/sci/ - Science & Math

[Anonymous No.5125875]

There is quite a large difference (conceptually) between the wavefunction as taught in introductory courses <span class="math">psi(x)[/spoiler]
and the wavefunction as taught in a more advanced QM course <span class="math">|\psi\rangle[/spoiler].

The later is more abstract and lives in a Hilbert space, of which the former is only a projection of onto a particular basis (i.e. <span class="math">\psi(x)=\langle x|\psi\rangle[/spoiler])

The thing is... They spend some time explaining why we need quantum theory over classical theory, and even attempt to "derive" the Schrodinger equation to kind of give us some justification for this new way of doing physics.
But, when you make the jump from <span class="math">\psi(x)[/spoiler] to <span class="math">|\psi\rangle[/spoiler] no explanation is given.

I do understand that it is the more powerful formalism and when we use it, the predictions made are very accurate and things like that, so it can be justified that way.. But is there any direct line of reasoning to show why <span class="math">\psi(x)[/spoiler] is inadequate and we need to make the jump to the more generalized and abstract <span class="math">|\psi\rangle[/spoiler]?

[Anonymous No.5125907]

>>5125875

(not OP) also interested in how /sci/ might explain this

[Anonymous No.5126027]

Or another way of wording it is the following:

It's easy to see how the wave mechanics comes out of the higher level formalism (that is going from <span class="math">|\psi\rangle\rightarrow\psi(x)[/spoiler])
But how can we see that it should go the other way? (<span class="math">\psi(x)\rightarrow |\psi\rangle[/spoiler])

Could someone explain the logical reasoning, or at least how was that conclusion _historically_ done? There must have been something... Even if it wasn't all that logical, how did we arrive at that?

Bonus points if you can say how it is logical that given what we had that we needed to have a more generalized wavefunction in a hilbert space.

[Anonymous No.5126048]

>>5126027
Hilbert Spaces are homomorphic to function spaces.

http://en.wikipedia.org/wiki/Functional_analysis

[Anonymous No.5126081]

Are you a troll? Both are taught in remedial elementary QM.

A quantum mechanical system is a *-algebra <span class="math">(A,(-)^*)[/spoiler] over <span class="math">\mathbb{C}[/spoiler].

Every *-algebra has a commutator, this is given by the Poisson algebra.

An observable is some element <span class="math">a\inA[/spoiler] compatible with the commutator

Any state is a linear function mapping <span class="math">A\to\mathbb{C}[/spoiler]

Take <span class="math">A[/spoiler] with a state <span class="math">\rho[/spoiler] on it, and you get an inner product space which is your Hilbert space, <span class="math">\mathcal{H}[/spoiler]

Rays in this Hilbert space correspond to state vectors, your <span class="math">|psi\rangle[/spoiler].

There it is a group of unitary operators on <span class="math">\mathcal{H}[/spoiler] induced from the observables in <span class="math">A[/spoiler], this is called the Heisenberg group.

Your <span class="math">\psi(x)[/spoiler] is the state vector in a position basis... do you know what an operator is? The vector space of bras is the dual space to the vector space of kets. This means that the self-adjoint linear operators output complex numbers when they act on a state vector, and this is equivalent to observation.

There are other operators besides position, DURRRRR. What if I want <span class="math">\psi[/spoiler] in momentum space? What if it has time dependence? What if it has spin?

[Anonymous No.5126084]

>>5125875
>2012
>thinking QM only has a shitty position operator
ISHYGDDT

[Anonymous No.5126088]

>>5125875
Hey Op, do you know that you can disprove your shit with a simple symplectomorphism of the Hamiltonian?

U a retard?

[Anonymous No.5126091]

>>5125875
How many cocks do you suck a day OP? Do you realize you need momentum space for any serious calculation in QFT?

[Anonymous No.5126094]

>>5125875
>babby's first QM
Did you ever get used to the taste of your fathers cum?

[Anonymous No.5126100]

>>5125875
OP is a sad pathetic troll

\thread

[Anonymous No.5126103]

>>5125875
It's cute that you want to learn QM, but you should read a textbook before making up this childish garbage

<<<

[sage No.5126108]

>>5125875
Let me guess OP, you're an engineer. It is evident by all your bullshit and lack of basic physics knowledge.

[Anonymous No.5126111]

>>5125875
LMFAO.

According to your reasoning, you only need position basis to do anything in QM WTF? How do you even have time evolution without a unitary operator?

Please, use real representations of CP(n) with actual functional analysis and different basis and operators, and not this retarded shit.

[Anonymous No.5126116]

I feel like you answered your own question, we use it because it is more general. Just like any derivation or theory you want to prove things in the most general sense, then apply it to specific cases if you need.

[sage No.5126114]

>>5126027
you sound like a faggot (engineer)

[Anonymous No.5126118]

Serioulsy OP, u a 10 year old girl?

[Anonymous No.5126122]

>>5126084
>>5126088
>>5126091
>6
>>5126094
>>5126100
>>5126103
>>5126108
>>5126111
For fuck's sake, quit samefagging and bumping this shitty thread. Jesus.

