What are the Different Formulations of Quantum Mechanics?

[sheaf]

Sorry mate - you can derive it eg see Chapter 3 of Ballentine or for an even deeper look the book by Varadarajan.

Its physical basis is space-time symmetry invarience and is not an axiom per-se. Specifically it's Galilean symmetries that leads to Schrödinger's equation. In fact of course its wrong because Galilaen relativity is wrong - the correct symmetries are relativistic and leads to the relativistic equations such as the Dirac equation.

That's one reason why I always recommend Ballentine as the textbook to learn QM from - he does the treatment correctly.

Thanks
Bill

Unfortunately I don't have Ballentine or Varadarajan, so I thought I'd ask:

I'm guessing (maybe wrongly?) that what they derive is the single particle Schrödinger equation in coordinate space. But can they really derive the functional Schrödinger equation which must be satisfied by the states (discussed in Hatfield's book). Isn't this fundamental/axiomatic?

 

[bhobba]

But can they really derive the functional Schrödinger equation which must be satisfied by the states (discussed in Hatfield's book). Isn't this fundamental/axiomatic?

Its derived for multi particle states. Whats fundamental and axiomatic depends on your choice of axioms. Ballentine derives it from the invarience of Born's rule under space-time rotations and translations which in implied by Galilean relativity. Most would consider that more fundamental than Schrödinger's equation because it is a general law of physics applicable to say EM as well - not just QM.

Specifically what he shows is the standard energy, momentum, and angular momentum operators must have the form of classical mechanics. In many cases of practical interest that's enough to quantize the system - its more general than the Poisson Bracket approach ala Dirac but not as powerful as the geometric approach - but that is mathematically quite advanced - still its the most powerful approach we have.

The standard cookbook methods in the usual undergrad texts like Griffiths skirt of over these issues. Applying Schrödinger's equation in many cases is not prescriptive. What these approaches attempt to do is rectify that - but unfortunately problems remain.

Thanks
Bill

 

[bolbteppa]

That paper http://www.mpipks-dresden.mpg.de/~rost/jmr-reprints/brro01.pdf above is quite frankly amazing, I wish I'd come across it before starting this thread. In it they begin with Schrödinger's original derivation of the TISE (albeit in more modern language, coincidentally in terms of the operators rubi said would prevent Schrödinger's derivation from holding true!), discuss Schrödinger's problems with the TDSE, go on to derive the TDSE from the TISE, discuss the importance of treating time as a classical quantity alien to quantum mechanics that arises in the TDSE as the result of a closed quantum system described by the TISE interacting with a classical external environment, speak about flaws in assuming time being an operator in QM, make a very interesting argument about the complex number in i\bar{h}\tfrac{\partial }{\partial t} as arising only due to interactions with the external environment & not always being necessary (I think) & they also make an argument for the necessity of complex solutions in the TDSE as arising from interactions with the external classical system. I'd think this were a crank paper if it wasn't by a person in the Max-Planck-Institute...

In the paper they derive the TDSE by assuming the TISE holds for the system & the environment together. In other words, they derive the TDSE from the TISE equation which was derived from fully classical principles, the point I've been making throughout this thread. Apparently it is within that complicated derivation they give that one will find the true reason why we all need to assume axioms. It's too difficult for me right now, but if anybody found this thread interesting & is interested in finding an answer this is the place to find it. If it's interesting enough I'd love it if that person would dumb it down for me, if not I'll eventually get there (I hope)

A less important point - they make a comment

In modern quantum mechanics textbooks little reference is made to Schrödinger's order of development or to his difficulties with the TDSE. Rather the TDSE is simply presented as the fundamental equation of wave mechanics from which the TISE (and hence a wavefunction with the exp(-iEt/(h/2pi)) factor) is derived as a special case for time-independent Hamiltonians. No mention is made of the fact that time is entering only from a classical interacting environment or that the TDSE does not correspond to energy conservation (the fundamental equation of wave mechanics violates the fundamental principle of physics)
Page 6

What do they mean by the fact that the fundamental equation of wave mechanics violates a fundamental principle of physics?

