What are the Different Formulations of Quantum Mechanics?

[bolbteppa]

Good idea:

My original questions were:
a) Why does one need to make an algebra out of mixing the Hamiltonian with Poisson brackets?
b) Why can't quantum mechanics just be modeled by extremizing a Lagrangian, or solving a H-J PDE?
c) How do complex numbers relate to this process?

These questions were motivated by the fact that Schrodinger's original derivation of the time-independent Schrodinger equation apparently assumes nothing more than classical mechanics. I was sloppy & missed the time-independence - after being shown this I thought the thread had ended, then in trying to find out about a time-dependent derivation I came across http://www.mpipks-dresden.mpg.de/~rost/jmr-reprints/brro01.pdf in which they purport to derive the TDSE from the TISE (among other things, well worth reading), however the derivation is pretty complicated & I don't understand it. But as it stands, the claim is that the TISE is derivable from classical mechanics, & that the TDSE is derivable from an equation derivable from classical mechanics. Therefore unless there is some step in this derivation that absolutely require quantum mechanical assumptions, it seems the TDSE may be derivable from classical mechanics also. Thus people are examining the strength of the claim that the TISE is derivable from classical mechanics, before examining the TDSE derivation in that article.

So how does this affect my questions?

a) If you can derive the TDSE from the TISE without axioms or quantum mechanical assumptions then it may be that all this algebra/vector space stuff is implicitly smuggling classical mechanics into it's very fabric. At present it seems that all the craziness results from the fact that eigenfunctions of the SE are, in general, complex-valued, that seems to be the only new thing going on thus far. An example of this is page 3 of the Max Planck article where Schrodinger's original derivation is given in terms of operators in a Hilbert space, it's nothing more than Schrodinger's derivation in disguise which means it may just be a classical mechanics derivation in the guise of algebra & arbitrary axioms & all this baggage was potentially added to the theory simply due to Schrodinger's inability to deal with the TDSE in the early years in the way the authors of the Max Planck article have done, that's what I'm trying to find out. It may be this machinery is not necessary, it's merely useful tricks the way the Hamilton-Jacobi equation could be seen as a trick for dealing with Lagrangians & Hamiltonians.

b) has an easy solution, it can be modeled by extremizing a Lagrangian, c.f. Landau QM Section 20. However as it stands we're only comfortable with the TISE, thus factoring in the notion that solutions are complex it makes sense to construct a Lagrangian in terms of complex functions as Landau does. The TDSE in the action is something we'd need to come back to later.

c) & it's Complex numbers are currently only justified in this by the fact that eigenfunction solutions of the TISE can in general be complex, however around page 16 of that article the authors make an interesting argument about the necessity of complex numbers in the TDSE as arising due to interactions of a system described by a TISE with a classical external environment, & that they not necessary when you ignore such an interaction, though to fully understand this I think one would need to follow the derivation completely.

Thus the question as it stands is to examine Schrodinger's original derivation, & to examine the derivation in that Max Planck article of the TDSE from the TISE & locate where (if any) new assumptions are forced on us & whether they are simply unavoidable, these two issues seem to be the crux on which everything rests.

 

[strangerep]

[...]

Thus the question as it stands is to examine Schrodinger's original derivation, & to examine the derivation in that Max Planck article of the TDSE from the TISE & locate where (if any) new assumptions are forced on us & whether they are simply unavoidable, these two issues seem to be the crux on which everything rests.

They're not. What matters is the dynamical algebra implicit in the equations of motion. The classical and quantum cases involve different representations of this algebra. The quantum case is what happens when one realizes that all practical measurements of an object system involve interaction with an apparatus, hence involve a nontrivial dynamics, hence one must model this more carefully with noncommuting dynamical variables, instead of the commuting dynamical variable from the classical case. Noncommuting dynamical variables also mean that one of the usual axioms of probability (the one involving joint and conditional probabilities) must be modified somehow -- Ballentine treats this issue in ch9(?) iirc.

My answer to the question "what is quantization?" then becomes a (lengthy) elaboration of the above, and I'm trying to figure out whether you really wanted an answer to this larger question (from a modern perspective), or prefer to stay with the basic (rather narrow, imho) historical derivation(s) of the Schrodinger equation(s) in which this thread seems to have become bogged down.

 

[rubi]

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

The point it that the introduction of that functional that has to be minimized is not valid mathematics. It is a lie that the eigenfunctions of the hydrogen atom solve the HJ equation. It just doesn't work out. No matter whether it satsfies any boundary conditions or not.

Here is a citation by Weinstock, the book you are referring to all the time (p. 262):

[...] Ignoring the problem of solving (2), Schrödinger instead considers the volume integral of the left-hand member carried out over all space [...]

And (2) is the HJE in that book. So Weinstock himself admits that this approach doesn't solve the HJE. The only person in the world who believes that it does, is you. You are living in a dream world. Wake up!

I told you multiple times how to check this with your own pencil and paper. Just take the time to insert one solution of the SE into the HJE. It's a simple calculation involving only derivatives. No fancy math here. You should be able to do it. But you didn't do it, so I'm forced to believe now that you are either a troll or a crackpot.

