 Report

Quoted By: >>12619445 >>12620870

$ \Psi = \text{e} ^{ \text{i}(kx- \omega t)}~~~(= \cos(kx- \omega t)+ \text{i} \sin(kx- \omega t)~) \\ \dfrac{ \text{d} \Psi }{ \text{d} x}= \text{i}k \, \text{e} ^{ \text{i}(kx- \omega t)}= \text{i}k \Psi \\ \dfrac{ \text{d}^2 \Psi}{ \text{d}x^2}= \text{i}k \, \text{i}k \Psi= -k^2 \Psi \\ k= \dfrac{p}{ \hbar}~~~~~ \left ( [k]=~ \frac{kg \cdot \frac{m}{s}}{J \cdot s}= \frac{kg \cdot m}{Nm \cdot s^2}= \frac{kg }{kg \cdot \frac{m}{s^2} \cdot s^2}= \frac{1}{m}~ \right ) \\ \dfrac{ \text{d}^2 \Psi}{ \text{d}x^2}= -\dfrac{p^2}{ \hbar^2} \Psi \\ - \hbar^2 \, \dfrac{ \text{d}^2 \Psi}{ \text{d}x^2}= p^2 \, \Psi \\ E= E_k+E_p= \frac{p^2}{2m}+E_p \\ E \Psi= \frac{p^2}{2m} \Psi+E_p \Psi=\frac{p^2 \Psi}{2m}+E_p \Psi=\dfrac{- \hbar^2}{2m} \, \dfrac{ \text{d}^2 \Psi}{ \text{d}x^2}+E_p \Psi \\ $

Anonymous

Anonymous Sat 23 Jan 2021 20:52:11 No.12619445 Report

Quoted By: >>12619465 >>12619641 >>12619647

>>12619439
Yes I know how to derive it. Why did Schrodinger win a Nobel Prize for this, am I missing something?

Anonymous

Anonymous Sat 23 Jan 2021 20:56:16 No.12619465 Report

Quoted By: >>12619585

>>12619445
Nobody did it before

Anonymous

Anonymous Sat 23 Jan 2021 21:25:46 No.12619585 Report

Quoted By: >>12619596

>>12619465
But why not? Was it the operators that he discovered? Or the fact that E = T + U is somehow related to this?

Anonymous

Anonymous Sat 23 Jan 2021 21:27:56 No.12619591 Report

Quoted By: >>12619595

>>12619320
>What is the double slit
It explained why a seemingly particulate experiment revealed a wave nature. It ascribed a wave function to particles, asserting that all particles have a wave property.
>Wave-particle duality
That's nobel prize worthy shit if I've ever seen it.

Anonymous

Anonymous Sat 23 Jan 2021 21:29:41 No.12619595 Report

Quoted By: >>12619602

>>12619591
But it was already know that light behaved as a wave.

Anonymous

Anonymous Sat 23 Jan 2021 21:30:00 No.12619596 Report

Quoted By:

>>12619585
>What did Schrödinger add that wasn't already obvious in Hamiltonian mechanics?
Hamiltonian mechanics for particles isn't a field theory. Before that there was no Psi function in space for describing particles, and in Hamiltonian mechanics, the predictions for X and P can be given simultanously.

Anonymous

Anonymous Sat 23 Jan 2021 21:32:24 No.12619602 Report

Quoted By:

>>12619595
True but then the photo-electric effect proved that light was make of particles and everything that came before had to be thrown out.

Anonymous

Anonymous Sat 23 Jan 2021 21:41:59 No.12619641 Report

Quoted By: >>12619650

>>12619445
Because it is something that was not derived from first principles. It was an educated guess that solved many of the outstanding problems of the time. Remember also that it is not just for the equation but the mathematical quantum mechanical framework around it - operators, complex valued probability wavefunctions etc and mapping all of that onto reality and measurements.

