/sci/ - Science & Math

I forgot what its about but you should know about hamiltonians from classical mechanics first, read SICM for example

I forgot what its about but you should know about hamiltonians from classical mechanics first, read SICP for example

>>5876829
fuck off

H(t) is temperature (hot)
w(t) is the orbital period of Neptune
i is you - what would you do in this equation?
h with dash is Planck's constant
d/dt is hertz since you cross out the d's and you are left with 1/time which is hertz

>>5876830
2edgy4me

>>5876832
What the fuck, dude.
It's not the orbital period of neptune.
It's the pitchfork constant. Get your head out of your ass.

Schrodinger's equation describes the status of a cat in a box.

>>5876832

Are you kidding me?

I haven't even studied quantum mechanics (yet) but I know for sure that "i" stands for the imaginary unit. You're right about Planck's constant but w(t) is definitely not the orbital period of Neptune; that has nothing to do with this equation which is far more fundamental than any specific planet's orbital period.

Schrodinger must be turning in his grave, idiot.

>>5876959
>>5876853
You guys suck. w(t) does refer to the orbital period of Neptune, however he was mistaken on the H(t), it doesn't refer to temperature but to the number of macroscopic stable states in which Uranus has a non-zero probability to be in.

>>5876959
Yes it is. Just look how close the w thing is to the symbol of Neptune . Perfect match.

it's a thingy

More important question is how to solve the Schrodinger equation.

>>5876959
>but I know for sure that "i" stands for the imaginary unit

Unfortunately you're wrong.

>>5876829
I believe you mean SICM.

What is to decipher on the S.E.? It's a postulate.

>>5880956
No, the Schrödinger equation follows from the postulate that the time-evolution is given by an unitary C0-semigroup.

>>5880975
Fair enough, I never dove into the details. Still nothing mysterious about the SE though: it merely states that some state <span class="math">\left|\psi \right>[/spoiler] containing all information on whatever you are describing evolves in time as prescribed by this equation.

Here's a fun fact, not many people actually know this, but this is how Schrödinger derived his equation:

At non relativistic speeds (disregarded external potentials and mass energy, this can of course be generalized to a more complicated Hamiltonian) the energy of a point particle is given by:
<span class="math">E = 1/2 m v^2 = \frac{p^2}{2m}[/spoiler]

But according to Einstein the energy for an (electromagnetic) wave is related to it's frequence:
<span class="math">E = h \nu = \hbar \omega[/spoiler]

Now assuming that at quantum level point particles behave like waves they should be equal:
<span class="math"> \hbar \omega = \frac{p^2}{2m} [/spoiler]

Now this is where things start to get a bit 'magical'. Schrödinger wondered: Could the above equation just be the result of a more general equation? What is the most simplest case of a wave? Of course a plane wave. So maybe if we assume this to be the result of some wave equation describing: (for simplicity choose a wave traveling in the x direction)
<span class="math"> \psi = exp(-i(k*x-\omega t))[/spoiler]

Now using debroglie's relation and the wave parameter wave length relation:
<span class="math"> \frac{h}{p} = \lambda = \frac{2\pi}{k}[/spoiler]

We can rewrite <span class="math">\psi[/spoiler]:
<span class="math">\psi = exp(-i (\frac{px}{\hbar} - \omega t)) [/spoiler]

Now a trained eye will see that the p^2 and the t in the equation can be obtained by a spatial and a time derivative respectively. Thus we get:
<span class="math"> i \hbar \frac{d}{dt} \psi = \frac{1}{2m} (-i \frac{d}{dx})^2 \psi [/spoiler]

>>5881013

Small mistake there. In the line on the right side of the equal sign it should be <span class="math">(-i/h \frac{d}{dx})^2[/spoiler], but you get the idea.

>>5880982
>the information itself somehow "evolves in time" >not regarding <span class="math">\Psi[/spoiler] to be a physical thing
Full retard.

>>5881013

In the last line on the rhs it should be <span class="math">(-i \hbar \frac{d}{dx})^2[/spoiler], but you get the idea.

$$\eps>0$$

\eps

$$\eps$$ asdasd is asdasd

>>5881071

It's {math} in [] (rectangular paranthesis)

>>5881081
like this????????????

<span class="math">\TeX<span class="math">[/spoiler][/spoiler]

>>5876959
please leave now.

>>5876853
Guys, can you spot a troll?

>>5881013


before i read that paper i always assumed the schrodinger was underivable. he even said so himself. crafty motherfucker, trying to make himself seem like he recieved a gift from god but in realiyy so simple an undergrad can do it

>>5882828
>assumed the schrodinger was underivable

You assumed wrong. Any physics grad student can derive schrodingers equation using the least action principle, and lagragian dynamics.

>>5876959
wut
Are you seriously THIS new?

