[ 3 / biz / cgl / ck / diy / fa / ic / jp / lit / sci / vr / vt ] [ index / top / reports ] [ become a patron ] [ status ]
2023-11: Warosu is now out of extended maintenance.

/sci/ - Science & Math

View post   
File: 56 KB, 639x323, ceiling feynman.jpg [View same] [iqdb] [saucenao] [google]
4166946 No.4166946 [Reply] [Original]

Hey guys, I just finished my first semester of Quantum (cute, I know), but I have a question.

Can it be shown that solutions to the (time-dep) Schrodinger equation are always travelling plane waves; ie of the form: <div class="math">\Psi (\mathbf x,t) = Ae^{i(\mathbf k \cdot \mathbf x - \omega t)}</div>
If so, how?
If not, why do we assume this? (Aside from it being necessary for the statistical interpretation of the theory.)

Pic always related

>> No.4167225
File: 136 KB, 670x893, 1319493347272.jpg [View same] [iqdb] [saucenao] [google]
4167225

bump with hot math chick?

>> No.4167249
File: 50 KB, 345x345, idontthinksotim.jpg [View same] [iqdb] [saucenao] [google]
4167249

>>4167225
Hot?
Math?

>> No.4167250

It can be shown that solutions to the time dependant equation are always plane waves in empty space.

Set your potential to zero and solve.

>> No.4167371

>>4167250
Well yeah, but I was looking for something more general than just the free particle.

I know that the infinite and finite square wells do it, and the Delta function potential does it.
The only ones that differ slightly are the Simple Harmonic Oscillator, which just comes out to Hermite polynomial multiples of an exponential; and the Hydrogen atom, whose radial part is just monomial multiples of <span class="math">e^{-\alpha r}[/spoiler], and whose angular part is similarly Laguerre and Legendre polynomial multiples of a complex exponential.

So I know that the solutions always typically come out that way, but is there some reason why the wavefunction is that way?
I mean, I see it sometimes just taken as an axiom that <span class="math">\psi ~ always[/spoiler] takes this form. Usually to show how or that the momentum operator, <span class="math">\hat p \to -i \hbar \partial / \partial x[/spoiler], works.
And that's nice and all, but wouldn't that argument fail if the wavefunction <span class="math">didn't[/spoiler] take the form of a complex exponential?
So there must be some reason that it takes that form according to the Schrodinger equation.

Can anyone help?
(Let me know if you actually want to help, but tl;dr)

>> No.4167408

>>4167225
MOOOOOAAAAARRRRR

>> No.4167491

>>4167408
I'm with you dude, but I don't have any more <span class="math">\ddot \frown[/spoiler]

>> No.4167507

it's a complete base of functions

>> No.4167529

>>4167371
read Schrodinger original paper for some background, or any book on the history of QM. the reason why is something like for a wave it must be of the form exp[i(ax-bt)], and the a and b numbers must transform like x and t respectively, the only vectors available are p and E(or k and omega). ill see if i get the source because i cant remember it anymore.

>> No.4167531

Learn about Fourier transforms.

>> No.4167532

>>4167507
Interesting, go on.
It can't form a complete basis of all functions. For example, polynomials. Or am I mistaken?

>> No.4167535

>>4167532
Fourier

>> No.4167546

>>4167532
it can

>> No.4167556

>>4167529
Thanks dude, that'd be awesome. I'll keep an eye out too.

>>4167531
I've actually been trying to for a while. But every book I've consulted either deals with Fourier analysis (ie assuming you already know the fundamentals of Fourier series and transforms) or simply says "You should know from your previous classes that...", so it assumes you already know it.
If you know of a book that teaches Fourier series/transforms, especially fundamentals and derivations of the subject (like Dirichlet's theorem and Plancherel's theorem), that would be awesome.

>> No.4167558

>>4167531
>>4167535
3rded.

>> No.4167561

>Usually to show how or that the momentum operator works.

I'm not sure i understand what you mean here...

>> No.4167575

>>4167561
He probably means the kind of piss poor "proof" of why you can represent the momentum operator as a derivative of position that is provided in every basic QM course.

Look, OP. The real proof requires group theory, and makes no assumptions on the form of the wavefunction.

>> No.4167587

As a chemist, thank fuck I have a computer to approximate this stuff for me.

>> No.4167625

>>4167587
>doesn't learn shit because you have a computer to do your work for you

Worse than engineers.

>> No.4167627
File: 37 KB, 800x600, 1312929258721.png [View same] [iqdb] [saucenao] [google]
4167627

>>4167561
>>4167575
Yeah, where they do some shit like
<span class="math">p \psi \to -i\hbar \, \partial \psi / \partial x = -i\hbar \partial e^{i(kx-\omega t)} / \partial = -i\hbar (ik) \psi = \hbar k \psi = p \psi[/spoiler].
Which is great and all, but it assumes <span class="math">\Psi[/spoiler] is always of that form.
And now that you guys say it, since any function can be broken down to linear combinations of complex exponentials, then I guess that would still work.

>>4167575
Hm, I actually might be able to loosely follow that kind of proof. I haven't had Group theory yet, but I've had a shitload of linear algebra; and as I understand it, a group is just a vector space with an additional associated operation -- I could be way off though.

