What are the Different Formulations of Quantum Mechanics?

[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 Schrodinger'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 Schrodinger & 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 Schrodinger'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 Schrodinger'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 Schrodinger'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 Schrodinger's equation from classical mechanics - its simply not possible regardless of what institute they come from.

I have seen Schrodinger'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 Schrodinger 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 Schrodinger'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 Schrodinger'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 Schrodinger'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 Schrodinger did so, & a discussion of the meaning of Schrodinger'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 Schrodinger 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 Schrodinger'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 Schrodinger did so, & a discussion of the meaning of Schrodinger'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 Schrodinger'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 Schrodinger 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 Schrodinger'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 Schrodinger 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 Schrodinger'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 Schrodinger'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 Schrodinger 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 Schrodinger 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 Schrodinger'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 Schrodinger 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 Schrodinger actually integrated the Hamilton-Jacobi equation.

And yet you think it somehow derives the Schrodinger 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 Schrodinger'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 Schrodinger 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 Schrodinger'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, & Schrodinger 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. Schrodinger introduces K but it needs to be -ihbar to give the Schrodinger 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 Schrodinger'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.

 

[atyy]

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.

I'm not sure whether the paper is right or wrong, but it's interesting. I think it's an open question of whether quantum mechanics can be derived from something like a classical system if one introduces many, many additional degrees of freedom, which is different from your original question where the number of degrees of freedom in the classical and quantum system are the same. However, even if this particular paper is correct, I'm not sure it provides support for a classical derivation, because in Eq 42 they use projection, which requires the inner product of the Hilbert space.

 

[bhobba]

I'm not sure whether the paper is right or wrong, but it's interesting.

Its wrong.

The paper I linked to explains what's going on - its the complex numbers thing.

You can derive it from the Hamilton Jacobi equation if you introduce complex numbers, but if you don't the sign of the resulting equation is incorrect. To correct it Schrodinger introduces an ad-hoc assumption - a quantity J he defines needs to be stationary - its pulled out of the hat without any justification whatsoever. By doing that he arrives at the correct equation.

Of course he then tacitly assumes complex numbers anyway because its well known solutions of the Schrodinger equation are complex. It never seems to occur to Schrodinger to go back to the derivations very start and allow K to be complex which easily gives his famous equation without the ad-hoc trickery. It looks like he missed discovering something really important - but hey - the beginnings of any game changing theory is often a morass. In GR Einstein based his original development of the principle of general covarience, but it was later shown by Kretschmann to be vacuous. Einstein was forced to admit Kretschmann was right and GR had to be put on firmer ground.

I read somewhere where some historian of science called these guys sleepwalkers - they had an intuitive idea of where they wanted to go but the steps that led them to it are often dubious.

I also learned something else very interesting in this thread - don't trust what textbooks say about original methods. Every textbook I have ever read on QM gives a completely different derivation of Schrodinger's equation and claims it closely follows what Schrodinger did - yea right.

Thanks
Bill

 

[atyy]

Its wrong.

The paper I linked to explains what's going on - its the complex numbers thing.

The paper you linked to http://arxiv.org/abs/1204.0653v1 was about the derivation of the TISE from the HJ equation, but http://www.mpipks-dresden.mpg.de/~rost/jmr-reprints/brro01.pdf which bolbteppa linked to was about deriving the TDSE from the TISE by considering a subsystem. Did you mean that Briggs and Rost's derivation in their section 4 is wrong?

 

[bhobba]

dresden.mpg.de/~rost/jmr-reprints/brro01.pdf[/url] which bolbteppa linked to was about deriving the TDSE from the TISE by considering a subsystem. Did you mean that Briggs and Rost's derivation in their section 4 is wrong?

My concern has never been with the derivation of the TDSE from the TISE - it was the claim the TISE can be derived from classical mechanics - which is obviously incorrect. This sent the thread on a sojourn to discover exactly the error in such a claim. The OP had posted about it before and it was pointed out you can only do it by introducing complex numbers - IMHO it's simply the fact a wiener integral with a wick rotation gives the path integral in another guise. The reason that works is the paths need a phase component so cancellation can take place to yield the PLA.