Here is a definite proof that this can't work out:
The HJE in the book is given by $(\nabla\Psi)^2+f\Psi^2=0$ (A) and the SE from the book is $\Delta\Psi-f\Psi=0$ (B), where $f=\frac{2m}{K^2}(V-E)$. Both depend on the energy through $f$. The only thing I've done was putting the factors into $f$. Now add (A)+$\Psi$(B):
$$(\nabla\Psi)^2 + \Psi\Delta\Psi = 0$$
Let's call it (C). Your claim is now that (C) is always true, whenever $\Psi$ is a solution to the SE. It should be obvious that this is wrong. Even the dependence on the potential and the energy is gone and thus irrelevant. But since I already know that you're not going to believe it and also not going to check it, let's just use $\Psi=\mathrm e^{-x^2}$, which is a ground state solution to some 1D quantum harmonic oscillator. A trivial calculation shows
$$(\nabla\Psi)^2 + \Psi\Delta\Psi = \mathrm e^{-2x^2}(8x^2-2) \neq 0$$
so your claim is definitely refuted.

 

[bolbteppa]

i) Alright I've done it explicitly using the first eigenvalue-eigenfunction given in Weinstock & you're right, it doesn't solve the H-J equation. I guess I was thinking the solution was supposed to solve both things the way the H-J equation implicitly assumes you've got the equations of motion built into the functional you're using to turn into a PDE to get H-J. In other words I was making a conceptual error, & now I have no idea in the world why extremizing the H-J equation should have any relevance whatsoever to this problem, or that the solution should have any meaning whatsoever. Interestingly in a note to Schrodinger's second paper he explicitly states that $\psi$ does not actually solve the H-J equation, wish I'd read it earlier but I was trying to ignore his time-dependent derivation (in that paper his explanation of the analogies with geometrical optics in that paper absolutely amazing!). Interestingly post 77 below seems to completely generalize this procedure.

Is it some way of saying that 'if we interpret the H-J equation as the Hamiltonian then solving the H-J equation gives us the energy, and plugging in functions that don't solve it are like deviations from the energy, thus minimizing this integral over space is basically giving us the minimal deviation from the true energy over all of space"? If that is not some roundabout way of encoding energy into what we're doing, then what the hell are we doing?

Calm down with the crank talk, I'm just a student in college about to start my first course on quantum mechanics, I barely understand what "ground state" even means apart from pop science descriptions, but my head will just not allow me to assume axioms unless I have some motivation for them, & Schrodinger's paper seems to be the way to do it. Also not knowing much I'm fearful to do calculations when I'm probably ultimately making a conceptual error, & it turns out here I was doing just that (I thought that since you got solutions only in some cases it explained why you were getting wrong answers in your example), I didn't need to do the calculation if I'd understood what Weinstock & Schrodinger were saying, but thanks for forcing me to do it since I didn't.

ii) I found what I think is the last chance at pulling this back to classical mechanics at the end of Schrodinger's paper, & it explains Weinstock's comment on page 263 about formulating the problem as an Isoperimetric problem as something more than a calculus of variations method. If I'd understood it at the beginning of this thread I'd say I'd have started here to avoid this whole integrating the H-J equation thing, so let's try to find the flaw in this:

In the case of conservative systems in classical mechanics, the variation problem can be formulated in a neater way than was previously shown, and without express reference to the Hamilton-Jacobi differential equation. Thus, let $T(q,p)$ be the kinetic energy, expressed as a function of the co-ordinates and momenta, V the potential energy, and $d \tau$ the volume element of the space, "measured rationally", i.e. it is not simply the product $dq_1$,..., $dq_n$, but this divided by the square root of the discriminant of the quadratic form $T(q,p)$. (Cf. Gibbs' Statistical Mechanics.)

Then let $\psi$ be such as to make the "Hamilton integral"

$$\int d \tau (K^2 T(q,\frac{\partial \psi}{\partial q}) + \psi^2 V)$$

stationary, while fulfilling the normalising, accessory condition

$$\int \psi^2 d\tau = 1.$$

The proper values of this variation problem are then the stationary values of the integral and yield, according to out thesis, the quantum-levels of the energy.
"Quantization as a Problem of Proper Values I" Addendum, Page 11

The integrand doesn't seem to be $T + V$ it's $T + \psi ^2 V$.

Also the volume element is "measured rationally", however in Weinstock Page 263 he completely ignores this & thus it may or may not be an issue. Schrodinger references http://archive.org/details/ElementaryPrinciplesInStatisticalMechanics & it seems as though Gibbs discusses this around page 21 & 22, if it does become an issue.

Finally I see constraints in mechanics calculus of variations problems as being formal ways of going from standard coordinates to generalized coordinates, at least that's what I was taught, so I don't know if the constraint here functions in an equivalent manner to that, so that might also be an issue here.

These seem to be the only three differences from classical mechanics per se, that I can see, so is this couched in classical mechanics or is there a fundamental difference when you start here? What do you guys think about all this?

 

[bhobba]

Calm down with the crank talk, I'm just a student in college about to start my first course on quantum mechanics, I barely understand what "ground state" even means apart from pop science descriptions, but my head will just not allow me to assume axioms unless I have some motivation for them, & Schrodinger's paper seems to be the way to do it.

At the start any student is best served in just accepting the status quo and learning the standard stuff instead of very fringe ideas like QM is based on CM - it isn't - and when you understand QM you will realize it can't be.

If you really want to understand QM, and I mean really understand it, the book for you is Ballentine - QM - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20
http://www-dft.ts.infn.it/~resta/fismat/ballentine.pdf

Most people starting out would be satisfied with Griffiths or something similar but you seem to want a deeper understanding. Ballentine is more advanced and difficult, but goes very deeply into exactly what's going on.

Thanks
Bill

 

[bolbteppa]

At the start any student is best served in just accepting the status quo and learning the standard stuff instead of very fringe ideas like QM is based on CM - it isn't - and when you understand QM you will realize it can't be.