>>5876959
>I haven't even studied quantum mechanics (yet) but I know for sure that "i" stands for the imaginary unit.

mysides.jpg

>>5882830
He didn't technically derive it the way most equations are derived. He just played around with equations till he got this out

check em

>>5882830
>Any physics grad student can derive schrodingers equation using the least action principle, and lagragian dynamics.

Prove it. Do it ITT.

>>5884132
I would also like to see the proof.

>>5881013
Not OP, but thank you. That makes things much clearer.

probably either Griffiths or Shankar

>>5882830
combining action and quantum mechanics leads to path integral formalism, which in fact, does not invoke the least action principle, but rather, summing over all possible paths.

>>5885804
And how do you derive Schrodinger's equation from there?

>>5885804
>summing over all possible paths
Isn't that the least action principle?

>>5887194
no.

>>5887241
How is it related to the least action principle?

>>5885804
For many systems (and specifically non-relativistic Hamiltonians like the one above) you can view <span class="math">\psi(t,\vec x)[/spoiler] both ways, as the quantization of trajectories <span class="math">q(t)[/spoiler], or as a classical field with Lagrangian

Someone has been keeping at least 35 threads alive
by bumping them twice a day. Compare the post times of
>>5878808 >>5879951 >>5880633 >>5882780 >>5884132 >>5884861
>>5886262 >>5888662
to the post times in other threads.
It's clear most of these are the same person.
The threads being bumped:
>>5858447 >>5861383 >>5862983 >>5863249 >>5865823 >>5866853
>>5867452 >>5867640 >>5868097 >>5868460 >>5868538 >>5869504
>>5869595 >>5869759 >>5869946 >>5872951 >>5873166 >>5873829
>>5874378 >>5874727 >>5875025 >>5876410 >>5876819 >>5878449
>>5878607 >>5878684 >>5878722 >>5880041 >>5880453 >>5880775
>>5881738 >>5883998 >>5884116 >>5884625 >>5885545
Write to moot@4chan.org if you want it to stop.

>>5886262
Construct a free particle path integral measure and use it to find the quantum master equation for the amplitude. Now all that is left is some manipulation with the result to recover the Schrodinger equation.

>>5888742
> or as a classical field with Lagrangian

How?

<span class="math">\mathcal{L}\left[\psi, \mathbf{\nabla}\psi, \dot{\psi}\right] = \mathrm i\hbar\, \frac{1}{2} (\psi^{*}\dot{\psi}-\dot{\psi^{*}}\psi) - \frac{\hbar^2}{2m} \mathbf{\nabla}\psi^{*} \mathbf{\nabla}\psi - V( \mathbf{r},t)\,\psi^{*}\psi[/spoiler]

>>5876819
I don't know if anyone has actually answered your question OP, and it's been a few years since QM, but I'll give it a go.

H is the Hamiltonian operator which represents the total energy of the system. It's basically kinetic plus potential energy. All the (t)'s in the EQ mean these things are functions of time.

\psi is the wave equation, which represents the physical state (like position) of something tiny like an electron. It can be written as a trig function because it bounces up and down like any other wave, but usually its written in exponential form to include the non-real parts which are sometimes important. Also it makes it easier to deal with the equation, and physicists are essentially lazy.

So you operate on the wave equation with the Hamiltonian, and out pops information about the wave function itself. The operation itself involves spacial derivatives, looking at how the function changes over some realm or "boundary". technically speaking, all the junk on the right that replaces the H(t) is an eigenvalue of H.

Anyway, the most basic result is to obtain something like the energy of the wave. However, it gets significantly more complex from there.

Again it's been years, and my interpretation is probably only marginally better than pedestrian at this point, but it's always fun to think about.

>>5890650
>represents the total energy of the system
That is not always the case.

>>5891345
when is it not? ive never seen it not be that.

>>5891367
Different anon; It's only not the case when your coordinate frame depends on velocity or if the lagrangian is an explicit function of time. For nearly all applications though neither conditions are true and you get the total energy. /sci/ trolls will always point out any lack of rigor in math-related posts with a trite and simple reply though, so don't worry about it.

>>5881013
this is a shitty derivation.
the best way to "derive" schrodinger's equation is to consider the equation as stating that the hamiltonian is the infinitesimal generator of time translation. the hamiltonian can be identified as this generator through classical mechanics, seeing through noether's theorem that energy is the generator of time translation, along with some support from de broglie's relations

>>5891416
Is this really a "superior" derivation?

I always thought the point was to get there really was no true derivation, only Heuristic ones. At least this is my memory from class.

>>5882816
can you?

>>5892675
No, I can't. How is it done?

>>5891606
The best derivation comes from realizing that energy-momentum conservation on CP(n) leads to a geodesic equation equivalent to the Schrodinger equation.

>>5893813
Please elaborate. Why do you need CP(n) when our world has only 3 dimensions?

>>5894928
Hast it something to do with string theory?

>>5895743
oh fuck off

>>5896643
No u. It's a serious question.