[Anonymous No.5126124]

>>5126114
>>5126084
>>5126088
>>5126091
>>5126094
>>5126100
>>5126103
>>5126108
>>5126111
OP asks a serious question and expresses his intention to learn about QM. What are you doing? You shit up his thread with insults and imbecilic reaction images. Please be a worthless piece of shit somewhere else. Yes, I'm mad.

[Anonymous No.5126134]

>>5126081
I was only using <span class="math">\psi(x)[/spoiler] as an example.

and yes I know it's the coefficient of the expansion of <span class="math">|\psi\rangle[/spoiler] into the position basis, I already said that in the OP, did you miss it?

You've just defined everything in that formalism of QM.
My point is we didn't start there did we? We started with those functions that we know look at as projections... The <span class="math">\psi(x)[/spoiler], <span class="math">\psi(p}[/spoiler], etc..


But I think I get it now, thanks for your reply.
I guess it comes from the fact that wavefunctions were written as linear combinations of eigenstates of some operators, and the fact that we could transform between sets of eigenstates that lead to the idea that a state could be described in a more abstract way and the eigenstates could act as eigenvectors in that space.

[Anonymous No.5126141]

>>5126134
You got it. And you're not an idiot, that guy is just an asshole. They probably should have covered it in your course, but it is one of the more complicated ideas to understand in intro QM.

[Anonymous No.5126150]

>>5126124
It's alright.

That samefag just wants to sound like he's smart because he can regurgitate the theory but he couldn't explain how to build up that formalism from physical observations and logic to save his life.

[Anonymous No.5126168]

>>5126150
I've seen him in other threads as well over the last few weeks. He does this whenever someone asks a question about theoretical physics. I don't know what severe mental illness he has, but this spam shit definitely has to stop. I'm okay with namefags shitposting in shit threads or elaborate trolls derailing threads, but the sick behaviour this guy shows is really going too far. Shitting up legitimate physics threads with such immature and utterly retarded spam is not acceptable anymore. Please help getting him banned as quick as possible by reporting his posts.

[Anonymous No.5126186]

>>5126134
We did, you take all classical observables (symplectomorphisms of the Hamiltonian) and promote them to operators.

This experimentally comes from the de Broglie relations and the uncertainty principle

The Hamiltonian generates time evolution with the Poisson bracket, this is what you replace

http://en.wikipedia.org/wiki/Deformation_quantization

And no, it does not come from the density matrix being written as eigenstates of operators. The main realization is that the classical Hamiltonian evolves with the Poisson bracket, and if we substitute this with a commutator expected from http://en.wikipedia.org/wiki/Matter_wave#Quantum_mechanics you get quantum mechanics

[Anonymous No.5126231]

I'm this guy:

>>5125907

And thanks fellas, this thread helped me quite a bit. I was fuzzy on this as well, perhaps because I took quantum before classical.

[Anonymous No.5128494]

Reading this thread I really feel dumb.

Can anyone link me to a good introduction to *-algebras and symplectomorphisms (whatever that is)?

[Anonymous No.5129639]

>>5128494
bump

Can anyone point me in the right direction? I don't want to start a new thread for this.

[Anonymous No.5131639]

bump
>pls respond

[Anonymous No.5131662]

>>5131639
I don't know man, I wish I could help, but most of us learned that in class.
Look in the sticky, there should be algebra resources.

[Anonymous No.5131679]

I think the main reason is because by the point that math started being used, there weren't any physicists capable or willing to bridge the gap between math and philosophy anymore. Fortunately, the philosophy comes back if you get to QFT.

[Anonymous No.5131683]

>>5131662
Can you give me some keywords? What branch of math is it? Is it algebra or topology or ...?

[Anonymous No.5131687]

>>5125875
So has anyone actually done R&D on the finglonger?

[Anonymous No.5132847]

Come on, guys.

[Anonymous No.5132851]

>>5131683
It's algebra. Just plain algebra.

[Anonymous No.5132858]

>>5131683
What are you trying to learn?

[Anonymous No.5132859]

>>5132851
What book covers these topics?

[Anonymous No.5132867]

>>5132858
I'm not sure. I'd love to understand everything posted ITT.

[Anonymous No.5132874]

>>5132867
Lecture notes on QM?

[Anonymous No.5132881]

>>5125875
I don't understand your problem.
The vector notation is just used when there are mutliple possible psi(x)'s.

What is hard to understand about that?

[Anonymous No.5132894]

>>5128494
>*-algebras
http://www.amazon.com/Algebra-2nd-Michael-Artin/dp/0132413779

>symplectomorphisms
http://www.amazon.com/Mathematical-Classical-Mechanics-Graduate-Mathematics/dp/0387968903


http://www-cip.physik.uni-bonn.de/~baehren/scripts/quantum.html
http://www.math.sunysb.edu/~mmovshev/MAT570Spring2008/syllabusfinal.html
http://math.berkeley.edu/~alanw/GofQ.pdf
http://www.amazon.com/Introduction-Mathematical-Structure-Quantum-Mechanics/dp/9812835229

[Anonymous No.5132971]

>>5132894
Thank you.