If you really want to understand QM, and I mean really understand it, the book for you is Ballentine - QM - A Modern Development:
https://www.amazon.com/dp/9810241054/?tag=pfamazon01-20
http://www-dft.ts.infn.it/~resta/fismat/ballentine.pdf

Most people starting out would be satisfied with Griffiths or something similar but you seem to want a deeper understanding. Ballentine is more advanced and difficult, but goes very deeply into exactly what's going on.

Thanks
Bill

Thanks, I can appreciate the point you're making, personally I'm trying to go down the Landau/Davydov/Griffith route, using a problems book associated to each, before worrying about the rest of the literature (probably won't happen like that in the end), so this thread is all about ignoring the textbooks & trying to find out why Schrodinger was forced out of classical mechanics as a means to motivate the necessity for quantization.

 

[bolbteppa]

I think I've found an answer to some of my questions in this book:

https://www.amazon.com/dp/9812381910/?tag=pfamazon01-20

http://books.google.ie/books?id=sojvOSNnTCwC&printsec=frontcover#v=onepage&q&f=false

In the book, from browsing, it appears that:

• He takes Schrodinger's original derivation as his starting point more or less,

• He defines a variational principle from which the H-J equation & a continuity equation associated to the H-J can be derived,

• Shows how Schrodinger's change of variables in the H-J equation implicitly introduces probability distributions into the discussion,

• Shows how introducing complex numbers adds an extra degree of freedom enabling that probability distribution to vary & is apparently responsible for the shift from CM to QM,

• Shows how the variational principle from which the H-J can be derived gives, when it involves complex numbers, the Schrodinger's equation (this seems to be the justification for what Schrodinger does in his paper, & so I think this explains why Schrodinger was getting complex eigenfunction starting from real numbers)

• Shows how the conjugate momentum contains the derivative of a probability density distribution, something not in classical mechanics

In other words the concept of a probability distribution is implicitly encoded into the Hamilton-Jacobi equation through the notion of a distribution of trajectories, where you can bring this out using Schrodinger's change of variables $S = k\ln(\psi)$, & selecting a solution to the H-J equation amounts to choosing a distribution with a fixed value. However allowing $\psi$ to be complex numbers adds an extra degree of freedom & so selecting a solution to the H-J equation amounts to choosing a distribution with a fixed value, but since classical dynamics comes from real-valued $\psi$ functions, we don't even need to worry about complex numbers. However allowing & accentuating complex numbers means that $|\psi|^2$ can genuinely take on the role of a distribution function & is not constant, & it links the functions that you derive the classical H-J equation from in the variational principle to get Schrodinger's equation.

This is based off browsing, but it looks pretty convincing. I've only looked at chapter 4, 7 & 8, these look completely rigorous & are fully mathematical, however some of the other chapters seem a bit crankish or at the very least overly philosphical in places, for example the section on the Ehrenfest relations in the last chapter, but that stuff has absolutely nothing to do with my question. In my ignorance I may be calling crankishness what is in reality part of the philosophical disputes about interpretation of QM, & he seems to have written legitimate books on QM, so the book might be completely legitimate in those sections, however chapters 4, 7 & 8 are just math & physics, no fluff, & looks very interesting. He does seem to criticize the Hilbert space approach as lacking generality, that & the Ehrenfest section are really all I can comment on so take it for what it's worth.

If we simply cannot make sense of Schrodinger extremizing the Hamilton-Jacobi equation as a valid mathematical trick, & cannot make classical sense of Schrodinger's derivation of the TISE as an isoperimetric problem in the addendum to his paper, then I'd say the above is the best explanation of what I've been looking for, unless you guys think it's crankish & see flaws

 

[rubi]

now I have no idea in the world why extremizing the H-J equation should have any relevance whatsoever to this problem, or that the solution should have any meaning whatsoever.

That's the whole point we were trying to explain to you: It's a completely unjustified ad-hoc assumption that basically axiomatically introduces the SE already. That's what other approaches to QM also do (axiomatically introduce the SE), but they don't try to hide this fact in nonsensical math.

my head will just not allow me to assume axioms unless I have some motivation for them

That's the basic mistake you're making. The only justification an axiom needs is that it describes the experimental data correctly. It's an experimental fact that the world is the way it is. You can't change the laws of nature; you have to accept them the way they are. And if they are not intuitive to you, then it's not the problem of nature. It's your problem. There are always axioms at the foundations of any theory and if nature is strange, then the axioms are strange as well.
Watch this: (from 4:20)

Schrodinger's paper seems to be the way to do it.

Schrödinger's paper is actually the worst thing to read if you want to understand quantum mechanics. It was written at a time when QM was still being developed and nobody had any clue what it was supposed to mean. It's a lucky accident that the paper even includes the Schrödinger equation, because they way he arrived there is completely nonsencial. We understand QM much better today and modern textbooks explain it the way it should be taught.

I found what I think is the last chance at pulling this back to classical mechanics [...] What do you guys think about all this?

You just figured out that your initial approach doesn't work and now you're trying another one which also won't work. It's simply impossible to arrive at QM if the only thing you accept is CM. You can only arrive at QM by making unjustified ad-hoc assumptions. QM is inequivalent to CM. It's entirely different and has nothing in common with it except for including it as a limiting case.

trying to find out why Schrodinger was forced out of classical mechanics as a means to motivate the necessity for quantization.

He wasn't forced into quantization. It's absolutely possible that there is a non-quantum theory that describes nature better than QM does. There's just no necessity for such a theory at the moment, because QM works really well. It's impossible to come up with an explanation for why we need quantum theory (as far as we know). Several generations of the smartest people in the world have been trying to understand this for almost a century now, dedicating their lives to this very question.