I think it relies on the fact time is involved in the equation (in the paper they stress the fact that time is a classical quantity whereas we're dealing with a quantum system) & it seems to me that they claim time only enters the TDSE by considering the TISE as fundamental & then imagining interaction of a closed system with a classical external environment as what changes in time represents:

the starting point is the TISE for a closed, energy conserving, quantum object comprised of two parts, called the system and the environment. In the limit that the environment can be treated classically, it provides a time variable with which to monitor the remaining quantum system whose development, as viewed from the environment, is governed by the TDSE for the system alone. This derivation shows explicitly that the origin of the classical time in H(t) is due to coupling with the classical environment, and that the parametric derivative  \partial/\partial t arises from the transition of environment variables from quantum to classical behaviour.
Page 2

Obviously time in a classical potential represents loss of conservation of energy, so I'm wondering how to make sense of all this here. In QM time in a potential allows "for transitions between one energy level & another" (Griffith QM P298), where the transition is caused by interactions with the external classical system as per the paper, yet I don't get what they mean by the TDSE violating fundamental principles, anybody have an idea?

 

[bhobba]

Schrödinger's equation is not derivable from classical mechanics. Every physicist working in quantum theory agrees with this. You can either postulate it directly or indirectly (using symmetry principles), but not derive it from classical mechanics. Please acknowledge this.

Of course it isn't - nor do I think it can ever be. Its just the physical principle that underlies CM is the same as in QM - the POR. But what the symmetries apply to is different - in CM its the PLA - in QM its the principles of QM (depending on what axioms you use). The PLA follows from those axioms so its hardly surprising there are analogies - but that's all there is. The Geometric approach to QM tries to pin down, as far as possible, exactly how QM and CM are related. A lot of progress seems to have been made, and I am surprised how far it has been taken, but there is zero doubt in my mind the program will always have issues. Still one never knows.

Just a personal comment. I have always been puzzled by Wigner famous paper:
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html

I now believe, along with Murray Gell-Mann, the reason for that is because each level is emergent from the level below it, so reproduces much of its formalism with a few twists.


QM and its relation to CM in an example of this. QFT and its relation to QM takes it further, and whatever theory lies below that (maybe string theory?) will as well.

Thanks
Bill

 

[bolbteppa]

Of course it isn't - nor do I think it can ever be. Its just the physical principle that underlies CM is the same as in QM - the POR.

Apparently that paper derives the TISE from the TDSE, in other words the TDSE is derived from an equation (the TISE) which itself was derived from classical mechanical principles. If this claim is incorrect it lies in the approximations in the derivation of the TISE, i.e. the derivation reaches a point at which classical mechanics simply disappears & becomes impossible to sustain. Up to this point, which I can't fathom in my persual of the derivation around page 12 of that article, I think I'm correct in saying that everything is completely classical & that hasn't been challenged by anybody yet.

Am I right in saying this? I think I am.

If it interests you enough to read that derivation please let me know where the point of no return lies & why it's unavoidable, maybe you'll see how it relates to what Varadarajan does.

 

[bhobba]

Apparently that paper derives the TISE from the TDSE, in other words the TDSE is derived from an equation (the TISE) which itself was derived from classical mechanical principles.

Sorry - must have missed the derivation of the TISE from classical mechanics principles.

The key thing is what you mean by classical mechanics principles

The POR is a classical mechanics principle and it can be used to derive QM and CM. But other things come into it as well - namely exactly what is the POR applied to - in CM its the PLA, in QM its the two axioms (or other equivalent ones) I gave. Those axioms are fundamentally different because CM and QM are fundamentally different. The PLA is a limiting case of the axioms of QM - the reverse is not true - nor can it be - there is no way one can derive QM from CM. The geometrical approach looks for formal connections at a deep level to elucidate exactly how you can figure out to quantize a classical system. But they are nothing but formal connections - QM is not derivable from CM.

No mate - I don't really have any zeal for finding the errors in claims like this. It's obviously not possible - its like the proofs of one equal zero - you know there is a division by zero somewhere - its the same here - they are making some assumptions about QM and apply CM principles to it - but those assumptions are different to CM - as they must be because QM is different to CM right at its foundations.

You know this because the axioms of QM and the PLA are different - one implies the other - but not conversely.