Anonymous

Anonymous Sat 23 Jan 2021 21:43:14 No.12619647 Report

Quoted By: >>12619650

>>12619445
wow anon you're so smart, even nobel-worthy things seem obvious to you. maybe you deserve a nobel too

Anonymous

Anonymous Sat 23 Jan 2021 21:43:46 No.12619650 Report

Quoted By: >>12619676

>>12619641
>operators, complex valued probability wavefunctions etc and mapping all of that onto reality and measurements.
Ah I thought Dirac did all that. Fair enough that's pretty impressive.

Way better than the matrix mechanics stuff Heisenberg did.

>>12619647
Oh shut up that's not what I said.

Anonymous

Anonymous Sat 23 Jan 2021 21:48:12 No.12619676 Report

Quoted By: >>12619732 >>12620615 >>12620621

>>12619650
> Ah I thought Dirac did all that
Many were working together on the foundations at the time but Schrodinger was the first to put it all together and make the theory work.

Frankly though what Dirac did after that was even more impressive with coming up with his Dirac Equation and his predication of anti-matter purely based upon the maths years before it was actually discovered. His work started the progress towards Quantum Field Theories and later on the Standard Model we have today.

Anonymous

Anonymous Sat 23 Jan 2021 21:53:51 No.12619709 Report

Quoted By: >>12619732

>>12619320
Dude, if there were Nobel prizes/fields medals in 4000BCE, you'd get one for showing that 2 * 2 = 4. You have to take into context what is or isn't already known.

Anonymous

Anonymous Sat 23 Jan 2021 21:55:09 No.12619717 Report

Quoted By:

>>12619320
Cute cunny.

Anonymous

Anonymous Sat 23 Jan 2021 21:57:15 No.12619732 Report

Quoted By:

>>12619676
Just found out Dirac and Schrodinger actually shared the Nobel Prize.

>>12619709
Yeah that's why I was confused because I thought in the 20s we already knew about the wave equation

Anonymous

Anonymous Sun 24 Jan 2021 00:08:04 No.12620218 Report

Quoted By:

>>12619320
Interference, because it's not an advection equation, it's an advection equation over the complex numbers.

Anonymous

Anonymous Sun 24 Jan 2021 01:54:30 No.12620615 Report

Quoted By: >>12621819

>>12619676
Schrodinger dealing with Heisenbergs douchebaggery by proving the equivalence of his PDE to matrix mechanics is an all time badass science moment.

That is all.

Anonymous

Anonymous Sun 24 Jan 2021 01:56:22 No.12620621 Report

Quoted By: >>12621322

>>12619676
Dirac was wild. A professor I once had described Dirac's mathematics as "building the zoo cages first, then finding animals that fit in them".

I still don't know what he meant.

Anonymous

Anonymous Sun 24 Jan 2021 03:06:29 No.12620870 Report

Quoted By:

>>12619439
can you spot the mistake?

Anonymous

Anonymous Sun 24 Jan 2021 03:07:54 No.12620872 Report

Quoted By: >>12621825

>>12619320
>Look at me I'm so smart by saying the Shrodinger equation is trivial.
It took over 20 years for them to derive an equation of motion to describe quantum behavior (before they used the "old quantum theory"), oddly enough, it ended up being universal. This equation is not obvious at all and the derivation was brute-forced. Plus it was all around more appealing to physicists than Hamilton's matrix mechanics, allowing them to visualize what's actually happening down there.

Anonymous

Anonymous Sun 24 Jan 2021 06:26:00 No.12621322 Report

Quoted By:

>>12620621
trying to make his maths relevant

Anonymous

Anonymous Sun 24 Jan 2021 11:06:36 No.12621819 Report

Quoted By:

>>12620615
They were both douchebags to each other desu. Schrodinger wasn't a good person in real life, and Heisenberg was practically a Nazi when it suited his career.

Anonymous

Anonymous Sun 24 Jan 2021 11:08:03 No.12621825 Report

Quoted By:

>>12620872
>doesn't read the thread
>doesn't add anything
>gets facts wrong
It's Heisenberg that invested matrix mechanics, not Hamilton. And I was asking about historical context, which has already been provided.

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