What was discovered, and its something the OP should have spotted, is once you get the full detail of Schrodinger's original derivation, which he only posted the outline of, is Schrodinger made an error. He defined a quantity K that only if its complex gives the TISE - the sign is wrong. Schrodinger then goes on a sojourn defining a quantity J that he claims should be stationary - its an ad-hoc assumption without any justification whatsoever. Using that device he gets the correct TISE. But he knew it in general has complex solutions so he should have really gone back and seen what happens if K is complex - but didn't.

Now regarding the TDSE, as the paper I linked to shows (it derives TDSE - see section 4), if you allow K to be complex there is no seed to derive the TISE then derive the TDSE from it - the TDSE falls out from it anyway. So really the whole thing is moot. If the derivation is done correctly you get the TDSE from the outset.

BTW this is the slickest derivation of Schrodinger's equation I have ever seen. Not physically transparent - but really slick.

Thanks
Bill

 

[bolbteppa]

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.

This is ridiculously beautiful, & it would be an absolutely stunning refutation of what I've been saying, however, as Weinstock says:

"For a given potential-energy function $V$, solutions $\psi$ of the Schrodinger equation which vanish sufficiently rapidly at infinity (for the existence of the volume integral) exist, in general, only for a privileged discrete set of values of $E$; that is to say, the solution of the Schrodinger equation under the "boundary" condition that the volume integral exist is an eigenvalue-eigenfunction problem in which the eigenvalues of $E$ are to be determined"
Page 263

In other words, what Schrodinger did is perfectly fine As a verification of this, take the Hydrogen atom example in Weinstock I referenced earlier, on page 275 he gives the first eigenvalues & eigenfunctions explicitly. Schrodinger says the same thing in his original paper if you want to check that, but Weinstock should be fine if you have access to that (it's on the page you were able to view, & the link to the derivation on the missing google books page is here). This is what I meant about the genius of Schrodinger...

I apologize if this wasn't clear but I'd assumed you'd read both my posting of Schrodinger's derivation & read the page I linked to in that derivation. I'm very interested in what you have to say next.

 

[bolbteppa]

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. Schrodinger introduces K but it needs to be -ihbar to give the Schrodinger 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 Schrodinger'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

Man you're not reading anything I'm writing carefully if you're going to post this - I've already discussed this in this thread here, you ignored it, & I also referenced this exact point in the Schrodinger's equation derivation thread that you were using to say I missed elementary things (you referenced this in your last post, again!)...

This is the second derivation Schrodinger offered, it is in his second paper, & it's an explicitly time-dependent derivation. The derivation in the first paper is time-independent, that is absolutely crucial. Please read that carefully.

My concern has never been with the derivation of the TDSE from the TISE - it was the claim the TISE can be derived from classical mechanics - which is obviously incorrect.

If the TISE equation does follow from completely classical principles, a claim that has only been challenged so far by the idea of solutions of the HJE not being solutions of the SE, something I've just responded to, then your "obvious" claim is incorrect. If it is the case that it derives from classical principles, & that the TDSE derives from the TISE as in the Max Planck article, then either the derivation of the TDSE absolutely forces quantum principles on us at some point in the complicated derivation, or else the derivation is itself purely classical - in which case it's all derivable from classical mechanics. That's just absolutely basic to this thread, can you even admit that much?

This sent the thread on a sojourn to discover exactly the error in such a claim. The OP had posted about it before and it was pointed out you can only do it by introducing complex numbers. What was discovered, and its something the OP should have spotted, is

I explicitly refuted that person in the other thread, was ignored, then I used that refutation to refute you as well - but you've ignored that & repeated the claims carte blanche, this is just not productive.

once you get the full detail of Schrodinger's original derivation, which he only posted the outline of, is Schrodinger made an error. He defined a quantity K that only if its complex gives the TISE - the sign is wrong.

I didn't post an outline I posted exactly what is in Weinstock & gave a link to it. Further I posted Schrodinger's justification for defining $K$ to be real, not complex - something that comes from experiment, something I think you've ignored.