Thanks
Bill

 

[bolbteppa]

I apologize if I gave that impression but the authors of that paper are not making any assumptions about QM & smuggling CM into it, they are merely using ideas of Born & Mott around 1931 to complete Schrödinger's 1926 derivation of the TDSE from the TISE, & they are analyzing the literature in the rest of the paper - that's it. I don't think it's fair to simply write off people in the Max Planck institute as just smuggling in division by zero into their papers & ignore them, you don't have to read it but there's no need for comments like that.

After this it's my question as to whether Schrödinger & those authors are, up to some assumed & as-of-yet unlocated point in the derivation of the TDSE from the TISE, in fact completely grounded in classical mechanical principles by virtue of the fact that Schrödinger's original derivation is all based on applying the calculus of variations to the Hamilton-Jacobi theory, I don't think there is an error in saying this - I don't know - but I'm not going to change my mind based you guys just telling me in a matter-of-fact fashion that it can't be done when apparently it can, or at least the reason it can't be done lies in a complicated derivation I've linked to. Thus far none of you guys have addressed the point about the TISE completely encoding classical mechanics in it's derivation & that it's only difference is complex-valued eigenfunctions, if you don't know how to address this point that's fine, honestly, & thanks for the help thus far, but remember your difficulties with this idea lie in the fact that the TDSE is apparently the reason why QM differs from CM & that the TDSE derives from the TISE so something about that derivation is important enough to force the entire theory of QM onto us (unless I've missed something about Schrödinger's derivation you guys can enlighten me about!).

 

[bhobba]

I apologize if I gave that impression but the authors of that paper are not making any assumptions about QM & smuggling CM into it, they are merely using ideas of Born & Mott around 1931 to complete Schrödinger's 1926 derivation of the TDSE from the TISE,

I don't know anything about the TDSE from the TISE thing - I have zero idea if you can get one from the other - nor am I particularly interested in it. My objection is you can't get any form of Schrödinger's equation from classical mechanics - its simply not possible regardless of what institute they come from.

I have seen Schrödinger's derivation and he did NOT derive it from classical principles but from the idea if you have a wave aspect to particles you should have a wave equation and proceeded to figure out what the most reasonable one would be. You can do it too - take the DeBrogle wave of a particle - transform any wave to its Fourier components via a Fourier transform then relate those components to the De-Brogle wave and you can easily show it obeys the Schrödinger equation. That's pretty much all there is to it - similar 'derivations' are found in most of the usual undergrad texts on QM - I seem to recall Griffith did something similar. However it is NOT a derivation from classical principles - nor can it be.

If you would like to post the derivation of any form of the Schrödinger's equation from classical mechanics you may get someone to look at it to find the error. I seem to recall one was discussed ages ago and the error was reasonably easy to spot - it must be there. But post away and we will see.

You seem to understand the fundamentals of QM - it should be pretty obvious you can't do this.

Added Later:'
Here is a derivation of Schrödinger's equation along the lines he used:
http://arxiv.org/pdf/physics/0610121.pdf

Notice the fundamental quantum assumptions it makes like the energy and momentum of a photon as well as de Brogle's assumptions. They are NOT classical assumptions of any form.

Thanks
Bill

 

[bolbteppa]

I posted a link to it in my OP, here is Schrödinger's original derivation again, furthermore it is derived http://www.mpipks-dresden.mpg.de/~rost/jmr-reprints/brro01.pdf in terms of operators completely analogously to the way Schrödinger did so, & a discussion of the meaning of Schrödinger's derivation & historical ignorance of this is also discussed in that article. All this contradicts your claims about it not being possible, at least in the time-independent case - and this is not a derivation involving DeBroglie wave-particle duality, that came after Schrödinger offered the derivation given twice above, & it only came about because he "sought to develop a connection between his own work and the wave theory of DeBroglie" (Weinstock P264). If you take the time to read this think about the fact it is derived solely from Hamilton-Jacobi theory and basic calculus of variations, nothing more, & that complex eigenfunction fall out of it as necessity. Then the TDSE derivation will hopefully seem more interesting.

 

[bhobba]

I posted a link to it in my OP, here is Schrödinger's original derivation again, furthermore it is derived http://www.mpipks-dresden.mpg.de/~rost/jmr-reprints/brro01.pdf in terms of operators completely analogously to the way Schrödinger did so, & a discussion of the meaning of Schrödinger's derivation & historical ignorance of this is also discussed in that article. All this contradicts your claims about it not being possible, at least in the time-independent case

Yea yea - know that one - you should as well. Here is the rub (from the thread):

You have to CHOOSE K to be the pure IMAGINARY number −iℏ.