>>5876959
"i" stands for ice cream

>>5894928
Many physical systems are of infinite dimension.

>>5891606
this one gets at the point better, which is unitary evolution operators. it generalizes easily to relativistic quantum mechanics and quantum field theory

>>5897839
But CP(n) is not of infinite dimension.

>>5899357
It could be. Set n to \infty.

>>5899848
What if I set it to a negative number?

>>5901085
It would become supersymmetric.

lol thanks for this thread guys
i almost forgot why i don't come to /sci/ anymore.

>>5901881
You left /sci/ because of the science threads?

>>5897564
The letter "i" is trademarked by Apple.

>yfw nobody noticed the error in OP's pic

>>5905483
What is the error?

>>5906278
He didn't cancel the d's in d/dt

>>5907936
Thanks, I did not see that.

>>5908615
There's another mistake. I'll give you some time to figure it out.

>>5910143
Did you find it?

>>5876819
Well,
<div class="math">H(t)</div> is the Hamiltonian,
<div class="math">|\varphi(t)\rangle</div> is the wavefunction,
<div class="math">i\hbar</div> should be evident, and finally
<div class="math">\frac{d}{dt}|\varphi(t)\rangle</div> is the time derivative of the wavefunction.

>>5911928
>is the wavefunction
>the time derivative of the wavefunction.
That is a ket, not a wavefunction.

>>5912894
....you cant be serious

>>5913666
if a ket and a wavefunction is the same thing, what is a bra?

>>5913713
the conjugate transpose of the ket

>>5913721
>hur

>>5913723
I don't know what you want
It IS the wavefunction. Just because it is in the bra-ket-notation doesn't make it something else

>>5913729
>It IS the wavefunction.

No, it is not. Did you fail babby's first QM?

>>5911881
I have not. I am not smart enough to find it.

>>5914930
everybody can tell that by how you ask stupid questions in every post

>>5914934
There are no stupid questions, only stupid answers.

>>5916090
*There are no stupid question, only stupid people.

FTFY

>>5914934
You sound highly intelligent. Please disclose to me what else is wrong with it.

Why don't you divide both sides of the equation by the weird greek letter?

>>5920436
You could, but the calculations would not add up. It is inserted twice to account for the inaccuracy.

Okay, so I divided the equation by (t), psi, |, > and cancelled the d's.

H = i * hbar / t

You should be able to solve this.

>>5922219
What do I need to solve for?

>>5923140
>What do I need to solve for?

i

>>5922219

lel

you can't divide like that

>>5924111
Why not? None of the terms is zero or else the equation would be zero.

>>5924111
You sure about that?

>>5925464
But it is zero. The cat was created from nothing.

>>5927375
How can something be created from nothing?

>>5929214
checkmate atheists

That's not the Schrodinger equation.

>>5929214
We just do not know.

>>5932197
But I thought science knows the answer to every question.

>>5876819
the universe is some made up bullshit from physisys from the 60's and relitivitys a croc

Non-Relitivty where it at; and we're outside the multiverse so hurdurdurhurrrrrrrr

>>5933656
>implying we don't know means wi'll never know

>>5933665
Can we know everything?

>>5882841
Jokes on you, it was a double troll

>>5934739
trollception

>>5934739
The joke is never on someone who isn't trolling.
Trolls are, by their very attempts, proving they are wasting their time and are being stupid.

>>5935960
This.

Trolls think they are epic ruse masters but in fact they only waste their time.

The solution has to be real but the right hand side contains an i.
Contradiction.

>>5939319
There are no solutions?

What letter do want to solve?

How do you solve an equation with a derivative in it?

This is Einstein's field equation.

Looks like physics.

Can't be shrodinger equatin. Shrodinger would have a laplacian in it.

>>5949129
This is true.

>>5949129
>>5949955
Fuck you, the Laplacian is in the goddamn Hamiltonian operator

>>5949972
OP did not tell us how his Hamiltonian looks like. Not every Hamiltonian is like the one you learned in high school,.

>not using units where hbar = 1

>>5951201
>Schroedinger equation
>H(t)
>"Durr where's the Hamiltonian guys"

How do you remember to breath?

>>5953351
>doesn't know what a Hamiltonian is

>>5953351
he asked WHAT the hamiltonian was, not WHERE it was. Retard.

>>5876965
Which based on your astrology sign and how many hairs are on your left big toe could be anywhere from 4 to 76
Ex. My sign is cancer and I have 16 hairs therefore the answer is 53.

>>5891416
as someone with virtually no background in quantum mechanics, it's like you guys are speaking another language. It's sorta cool.

>>5954488
You can learn it from any intro to physics.

It's been over a month and this fucking thread is still there?

>>5955970
The original question remains unanswered.

>>5954428
Astrology is interesting. Neil deGrasse Tyson is my favorite astrologist.

That's the position space schroedinger equation

>>5957956
This is wrong. Nowhere does it say position space.