I think I've found an answer to some of my questions in this book:
[...]
If we simply cannot make sense of Schrodinger extremizing the Hamilton-Jacobi equation as a valid mathematical trick, & cannot make classical sense of Schrodinger's derivation of the TISE as an isoperimetric problem in the addendum to his paper, then I'd say the above is the best explanation of what I've been looking for, unless you guys think it's crankish & see flaws

This book is reeeeaaally cranky as far as I can tell. It makes even more ad-hoc assumption than Schrödinger did and is full of unjustified vague statements, just to give you the impression that it was possible to go from CM to QM in a meaningful way. I really don't want to waste my time on debunking all the unjustified steps in the book. Luckily the auther is a chemistry professor, so he isn't really supposed to understand QM deeply.

 

[bolbteppa]

That's the whole point we were trying to explain to you: It's a completely unjustified ad-hoc assumption that basically axiomatically introduces the SE already. That's what other approaches to QM also do (axiomatically introduce the SE), but they don't try to hide this fact in nonsensical math.

When I thought the solution of the Hamilton-Jacobi equation was built into extremizing that integral I seen it as merely integrating a PDE implicitly containing the EOM within. If that were true then I'd be right, unfortunately I'm an idiot...

You just figured out that your initial approach doesn't work and now you're trying another one which also won't work. It's simply impossible to arrive at QM if the only thing you accept is CM. You can only arrive at QM by making unjustified ad-hoc assumptions. QM is inequivalent to CM. It's entirely different and has nothing in common with it except for including it as a limiting case.

If you don't have any interest in analyzing the issues with Schrodinger's addendum that's fine.

This book is reeeeaaally cranky as far as I can tell. It makes even more ad-hoc assumption than Schrödinger did and is full of unjustified vague statements, just to give you the impression that it was possible to go from CM to QM in a meaningful way. I really don't want to waste my time on debunking all the unjustified steps in the book. Luckily the auther is a chemistry professor, so he isn't really supposed to understand QM deeply.

Schrodinger made one ad-hoc assumption in that paper, one which this book claims to explain. He explicitly says:

"His contribution in his epoch-making first paper may be summarised in two steps; one apparently trivial and one boldly new. ... In looking at Schrodinger's work we must, of course, guard against the idea that his mechanics can be "derived" from the Hamilton-Jacobi equation; it cannot. Schrodinger's mechanics is a new creation, it contains new intuition about reality which mathematical manipulation can never supply"
P127

This is what you guys have been saying all along. Furthermore as far as I can see at the moment he basically just uses dimensional analysis, the continuity equation & the functional that generates the H-J & continuity equation to motivate going over to complex numbers which allows him to derive the Schrodinger equation as nothing but it's extremal (along with intuition derived from the probability interpretation of $|\psi|^2$ as the original means to motivate the interpretation of a term in the functional, though it's not logically dependent on knowing the probability interpretation in advance). Obviously this isn't deriving things from classical mechanics, as is explicitly stated, but it is deriving things from classical mechanics as much as possible, as opposed to memorizing a bunch of axioms. I'd love to test your statements by asking you to provide some crankiness in sections 7.2 to 7.4 explicitly, but if you're fed up that's fine.

 

[rubi]

If you don't have any interest in analyzing the issues with Schrodinger's addendum that's fine.

I really don't have too much interest in doing that. It's quite clear that we won't find anything enlightening in there apart from even more weird reasoning. The beginnings of QM were full of unjustified math and ad-hoc assumptions. You're on the wrong track if you think there is any justification to be found in those papers. Back then, people were already satisfied if any of their formulas made correct predictions. They didn't care about consistency very much. It's all heuristics.

Schrodinger made one ad-hoc assumption in that paper, one which this book claims to explain.

There is no explanation for it in the book. It's all just beating around the bush. If he didn't know QM already, he couldn't have come up with all his "explanations". He's working backwards: Indeed, he starts from CM, but he knows what he wants to end up with, so he just makes all the random, unjustified assumptions he needs, in order to arrive there. That's not valid reasoning.

Obviously this isn't deriving things from classical mechanics, as is explicitly stated, but it is deriving things from classical mechanics as much as possible, as opposed to memorizing a bunch of axioms.

No, it's not. He's introducing much more random axioms. The only reason why you think that it has anything to do with CM is that he uses the same symbols. But as soon as he introduces the ad-hoc assumptions, their meaning change to something completely different! And he's even making wrong analogies, just to give you fake impressions. The probability density from CM has nothing to do with the QM $|\Psi|^2$. In QM, we already have an analogue to the classical probability density, which is given by density matrices. It's really really bad to compare the classical $\rho$ to the QM $|\Psi|^2$! They are as unrelated as they could possibly be! The classical analogue of $\Psi$ isn't $\sqrt\rho$, but rather $(x,p)$.

 

[bolbteppa]

I really don't have too much interest in doing that. It's quite clear that we won't find anything enlightening in there apart from even more weird reasoning. The beginnings of QM were full of unjustified math and ad-hoc assumptions. You're on the wrong track if you think there is any justification to be found in those papers. Back then, people were already satisfied if any of their formulas made correct predictions. They didn't care about consistency very much. It's all heuristics.

As long as you know this doesn't address Schrodinger's arguments at all then this is fine.

There is no explanation for it in the book. It's all just beating around the bush. If he didn't know QM already, he couldn't have come up with all his "explanations". He's working backwards: Indeed, he starts from CM, but he knows what he wants to end up with, so he just makes all the random, unjustified assumptions he needs, in order to arrive there. That's not valid reasoning.