It's a wick rotation from a classical Wiener process. That this gives QM is a very interesting but well known fact. From classical principles it aren't.

I am surprised you didn't see it - it was more or less pointed out in the thread.

Thanks
Bill

 

[bolbteppa]

First my last response in that thread challenged him on his assertion about K being imaginary, read my response. Second refer to Weinstock page 262 to see K is most explicitly not imaginary. Third refer to Schrödinger's original paper "Quantization as a Problem of Proper Values I" page 2 to see even he defines K to be the real h/2pi. Fourth refer to that Max Planck article, page 16, to read them "argue that this term only arises in a classical approximation to the environment" which is most explicitly part of the TDSE equation exclusively & to justify this in their derivation. In other words I did see it (refer to that thread), I've offered two justifications for it in the past, I mentioned it in this thread not 10 posts ago to someone else & here I've provided two more reasons, now that's four objections.

 

[bhobba]

First my last response in that thread challenged him on his assertion about K being imaginary, read my response. Second refer to Weinstock page 262 to see K is most explicitly not imaginary.

Well if you know a derivation that doesn't use complex numbers - post away. Not a page in some book - but the actual derivation.

It must - if not its a contradiction to wick rotation which is a very well known mathematical procedure.

Also if you really want to continue that discussion, its probably better to do it in that thread, not start another one that eventually gets around to it.

Thanks
Bill

 

[bolbteppa]

Well if you know a derivation that doesn't use complex numbers - post away. Not a page in some book - but the actual derivation.

It must - if not its a contradiction to wick rotation which is a very well known mathematical procedure.

Also if you really want to continue that discussion, its probably better to do it in that thread, not start another one that eventually gets around to it.

Thanks
Bill

I linked to a separate post, which you apparently read, where I posted the derivation in detail - here it is again... Further your language here is impossible to satisfy, you ask me to post a derivation not using complex numbers (I've linked to it maybe 5 times now) yet then you tell me I simply cannot do this - am I wasting my time? Have you now conceded that K need not be imaginary in my derivation, or is this a game of just catching me out with any weapon possible?

 

[bhobba]

I linked to a separate post, which you apparently read, where I posted the derivation in detail - here it is again... Further your language here is impossible to satisfy, you ask me to post a derivation not using complex numbers (I've linked to it maybe 5 times now) yet then you tell me I simply cannot do this - am I wasting my time? Have you now conceded that K need not be imaginary in my derivation, or is this a game of just catching me out with any weapon possible?

In that derivation, as was pointed out, K must be complex. You posted 'Apparently Schrödinger was able to do what I have posted using real-valued functions & have K as I have defined it, without i.' Well where is it? The actual derivation - not saying someone was able to do it.

Its nothing more than a wick rotation -whether you can see it or not.

No use arguing any further - this is well known.

Thanks
Bill

 

[bolbteppa]

Okay that is fair enough. From Schrödinger's original paper:

"First, we will take for H the Hamilton function for Keplerian motion, & show that \psi can be so chosen for all positive, but only for a discrete set of negative values of E. That is, the above variation problem has a discrete & a continuous spectrum of proper values.
The discrete spectrum corresponds to the Balmer terms & the continuous to the energies of the hyperbolic orbits. For numerical agreement K must have the value h/2\pi"
Page 2

Then he spends 6 pages solving this problem & eventually derives on page 8 that E_l = \tfrac{mc^4}{2k^2l^2} & says:

"Therefore the well-known Bohr energy-levels, corresponding to the Balmer terms, are obtained, if to the constant K, introduced for reasons of dimensions, we give the value K = \tfrac{h}{2\pi}

This was obvious on a basic level from what I'd written, but here it is explicitly. I don't see a Wick rotation, but I do see pages & pages of justification for what I've been saying all along, which is why I haven't written this off so quickly...

 

[bhobba]

You are missing the point. As was pointed out the sign is wrong in the equation you posted. There is a negative sign in front of K. To get the negative value K must be imaginary. Also you have the wave function squared - but I assume that is a mistake.

I think that thread died for good reason - its just wrong on so many levels.