Schrodinger then goes on a sojourn defining a quantity J that he claims should be stationary - its an ad-hoc assumption without any justification whatsoever.

I posted Weinstock's justification for this here.

I also learned something else very interesting in this thread - don't trust what textbooks say about original methods. Every textbook I have ever read on QM gives a completely different derivation of Schrodinger's equation and claims it closely follows what Schrodinger did - yea right.

If you' read that Max Planck article you'll find they go through three derivations you find in textbooks, & they point out the assumptions.

 

[bolbteppa]

I'm not sure whether the paper is right or wrong, but it's interesting. I think it's an open question of whether quantum mechanics can be derived from something like a classical system if one introduces many, many additional degrees of freedom, which is different from your original question where the number of degrees of freedom in the classical and quantum system are the same. However, even if this particular paper is correct, I'm not sure it provides support for a classical derivation, because in Eq 42 they use projection, which requires the inner product of the Hilbert space.

That's an interesting point, however they give a derivation of the TISE on page 3 involving operators & Hilbert spaces, yet it's nothing but Schrodinger's original derivation applied to the formalism of Hilbert spaces. Maybe their derivation does invoke Hilbert spaces, however it may be possible to do the whole thing without Hilbert spaces & get the same result, i.e. to use a derivation assuming nothing but classical mechanics. I don't know, but if this does hold & someone does it before me I explicitly want some credit for having the idea *evidence*

And in fact, my original question was about whether this Hilbert space stuff is doing nothing but smuggling classical mechanics into a vector space, so even as it stands the derivation may indeed encode classical mechanics within it's very fabric - especially if those operators are defined on the basis of only knowing the TISE!

 

[bhobba]

Man you're not reading anything I'm writing carefully if you're going to post this - I've already discussed this in this thread

You keep pointing to that thread saying I have not read it. I have innumerable times.

You do not do what you claim - simple as that. You can squirm, say I didn't read or the inumerable things you keep doing, but facts are facts.

To get the Schrodinger equation CORRECTLY from the Hamilton-Jacobi equation you need complex numbers. Schrodenger got it by by an unjustified ad-hoc assumption that's totally unnecessary if you do it correctly from the start ie use complex numbers. The key physical assumption is that you must allow psi, defined from the action to be complex - nothing else will allow you to do it correctly ie without ad-hoc unjustified assumptions. And not only that you get the TDSE, not just the TISE.

This was pointed out in the thread you claim I didn't read but you still refuse to accept it.

It was explained in full detail in the paper I linked to - but again you fail to see.

And even if you adhere to that fiction at the end of the day, the way psi was defined by Schrodinger, it must be real. Yet the resultant equation admits complex solutions which should have been a red flag something was amiss. Go back and do it correctly, this time allowing psi to be complex and it works perfectly. Basically what Schrodinger did is an inconsistent hopeless mess.

Sometimes in discussions there comes a point where one side will not see the bleeding obvious - I am afraid its been reached here. You can continue with the fiction that somehow classical mechanics can be used to derive Schrodinger's equation, which every single textbook I have read, every single one, says is impossible, and when you actually understand QM its easy to see its impossible, - or you can continue on in blissful ignorance of the facts.

The choice is yours.

Thanks
Bill

 

[bhobba]

If the TISE equation does follow from completely classical principles

Yea right. A quantity J is introduced and an ad-hoc assumption made that it must be stationary is a completely classical valid argument

Its not even a logically valid way to do a derivation.

And since psi by definition in Schrodinger's derivation must be real what of the complex solutions to the equation? They must be nonphysical and dismissed. Down the gurgler goes any hope of applying the thing unless you admit it can be complex. But once you do that then you have the CORRECT derivation.

Basically as it stands its an inconsistent mess that points to its own rectification - all it takes is a little thought.

Thanks
Bill

 

[strangerep]

Bolbteppa,

I was about to reply to the original questions that started this thread, but after reading the whole thread I see the discussion has become rather heated, making me reluctant to get involved.

But I'm curious... do you think you have found answers to your original questions? Or do questions remain? If so, could you please restate/resummarize them?