Notice the lack of detail in your response here, I literally predicted this in my last response:

Furthermore as far as I can see at the moment he basically just uses dimensional analysis, the continuity equation & the functional that generates the H-J & continuity equation to motivate going over to complex numbers which allows him to derive the Schrodinger equation as nothing but it's extremal (along with intuition derived from the probability interpretation of $|\psi|^2$ as the original means to motivate the interpretation of a term in the functional, though it's not logically dependent on knowing the probability interpretation in advance).

You just ignore this & go for the generalizations, you're obviously uninterested at this stage & that's fine.

No, it's not. He's introducing much more random axioms.

?

The only reason why you think that it has anything to do with CM is that he uses the same symbols.

There's more to his argument than mere symbol pushing, he explicitly discusses how selecting a solution to the classical H-J equation amounts to choosing a distribution with a fixed value, due to sole dependence on real numbers. This is just based off of dimensional analysis after the change of variables Schrodinger used $S = K \ln(\psi)$, refer to pages 131-132, (about 10 sentences). Already this is an indication that there's more to what he's doing than mere symbol pushing, this is just classical mechanics & he's ending up with notions of probability distributions using units, i.e. by dimensional analysis, alone. He goes on to show how allowing complex numbers removes the possibility of selecting a path with a fixed value thus adding complex numbers removes the dependence on paths, changes the emphasis to means & introduces variable probability densities associated with what will become solutions of the extremized functional (the Schrodinger equation).

But as soon as he introduces the ad-hoc assumptions, their meaning change to something completely different! And he's even making wrong analogies, just to give you fake impressions.

The only ad-hoc assumption I see is going to complex numbers, however it's not ad-hoc if you read it you see it's motivated by dimensional analysis & by the desire to relate the two functionals in the extremal that gives the H-J equation to accentuate the consequences of a variable probability distribution (instead of a constant one in the classical H-J equation). Furthermore the meaning of the Schrodinger equation (Pages 136-137) becomes explicit before the equation is even defined using this approach.

The probability density from CM has nothing to do with the QM $|\Psi|^2$.

I don't know what you mean by this, but it's clear you don't know what he's talking about. If you're not willing to bother there's no point, here he's talking about a specific probability distribution function on the space of trajectories in the Hamilton-Jacobi equation & using Kolmogorov's probability theory to justify that $\rho = |\psi|^2$ is indeed a probability distribution function (pages 58-60). I don't know how he modifies his domains etc... when allowing it to be complex, but I don't think anyone knows the answer to that.

In QM, we already have an analogue to the classical probability density, which is given by density matrices. It's really really bad to compare the classical $\rho$ to the QM $|\Psi|^2$! They are as unrelated as they could possibly be! The classical analogue of $\Psi$ isn't $\sqrt\rho$, but rather $(x,p)$.

You have an analogue defined within the context of matrix mechanics in a Hilbert space - here he is working within Schrodinger's wave mechanics. Furthermore it's clear you do not understand how he relates any form of classical density function with a quantum density function.

As I see it the only ad-hoc assumption is going to complex numbers, however it's not really ad-hoc if you spend 5 minutes reading it. The claim is that this leads to all the crazy stuff, for example he makes a point about the conjugate variable being a partial derivative of a probability distribution & emphasizes this as distinguishing classical from quantum mechanics, as well as going from classical to complex numbers. If this is too quackish/crankish that's fine, it doesn't look bad to me as it stands, I'd rather not get help if it's causing anguish.

 

[rubi]

As long as you know this doesn't address Schrodinger's arguments at all then this is fine.

It does address Schrödinger's argument.

Notice the lack of detail in your response here, I literally predicted this in my last response:

You just ignore this & go for the generalizations, you're obviously uninterested at this stage & that's fine.

Look. I just spend about ten posts convincing you that two obviously non-equivalent differential equations are non-equivalent. You finally admitted that you were wrong, but immediately came up with more claims that you don't even understand. You made it clear that you don't have a clue about how modern quantum mechanics works, yet you believe that you understand it better than everyone else. I really don't have the time to spend another 30 posts on convincing you. It's obvious to me that you will never accept the fact that QM can't be motivated using only CM.

There's more to his argument than mere symbol pushing, he explicitly discusses how selecting a solution to the classical H-J equation amounts to choosing a distribution with a fixed value, due to sole dependence on real numbers. This is just based off of dimensional analysis after the change of variables Schrodinger used $S = K \ln(\psi)$

It's already nonsensical to do a change of variables like this in the first place if you stop using valid math afterwards and instead choose to transform the resulting expression into an action that should be minimized, because then the resulting formula depends on what choice of variables you made. Why $S=K\ln\psi$? Why not $S=K\psi^2$? The answer is that he chooses the transformation in such a way that the equation that comes out will be the Schrödinger equation. Everything is set up in order to yield the result that he wants.

The only ad-hoc assumption I see is going to complex numbers, however it's not ad-hoc if you read it you see it's motivated by dimensional analysis & by the desire to relate the two functionals in the extremal that gives the H-J equation to accentuate the consequences of a variable probability distribution (instead of a constant one in the classical H-J equation). Furthermore the meaning of the Schrodinger equation (Pages 136-137) becomes explicit before the equation is even defined using this approach.

There is no valid argument for introducing complex numbers here, especially not if the quantity that is made complex is part of any non-linear expressions and must be real in order to be meaningful. But even if this were okay, you introduce an additional degree of freedom that wasn't there before. No matter how hard you try, this can't be a meaningful modification.