Anyway I will leave it to others to take up with you - its pretty obvious what's going on.

Added later:
I shouldn't have to post this - but the following contains the real time independent Schrodenger equation:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/scheq.html

What you posted aren't it.

Thanks
Bill

 

[bolbteppa]

There's no sign error in anything I've posted, at this stage you're not reading what I'm writing, ignoring every correction I make of your claims & looking for any old excuse to contradict me, which isn't getting us anywhere useful, so thanks for the help thus far but I see this going no further unless you're actually interested in the problem at hand, which you've said you aren't.

 

[rubi]

bolbteppa,

Here's why Schrödinger's derivation as given in the thread you linked earlier is wrong:

Now instead of solving this he, randomly from my point of view, choosed to integrate over space

I \ = \ \int\int\int_\mathcal{V}(\frac{K^2}{2m}[(\frac{\partial \Psi}{\partial x})^2 \ + \ (\frac{\partial \Psi}{\partial y})^2 \ + \ (\frac{\partial \Psi}{\partial z})^2] \ + \ (V \ - \ E)\Psi^2)dxdydz

& then extremizes this integral which gives us the Schrödinger equation.

This is a completely unjustified, random step. In a derivation, there are no random steps, however. Every step must be justified either by an axiom or by an already proved theorem. It's not the case for this step, so this "derivation" is flawed.--
Added later:
By the way, geometric quantization is just canonical quantization done right. Canonical quantization as proposed by Dirac can't work, because it's impossible to have $[\widehat A,\widehat B] = \mathrm i\hbar \widehat{\{A,B\}}$ for all observables (due to the Groenewold-van-Hove theorem). In geometric quantization, you choose the observables for which this should hold exactly and then allow additional $O(\hbar^2)$ terms for all other observables (very roughly speaking).

 

[bhobba]

There's no sign error in anything I've posted, at this stage you're not reading what I'm writing, ignoring every correction I make of your claims & looking for any old excuse to contradict me, which isn't getting us anywhere useful, so thanks for the help thus far but I see this going no further unless you're actually interested in the problem at hand, which you've said you aren't.

I read what you wrote.

You claim the equation you posted, which was NOT Schrödinger's equation, gives it on variation.

You didn't give this step - or many others for that matter - including why you should take the variation anyway - but simply made claims.

Now without doing that variation its pretty obvious it won't change a positive to a negative - if you think it does post the details.

Thanks
Bill

 

[bolbteppa]

bolbteppa,

Here's why Schrödinger's derivation as given in the thread you linked earlier is wrong:This is a completely unjustified, random step. In a derivation, there are no random steps, however. Every step must be justified either by an axiom or by an already proved theorem. It's not the case for this step, so this "derivation" is flawed.

I like that idea, however I don't see why it holds water. All you're doing is integrating an equation, there's nothing illegal in that. Then, as Weinstock say:

"He then poses the question: What differential equation must the function \psi satisfy if *the volume integral* is to be an extremum with respect to twice differentiable functions \psi which vanish at infinity in such fashion that *the volume integral* exists?

There's absolutely nothing wrong with doing this, nothing except genius as far as I can see.

Interestingly in Schrödinger's original paper I think he justifies this in the context of the Keplerian problem I mentioned above, i.e. I think he has good reason for this. Also in his paper I think he even justifies the substitution S = K\ln(\psi) as some form of converting an additive separation of variables problem (since we've started with the Hamilton-Jacobi equation) to a multiplicative one (in the Schrödinger equation). Crazy/genius... In other words, here are (I think) two justifications, one being that it's not illegal, per se, to do this, & second I think it's in the context of a physical problem that he can do this, but I'm not too sure about the second idea.

 

[bhobba]

By the way, geometric quantization is just canonical quantization done right.

Actually that's pretty much it stripped of its mind numbing math (I shouldn't be that uncharitable so is the math in QFT in my view) - good point.

