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vincentchan
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Newton's equation (F=ma) could derive from Lagrangian, My question is, could we derive the schrodinger equation from the more fundemantal principle in Physics...
I'm not sure, but I don't think so. I've only seen it used as a postulate in quantum mechanics. I've never seen anyone derive it from more fundamental principles.vincentchan said:Newton's equation (F=ma) could derive from Lagrangian, My question is, could we derive the schrodinger equation from the more fundemantal principle in Physics...
Kane O'Donnell said:On the other hand, it would be *very* nice to have a theory which shows *why* Dirac's commutator postulate is correct.
Kane O'Donnell said:I tried to do a simple summary of what Stone's theorem says but it got a bit technical. Basically it says that some of the unitary operators on a Hilbert space have an associated self-adjoint operator. The Hamiltonian is the self-adjoint operator associated with the time-evolution operator, and the technical details of the association give the time-dependend Schrodinger equation.
Since the quantum wavefunction [itex]\psi(x,t)[/itex] behaves like a classical field, it's equations of motion (Schrödinger's equation) can be derived by making extremun certain action (see for example, José, J. V. and Saletan, E. J., "Classical Dynamics: A Contemporary Approach", Cambridge University Press, 1998, chapter 9 (I really like this book )), but hey, that's just mathematics, remember that even if the wavefunction behaves like a classical field, it's not one, it's physical interpretation is not classical at all.vincentchan said:I deeply believe the laplacian in schrodingers' equation is came from the hamilton's principle...
I just want to know am I correct...
You mean (graded) Dirac brackets,right?
vincentchan said:Newton's equation (F=ma) could derive from Lagrangian, My question is, could we derive the schrodinger equation from the more fundemantal principle in Physics...
Is it really "lies to students", or just a simple confirmation that whatever the bigwigs set to be their axioms are actually compatible/in accordance with basic physics that came before? Because if those "justifications" are just BS, and we're actually being lied to, then that's disillusioning...
dextercioby said:Well,Marlon,i guess your idea works iff you postulate the (quantum spinless nonrelativistic) free particle's wavefunction and then u'd still have to derive the equation which has the advantage to be useful in any occasions,while the plane wave,not...
dextercioby said:then u'd still have to derive the equation which has the advantage to be useful in any occasions,while the plane wave,not...
dextercioby said:What do you mean,that u can use the plane waves without getting Schroedinger's equation??
What is wrong there??
Daniel.
dextercioby said:Hold on,what are you trying to say?The SE is indeed postulated in the general form it has (Dirac formalism).Sakurai shows how it can be derived...As a consequence of other principles...
The particularity of Sakurai's book is the absence of AXIOMATIZATION...
Logically, correct. I would diagree, though i don't understand what you are saying here. What is unlogical about deriving the SE from the wavefunctions. In my opinion, it is the best and most logical way to explain why the SE has the particular structure that it hasI simply rejected the way of deriving the SE from the wavefunctions...It's the other way around,logically correct.
Daniel.
dextercioby said:1.Conservation of sum of probabilities (the sum of the square modulus of the Fourier coefficients) with time evolution (similar,the conservation of normalization in time) => unitary operator of time evolution => inifinitesimal time evolution operator=>=>(on the basis of Stone's theorem) pops up the self-adjoint generator of infinitesimal time translations=> (analogy with CM) =>Hamiltonian (the operator) as the generator of time translations.Then the ODE for the time evolution operator and then SE for state vectors...
2.How on Earth would u find the wave-functions without GETTING THE SE FROM SOMEWHERE (either by postulating it,or as Sakurai "digs" it)?
P.S.What do you mean "particular structure"...It's DAMN GENERAL AND ABSTRACT,at least in the Dirac formulation and no choise of representation whatsoever...
Loren Booda said:Reasonable assumptions concerning the Schroedinger equation include the de Broglie (momentum-displacement) and Einstein (energy-frequency) postulates, along with the conditions of linearity, constant potential for free particles and reliance on the Hamiltonian energy formulation.
marlon said:Very easy,...the wave function (or at least the concept thereof) comes from the double slit experiment.
marlon said:. I strongly urge you to check out a QM-textbook and just look at how the SE is introduced.
marlon said:I have heard people whinning about the fact that this way is just the introductory way if introducing the SE but there ain't no other way...C'est tout...
marlon said::rofl: This clearly proves you don't know what we are talking about. I am just referring to the actual SE that is all, i am sure you know its equation. :rofl:
marlon said:No need to bring such vague terms as Dirac-formulation and representation of the SE...we are ONLY :rofl: talking about the SE :rofl:
That is what i said, so let's just drop thisdextercioby said:How do we get the mathematical expression for the wavefunction?Analogy with optics + DE BROGLIE's CONDITION...without the latter,we wouldn't make any difference between light waves & matter waves...
yes that is the one...so my answer still stands.So you were talking about this:
[tex] i\hbar \frac{\partial \Psi(\vec{r},t)}{\partial t} =-\frac{\hbar^{2}}{2m}\nabla^{2}\Psi(\vec{r},t) +V(\vec{r},t)\Psi(\vec{r},t) [/tex]
In that case,i wasn't talking about that...I was stalking about the one that's postulated...Not derivable...But i know what I'm talking about...
marlon said:yes that is the one...so my answer still stands.
Can you give me the equation of "the one that is postulated" because i don't know what you are talking about here.
marlon
dextercioby said:Surely.
[tex] \frac{\partial |\Psi (t)\rangle}{\partial t} =\frac{1}{i\hbar} \hat{H} |\Psi (t) \rangle [/tex]
This one of the 2 (completely equivalent) equations which CONSTITUTE THE IV-TH POSTULATE OF QM IN THE DIRAC/TRADITIONAL FORMULATION IN THE SCHROEDINGER PICTURE...
marlon said:Look this is all very true...But where is the proof ?
Besides i still wonder where the actual proof for the SE remains ?
Why is it that this equation is valid ? Besides, why is the time-evolution operator necessarily unitary ? These questions have very clear answers but i have not seen them in your posts. Now, if you don't want to answer them, i will do it myself. You got one more chance
marlon
dextercioby said:I believe that all these questions have been answered by Sakurai in his book and i gave them a few posts ago,when you asked me for a derivation of the equation is posted as being traditionally postulated.And i gave you exactly the one from Sakurai...
I still like the postulates...They give an unitary axiomatical (hence logically rigurous) structure to QM...If you want to erase them,be my guest...But,as i probably may have said before,i wouldn't want to be taught QM by you... :tongue2:
Daniel.