I don't know what you mean by this, but it's clear you don't know what he's talking about. If you're not willing to bother there's no point, here he's talking about a specific probability distribution function on the space of trajectories in the Hamilton-Jacobi equation & using Kolmogorov's probability theory to justify that $\rho = |\psi|^2$ is indeed a probability distribution function (pages 58-60). I don't know how he modifies his domains etc... when allowing it to be complex, but I don't think anyone knows the answer to that.

You have an analogue defined within the context of matrix mechanics in a Hilbert space - here he is working within Schrodinger's wave mechanics. Furthermore it's clear you do not understand how he relates any form of classical density function with a quantum density function.

On p. 131 he clearly states that the $\Psi^2$ is to be identified with the classical $\rho$, which satisfies the continuity equation that is written above and thus is clearly meant to be the density in phase space. This is however the wrong identification if $\Psi$ is to become the wave-function of QM, since the QM analogy of $\rho$ isn't $\Psi^2$, but rather the density matrix. Once again you make it obvious here that you have no idea about quantum mechanics. The density matrix is a completely valid tool in standard wave-mechanics and you would be unable to do quantum statistical mechanics without it.

As I see it the only ad-hoc assumption is going to complex numbers, however it's not really ad-hoc if you spend 5 minutes reading it.

I really can't help you if you don't see all the ad-hoc assumptions and the invalid reasoning here. I suggest you grab a math book. It's clear that this is nothing more than a failed attempt to justify Schrödinger's original derivation in retrospect. This must fail and it would be obvious to you why this is the case if you did understand a tiny bit of real quantum mechanics. There is a reason for why this "derivation" isn't even mentioned in any textbook on QM.


Maybe this convinces you:
It is expected that there are some quantities in QM and CM that have approximately the same numerical values in some situations. This is due to the fact that QM is supposed to include CM as a limiting case (correspondence principle!). The presence of these numerically almost identical quantities doesn't make it possible to motivate any of the axioms of QM from just CM, though. There is in principle an infinite number of generalizations of CM that all agree with CM in some situations. Thus you can't use the fact that there is some agreement to motivate the correct generalization! If you have just CM and nothing more, you are unable to guess the correct generalization of CM! The only way to make progress is to take experimental data into account. That's what we did and it led us to QM!

 

[bolbteppa]

It does address Schrödinger's argument.

I mentioned what I seen as the three issues I had with his addendum argument, ignoring this & going to generalities doesn't address that. Thanks I'd rather we ended this.

Look. I just spend about ten posts convincing you that two obviously non-equivalent differential equations are non-equivalent. You finally admitted that you were wrong, but immediately came up with more claims that you don't even understand. You made it clear that you don't have a clue about how modern quantum mechanics works, yet you believe that you understand it better than everyone else. I really don't have the time to spend another 30 posts on convincing you. It's obvious to me that you will never accept the fact that QM can't be motivated using only CM.

If this is the way you respond to someone asking questions I'd rather we ended this. For instance "yet you believe that you understand it better than everyone else" signifies this conversation is over because I've explictly called myself an idiot in my last post, said I was only beginning Landau/Davydov/Griffith etc... & that the point of this thread was to ignore the textbooks & analyze Schrodinger. This: "It's obvious to me that you will never accept the fact that QM can't be motivated using only CM" further signifies this should end, I'd rather someone else helped me thank you. Finally I've explained my reasons why I thought one could get QM from CM based off of Schrodinger - I thought he was implicitly encoding the EOM in what he was doing, it was a mistake, so I've asked what I see as my final issues to see if anything can be salvaged, however to you this is tantamount to cheating so I'd rather not get your help thank you.

It's already nonsensical to do a change of variables like this in the first place if you stop using valid math afterwards and instead choose to transform the resulting expression into an action that should be minimized, because then the resulting formula depends on what choice of variables you made.

Again I've mentioned the reason for this at least three times now, it's nothing more than dimensional analysis. Ignoring that & preferring generalities means this is finished.

Why $S=K\ln\psi$? Why not $S=K\psi^2$?

Because of dimensional analysis, this has been mentioned at least three times now. Furthermore since $\psi^2$ has dimensions you'll have to modify $K$ to get units of action on the R.H.S. & I don't know how that will play out, but you won't get the Schrodinger equation as it looks, it looks like your result will either be singular or non-linear. Furthermore on the basic mathematics I wonder do you understand what it means to reparametrize something? Just because it looks arbitrary it is absolutely fine to do since it just represents your original quantity anyway.

The answer is that he chooses the transformation in such a way that the equation that comes out will be the Schrödinger equation. Everything is set up in order to yield the result that he wants.

Where is the problem with that? That was the motivation for mathematicians developing distribution theory, for example, & the motivation for the early practitioners of statistical mechanics in seeking to derive classical thermodynamics.

There is no valid argument for introducing complex numbers here, especially not if the quantity that is made complex is part of any non-linear expressions and must be real in order to be meaningful. But even if this were okay, you introduce an additional degree of freedom that wasn't there before. No matter how hard you try, this can't be a meaningful modification.

How is it not meaningful if it ends up giving results confirmed by experiment? Furthermore his claim is that this degree of freedom is there in the classical case however choosing the EOM of a system is tantamount to choosing a value for that degree of freedom, and again motivates it's existence by mere dimensional analysis on the classical H-J equation. Obviously emphasizing the extra degree of freedom is the crux of why this method is a generalization of CM & not just CM, further it gives the right answers. I don't see how this is any different to the procedure followed in relativity books, you find the classical kinetic energy assumes infinite velocity, & the potential energy assumes instantaneous velocity of propagation of interaction, thus it has to be modified. One has to change the geometry of space itself in this modification, by your logic "that wasn't there before. No matter how hard you try, this can't be a meaningful modification" I don't think it's worth following this point up though, "It's obvious to me that you will never accept the fact that QM can't be motivated using only CM".