Thanks
Bill

 

[bolbteppa]

This might be relevant:

The Schroedinger equation - Shortly after Heisenberg's work, Schroedinger came up with the equation that now carries his name. The essential idea was to start from the Hamilton-Jacobi equation, claim the action is the logarithm of some wave function psi (think WKB!), and derive a quadratic form of psi that is to be extremized (Schroedinger equation from the variatonal principle). This leads to the stationary Schroedinger equation, which he then solves for the hydrogen atom, as well as for the harmonic oscillator, the rotor and the nuclear motion of the di-atomic molecule (Schroedinger 1926a and Schroedinger 1926b).
http://theorie2.physik.uni-erlangen...ntum_Mechanics_(Lecture_by_Florian_Marquardt)

I don't know what it means to "derive a quadratic form of psi that is to be extremized", but I think it justifies why Schrödinger actually integrated the Hamilton-Jacobi equation.

However he does say in his paper:

"We now seek a function \psi, such that for any arbitrary variation of it the integral of the said quadratic form, taken over the whole co-ordinate space (I am aware this formulation is not entirely unambiguous) is stationary, \psi being everywhere real, single-valued, finite & continuously differentiable up to the second order. The quantum conditions are replaced by this variation problem".

It's ambiguous alright, but not illegal or flawed.

 

[bhobba]

There's absolutely nothing wrong with doing this, nothing except genius as far as I can see.

What genius? Why does doing that give an equation describing anything? It seems just like formal manipulations to me.

Thanks
Bill

 

[rubi]

I like that idea, however I don't see why it holds water. All you're doing is integrating an equation, there's nothing illegal in that. Then, as Weinstock say:


There's absolutely nothing wrong with doing this, nothing except genius as far as I can see.

Interestingly in Schrödinger's original paper I think he justifies this in the context of the Keplerian problem I mentioned above, i.e. I think he has good reason for this. Also in his paper I think he even justifies the substitution S = K\ln(\psi) as some form of converting an additive separation of variables problem (since we've started with the Hamilton-Jacobi equation) to a multiplicative one (in the Schrödinger equation). Crazy/genius... In other words, here are (I think) two justifications, one being that it's not illegal, per se, to do this, & second I think it's in the context of a physical problem that he can do this, but I'm not too sure about the second idea.

Please just do the following: Take a solution to the Schrödinger equation of the hydrogen atom for example and just insert it into the Hamilton-Jacobi equation of the hydrogen atom. Just do it. You will find that it does not solve the HJ equation! So the derivation must have been flawed!

Schrödinger is just using illegal steps in his "derivation". Please acknowledge this! I'm not going to argue about this anymore. You've been told by multiple people now that you can't derive the Schrödinger equation from classical mechanics.

 

[bhobba]

I don't know what it means to "derive a quadratic form of psi that is to be extremized", but I think it justifies why Schrödinger actually integrated the Hamilton-Jacobi equation.

And yet you think it somehow derives the Schrödinger equation and you don't even know what it means to carry out one of the important steps in its derivation?

Look this Hamilton Jacobi stuff is well known to give Schrödinger's equation - many textbooks do it - but you have to start from Feynman's path integral equation with its functional integral eg:
http://hitoshi.berkeley.edu/221a/pathintegral.pdf

But the key to its derivation is the complex numbers in the integral. That's the reason for my comment about wick rotation - you get a Wiener integral without complex numbers - and that is one of the basic equations of statistical mechanics - which is probably why entropy was introduced - to sneak this in via the back door.

Thanks
Bill

 

[bolbteppa]

Please just do the following: Take a solution to the Schrödinger equation of the hydrogen atom for example and just insert it into the Hamilton-Jacobi equation of the hydrogen atom. Just do it. You will find that it does not solve the HJ equation! So the derivation must have been flawed!

Schrödinger is just using illegal steps in his "derivation". Please acknowledge this! I'm not going to argue about this anymore. You've been told by multiple people now that you can't derive the Schrödinger equation from classical mechanics.

This just can't be true, & hilariously you picked the Hydrogen atom - go to page 271 of Weinstock, he quite literally solves the Hydrogen atom by first considering it as a volume integral over space & extremizes it with the explicit potential plugged into solve the problem - this couldn't be a more perfect refutation of your statements if I'd prayed for it.

At this stage you guys have to cut out the "you've been told multiple times" innuendo's & the insinuations that I'm ignoring people, or the 'we know what you're up to' stuff. I've refuted just about every issue you guys have thrown at me, sometimes in 2 if not 4 ways, so please end the character defamation & follow the logic of the argument here, I'm doing my best...