>>4167587
Pic related! It's a joke of course... but still

>> No.4167641

>>4167627
That LaTeX came out really ugly, but you get the point.
(Also, testing... <span class="math">\LaTeX[/spoiler])

>> No.4167645

>>4167627
Haha, can't believe someone saved that

>> No.4167653

>>4167645
Ha, did you make that? I didn't even know that, but I've seen it posted around a few times :P

>> No.4167654

>>4167627
Haha, yeah I do wish I could do the actual mathematics behind it but for the size of things we work with it's hugely impractical, and we have so much other theories and approximations and reactions to learn I think I'd go crazy if i had to understand the math as well. But that's what you guys are for!

We learn shit like left hand side is physical chemistry btw.

>> No.4167667

ok, the complete argument is:
>the wave-function is a wave, thus it will be better to represent it as a sum of periodic function (if its not periodic, just set the period to infinity)
>the simplest functions are sin(ax-bt) and cos(ax-bt) or using Euler identity e^i(ax-bt)
>it can be shown that e^i(ax-bt) form a complete bases thus any wave-function can be writen as a sum of it.
>a and b must transform like x and t and the only 4vector available is (p,E) so except for some constants a = p and b = E

the reason why it must be of the form ax-bt can be found in some wave mechanics book, but its pretty simple:
>after a time t the point at x should be at x+vt with v the speed of propagation, so at time t, the wave will have the value x-vt at x. this can be multiplied my a constant if you want to get ax-bt with b = av.

dont know if this is wat you wanted.

>> No.4167668

>>4167654
Yeah that's what I've heard. Physical chemistry is pretty legit.
As much as physicists bag on chemists, you guys are doing the right thing. If everyone was doing physics, we wouldn't even know shit about basic shit like molecules.

>> No.4167669

>>4167653
<span class="math">\ddot\smile[/spoiler]

>> No.4167698

>>4167667
Yeah that's nice, but I always get hung up on why the wavefunction has to be a wave ("blah blah, otherwise they wouldn't call it a <span class="math">wave[/spoiler]-function, etc...").
I mean there is no reason you couldn't (at least initially) set up a particle in the state <span class="math">\Psi (x,0) = Ax (a-x)[/spoiler] (on the appropriate interval and in the ISW for example).
I understand that <span class="math">\Psi[/spoiler] will evolve with time to resemble sinusoids, but I've never seen an explicit reason <span class="math">\Psi ~ has[/spoiler] to be that way.

I've heard a lot that it's because the Shrodinger equation is a 'wave'-equation. But it doesn't like your typical wave equation to me (ie only one time derivative and then there's the potential smack dab in the middle of it).
Can anyone explain this more?

>> No.4167699

>>4167668
It's okay, we have biologists to rip on for the exact same reason.

I still feel like a halfwit when I look at some of that shit in my textbooks though. Same with deriving the mass-transfer stuff the chemical engineers do, though their approximations are particularly horrific because if the volumes of stuff they're working with.

Apologies for thread derailment.

>> No.4167730

>>4167698
i can give you the historical reason, light travels as waves, then light was particles, then debrogly decided to do the reverse with particles, now they are waves with a frequency of whatever. so particles and light is the same and light has the Maxwell equations so particles need such an equation, that's the Schrodinger equation. so basically it started because light travels as a wave. but as in >>4167667

>> No.4167793

>>4167730
That's actually pretty interesting, I hadn't considered that.
So the argument is not mathematical, but rather physical<span class="math">{}^1[/spoiler] -- that <span class="math">\Psi[/spoiler] should be wave-like because particles represent deBroglie waves of probability density.
But isn't the deBroglie wavelength supposed to represent a real (as opposed to probabilistic) wave? I could be mistaken on that and, come to think of it, I've never had a teacher who expounded on what kind of waves the deBroglie matter waves are.
(<span class="math">{}^1[/spoiler]I'm sure there still is a rigorous mathematical justification though.)

>>4167699
No problem dude, I think we're still on topic for the most part!
I actually don't know what kind of math chemists do. But it must be nice that it's not as bad as the shit in physics or ChemE, like you say.

>> No.4167847

>>4167793
the deBroglie waves come from the "old quantum theory", id suggest you look into it if you are interested in the whys of QM. old quantum theory didnt realy make mutch sense thought, at first the deBroglie waves were thought to be real waves and anly later did the probability interpretation come in. it
at first the action of a particle was thought to be quantized, this gives the correct electron orbitals (sometimes) but didn't work great
then deBroglie found that if its waves you get the same answers as if thr action was quantized
after that Schrodinger made a wave equation to explain how matter waves propagate

meanwhile Heisenberg made matrix mechanics that handled the profitability of how electrons jumped between energy levels, but said nothing of how they move. (thats where the eigenvalues and vectors started)

the wave equation was seen to be able to be used in the same way as the matrix in matrix mechanics and thus they combined it and say the wave is a probability wave

>> No.4167854

>>4167847
profitability -> probability

>> No.4167862

>>4167847
Haha, I literally just read about old quantum theory.
I'll go back and reread it now for clarity and whatnot. It's interesting to see the first attempts at constructing QM.

I'm also very interested in how Shrodinger came up with the wave equation. I found this paper on a derivation
http://www.ks.uiuc.edu/Services/Class/PHYS480/qm_PDF/chp3.pdf
but it's really messy and there is some notation I don't understand.

I'll keep looking though, thanks for your help!