On p. 131 he clearly states that the $\Psi^2$ is to be identified with the classical $\rho$, which satisfies the continuity equation that is written above and thus is clearly meant to be the density in phase space. This is however the wrong identification if $\Psi$ is to become the wave-function of QM, since the QM analogy of $\rho$ isn't $\Psi^2$, but rather the density matrix.

When he says $\Psi^2$ is to be identified with the classical $\rho$, he clearly defines what he means on pages 103-104, & I'm almost sure he makes a distinction that takes him out of full-blown phase space to focus on what the meaning of the Action is, i.e. I think he only concentrates on positions not position+momentum as in phase space, so you'd have to be more careful than that. Furthermore:

Once again you make it obvious here that you have no idea about quantum mechanics. The density matrix is a completely valid tool in standard wave-mechanics and you would be unable to do quantum statistical mechanics without it. I really can't help you if you don't see all the ad-hoc assumptions and the invalid reasoning here. I suggest you grab a math book.

If the density matrix is a representation of a linear operator on a Hilbert space, & in chapter 10 he shows how everything he does fully applies to Hilbert spaces:

"That is, the structure of a Hilbert space may be abstracted from the solutions of the Schrodinger equation. These solutions have, of course, much more content and meaning than that particular algebraic structure; they are functions of 3-space which carry the probabilistic interpretation of the whole mechanics"

then I don't see how what he does doesn't, at least in principle, fully imply the applicability of density matrices

It's clear that this is nothing more than a failed attempt to justify Schrödinger's original derivation in retrospect. This must fail and it would be obvious to you why this is the case if you did understand a tiny bit of real quantum mechanics. There is a reason for why this "derivation" isn't even mentioned in any textbook on QM.

Well you can use assertion here if you want, but I'd wager it's because Schrodinger's derivation was of the time-independent equation, not the more general time-dependent equation, though I've come across multiple papers discussing it in the past two days now that I know to look for it. However the guys derivation in that book is explicitly not Schrodinger's derivation, he generalizes it to the time-dependent schrodinger equation, generalizes the logic by explaining the necessity of complex numbers, & to me it explains why Schrodinger's original derivation ended up with him getting complex eigenfunctions even though he thought he was working with real functions, & again he ends up with results very different from classical mechanics which agree with experiment. Of course it is a retrospective derivation, nobody said it wasn't, however you using the word "failed" indicates this discussion is over as you've made numerous errors & assumptions I've already addressed.

There is in principle an infinite number of generalizations of CM that all agree with CM in some situations. Thus you can't use the fact that there is some agreement to motivate the correct generalization!

This seems to be the thrust of the anger I'm dealing with - just because I was wrong about one thing (the H-J equation containing the EOM implicitly in Schrodinger's equation) I must also be wrong about everything else, similarly just because a theory parallels CM in some cases it must be wrong because another theory claiming the same thing was wrong. I can't argue with that logic, it's all-encompassing. I came here to get help, you clearly have no interest in that anymore.

If you have just CM and nothing more, you are unable to guess the correct generalization of CM! The only way to make progress is to take experimental data into account. That's what we did and it led us to QM!

I've explicitly pointed out that this is what he's doing, at least to motivate what he's doing, however I don't see anything wrong with what he's done - nothing illegal, he merely provides motivation for why he should generalize to complex numbers, & in the end it's confirmed by experiment. Nobody ever said we had CM & nothing more, you're not listening to me, thus I think this is done.

 

[bolbteppa]

Here is an article that seems to verify everything I've been saying (referring to textbooks that take this approach and all), oh man, I think I've found a research project - please just write me off as a crank & forget everything I've been saying...

 

[bhobba]

Here is an article that seems to verify everything I've been saying (referring to textbooks that take this approach and all), oh man, I think I've found a research project - please just write me off as a crank & forget everything I've been saying...

Yea - interesting project.

But its long been known what the significance of the HJ equation is - Feynman sorted it out ages ago. Particles take all paths but most paths have close paths that are the same except are 180% out of phase so cancel - the only exception are paths whose close paths are the same ie only paths stationary in the action are left. This leads to the HJ equation - the exact detail can be found in Landau - Mechanics for example.

The paper I linked to gives a very slick derivation of Schrodingers equation from the HJ equation - normally one needs to use the method of steepest decent.

Thanks
Bill

 

[rubi]

Again I've mentioned the reason for this at least three times now, it's nothing more than dimensional analysis. Ignoring that & preferring generalities means this is finished.

Because of dimensional analysis, this has been mentioned at least three times now. Furthermore since $\psi^2$ has dimensions you'll have to modify $K$ to get units of action on the R.H.S. & I don't know how that will play out, but you won't get the Schrodinger equation as it looks, it looks like your result will either be singular or non-linear.

$S=K\Psi^2$ is perfectly fine from the point of view of dimensional analysis. It has the same dimensions as $S=\ln\Psi$. You can't use dimensional analysis to find the correct change of variables.

Furthermore on the basic mathematics I wonder do you understand what it means to reparametrize something? Just because it looks arbitrary it is absolutely fine to do since it just represents your original quantity anyway.

Yes, it would be perfectly fine to do a change of variables if you were going to use valid mathematics afterwards as well. However, if you are going to using the resulting expression as a Lagrangian, this is not valid mathematics and a different choice of variables would have yielded a different Lagrangian!