 

[rubi]

This just can't be true, & hilariously you picked the Hydrogen atom - go to page 271 of Weinstock, he quite literally solves the Hydrogen atom by first considering it as a volume integral over space & extremizes it with the explicit potential plugged into solve the problem - this couldn't be a more perfect refutation of your statements if I'd prayed for it.

You haven't done what I told you: Pick a solution of the SE and insert it into the HJE. It doesn't work out! The SE is inequivalent to the HJE! It has a different set of solutions.

 

[bolbteppa]

And yet you think it somehow derives the Schrödinger equation and you don't even know what it means to carry out one of the important steps in its derivation?

First off, in Weinstock he never mentions that thus my whole argument completely ignores it. It merely addresses a potential motivation for doing something completely legal, so I'm sorry this is not a weapon to wield against me, though I'm glad you find the problem interesting enough to comment on again.

Look this Hamilton Jacobi stuff is well known to give Schrödinger's equation - many textbooks do it - but you have to start from Feynman's path integral equation with its functional integral eg:
http://hitoshi.berkeley.edu/221a/pathintegral.pdf

Apparently not, we have Weinstock deriving it straight from a volume integral of the Hamilton-Jacobi equation, & Schrödinger deriving it from a volume integral of a Hamilton-Jacobi equation which he justifies by this quadratic form stuff, which I'm thinking might just be a small-angle approximation or something, but I don't see how it even matters quite honestly.

But the key to its derivation is the complex numbers in the integral. That's the reason for my comment about wick rotation - you get a Wiener integral without complex numbers - and that is one of the basic equations of statistical mechanics - which is probably why entropy was introduced - to sneak this in via the back door.

Thanks
Bill

Again, no complex numbers feature thus far & no Wick rotations. They may very well be necessary but I quite simply do not see why & would love to see this without blindly assuming anything unless there's no other reason, I think that's reasonable enough. When I thought my post was incorrect due to this all deriving from time-independent potentials I was then willing to start accepting axioms, but now as it stands you can apparently derive it all from time-independent potentials thus I may not need to accept anything on faith, I won't know until I have my issues addressed.

 

[bhobba]

Ok - I have got to the bottom of it and found a paper examining Schrodenger's original derivation:
http://arxiv.org/pdf/1204.0653v1.pdf

See section 8. Schrödinger introduces K but it needs to be -ihbar to give the Schrödinger equation - as you can see in section 8 his reasoning is round about, tortuous and incorrect. This is exactly what was pointed out to you right from the start. As the article states 'This ansatz is the same as the fundamental postulate II of Feynman’s formulation of quantum mechanics, for the spatially-dependent part of the path amplitude, on making the replacement'.

The reason Schrödinger's derivation works is complex numbers introduce phase so we get path cancellation - its the same reason a wick rotation from a wiener process works and one of the deep reasons you need complex numbers in QM. But he didn't get it right so had to introduce a 'compensating' step - the variation step - but two wrongs, while giving the right answer - don't make a right derivation.

Thanks
Bill

 

[rubi]

Here's a simple example that shows you that the step that Schrödinger did in his "derivation" is not valid mathematics:

Let's assume we want to solve
\left(\frac{\mathrm d x}{\mathrm d t}\right)^2-x^2=0
by his method (let's call this equation A). We set:
I=\int\left(\left(\frac{\mathrm d x}{\mathrm d t}\right)^2-x^2\right)\mathrm d t = \int L\mathrm d t
Minimizing this using the Euler-Lagrange equations yields
0=\frac{\mathrm d}{\mathrm d t} \frac{\partial L}{\partial\dot x} - \frac{\partial L}{\partial x} = 2\frac{\mathrm d^2 x}{\mathrm d t^2} + 2x
Let's call this equation B.

Now $x(t)=\mathrm e^t$ ($\dot x(t)=\mathrm e^t$, $\ddot x(t) = \mathrm e^t$) is a solution to the original equation A, but it's not a solution to equation B. On the other hand, $x(t)=\sin(t)$ ($\dot x(t) = \cos(t)$, $\ddot x(t) = -\sin(t)$) is a solution to equation B, but it's not a solution to equation A.

I think this unmistakably shows that integrating the equation and then minimizing the integral is not a valid mathematical technique and thus Schrödingers "derivation" is flawed.