Where is the problem with that? That was the motivation for mathematicians developing distribution theory, for example, & the motivation for the early practitioners of statistical mechanics in seeking to derive classical thermodynamics.

The problem is that you claim to logically arrive at QM, but your reasoning is circular, because you are adjust your argument in order to give you the result that you actually want to derive. It's invalid reasoning.

How is it not meaningful if it ends up giving results confirmed by experiment? Furthermore his claim is that this degree of freedom is there in the classical case however choosing the EOM of a system is tantamount to choosing a value for that degree of freedom, and again motivates it's existence by mere dimensional analysis on the classical H-J equation. Obviously emphasizing the extra degree of freedom is the crux of why this method is a generalization of CM & not just CM, further it gives the right answers.

Your claim was that it is possible to arrive at complex numbers here using nothing but logic! Now you emphasize that you need the experiment, which is a step in the right direction. It's still problematic though, because a complex valued action is nonsensical from the point of classical mechanics (it gives you complex positions and momenta), so it's not really a generalization of CM. In order to introduce complex numbers here, you must admit that you have already given up CM comletely to the point of no return and your theory is already utterly different.

I don't see how this is any different to the procedure followed in relativity books, you find the classical kinetic energy assumes infinite velocity, & the potential energy assumes instantaneous velocity of propagation of interaction, thus it has to be modified. One has to change the geometry of space itself in this modification, by your logic "that wasn't there before.

Nobody in relativity tries to deduce it from CM! Relativity starts from clearly stated axioms. This is what you refuse to do here!

When he says $\Psi^2$ is to be identified with the classical $\rho$, he clearly defines what he means on pages 103-104, & I'm almost sure he makes a distinction that takes him out of full-blown phase space to focus on what the meaning of the Action is, i.e. I think he only concentrates on positions not position+momentum as in phase space, so you'd have to be more careful than that.

The correct quantum analogy of a distribution of states is still given by a density matrix. You just evaluate it in position space.

If the density matrix is a representation of a linear operator on a Hilbert space, & in chapter 10 he shows how everything he does fully applies to Hilbert spaces [...] then I don't see how what he does doesn't, at least in principle, fully imply the applicability of density matrices :confused

Erm.. The density matrix is of course applicable. I never said it isn't. The point is that it must be applied! (Which he isn't doing.) The analogy of an ensemble in CM is an ensemble in QM, not a single state. So if you want to carry over an ensemble of classical particles into QM, you need to describe the state using a density matrix. The quantum version of the Liouville equation is the Von-Neumann equation, not the Schrödinger equation.

However the guys derivation in that book is explicitly not Schrodinger's derivation, he generalizes it to the time-dependent schrodinger equation, generalizes the logic by explaining the necessity of complex numbers

It's almost the same derivation, only with some additional claims. He explains the necessity of complex numbers? How come they aren't necessary then in real QM? There are situations that can be described using only real numbers.

Of course it is a retrospective derivation, nobody said it wasn't, however you using the word "failed" indicates this discussion is over as you've made numerous errors & assumptions I've already addressed.

You are the one making errors and assumptions all the time. The attempt is failed, because it doesn't archieve what it claims to archieve: Give a logically consistent answer to the question of why QM is necessary.

however I don't see anything wrong with what he's done

And that is the problem. I've tried long enough now to explain to you that it's full of unjustified ad-hoc assumptions. Bhobba told you how the HJ formalism relates to QM (using the path integral). It doesn't make sense to argue any longer. You said that you "are about to start your first course on QM" and I infer from that (and this is consistent with my observations) that you don't have a basic understanding of QM, which is needed in order to understand what's going on here. You don't see the drastic shift of conception that is inherent to QM. This drastic shift can't be motivated by just modifying CM a little bit.

 

[bolbteppa]

Yes, it would be perfectly fine to do a change of variables if you were going to use valid mathematics afterwards as well. However, if you are going to using the resulting expression as a Lagrangian, this is not valid mathematics and a different choice of variables would have yielded a different Lagrangian!

The change of variables occurs in the extremal of a particular Lagrangian (the Hamilton-Jacobi equation), not in the Lagrangian, http://www.mth.kcl.ac.uk/~llandau/231a/GenCoords . We're not going off of Schrodinger's arbitrary choice to integrate the Hamilton-Jacobi equation anymore, we're going off of the notion that extremizing a particular functional gives you both the Hamilton-Jacobi equation & an associated continuity equation, & that when you allow complex numbers the exact same process gives you the Schrodinger equation, thus you must be thinking I'm still talking about Schrodinger's original derivation when I'm not I'm going off the derivation in that book & the paper I've linked to.

The rest is fine, I take it on board, the substance will stay with me, thanks.

 

[bhobba]

If you want to see a derivation of Schrodinger's equation using the method of steepest decent check out:
http://www.phys.vt.edu/~ersharpe/6455/ch1.pdf

It justifies the intuitive argument I gave about close paths cancelling.

Thanks
Bill

 

[atyy]

You may also like to read about the different formulations of quantum mechanics:

https://www-physique.u-strasbg.fr/cours/l3/divers/meca_q_hervieux/Articles/Nine_form.pdf .

Some cases, such as second quantization are easily generalized to relativistic quantum field theory, but whether that's the case for the de Broglie - Bohm formulation is still being researched. It's interesting to ask in each case what formulation of classical mechanics is obtained when the classical limit is taken, eg:

http://www.physics.ohio-state.edu/~mathur/821hj.pdf.