Deriving Schrödinger's Equation From My Fundamental Equation

[Boof-head]

[math] \psi_{abc}\, =\, \alpha |0 \rangle\, +\, \beta |1 \rangle [/math]...this equation is too general; there are 3 colors (red,green,blue) in each stripe; each 3-stripe is sectioned so that s(1)(r,g,b),...,s(n)(r,g,b); are the colored subspaces in each stripe and over the "screen" S of the T and V (TV)

 

so that the anzatz color distribution function [math] \psi_c\, [/math]. should take 'colors to colors' over phases of (r,g,b) for each section s(r,g,b); there's a way to fold up the colors into just 2 color 'separation' measures, viz: red-green,green-blue, since the coloring is circular, then blue-red is a redundant color difference measure and 2 are sufficient to encode the wavefunction, or color distribution over S - time ind. Hamiltonian;

 

Then [math] \psi_c(ab)\, =\, \alpha |0 \rangle\, +\, \beta |1 \rangle [/math], where a,b are the transformed r-g,g-b measures.

 

Each contribution to (r,g,b) in projected space (projective map B) is from 2 color phases which are fixed by the striping (coloring) over S; now we can derive a general formulation of 'what really happens when TV is 'on' your tangent space T'.

[Boof-head]

Assume a circular measure in a manifold M; w/tangents t,t' touching at right angles and forming a vertex, z at the upper right of C; this is the involution that finds a cylindrical surface - where you are decoding the azimuth of color-angles and differences in your tangent space T, xy plane aligned || to S, the surface of "color".

 

This measure u (mu) from M is a circle w/radius 1; a unit that corresponds to an n-ary vector space V of unit vectors, so that "units are mapped to units" in terms of units of color - the transfer has "color" in it, there is a transfer function H, and a unitary space V of v(x,y,z); we make the step of 'locating' for events e,e' which are edges in the graph G(E,V) -> G'(V,E) (transform G) is the way into the color-graph = TV with brightness functor (B), colors c, sections s, tangent spaces and a bundle map B -> B(s); the algebra A is of states (or color events = bits of red,green,blue), and we have an exponential phase generator that maps from H(s) to H(t) in real time.

[Boof-head]

Comment: I would like to review what has been done here, as DD has also showed, there is an abstract "wavefunction" or wave of functions - that looks like a circle of them in our "field of view"; this view is congruent with our location at x,y,z in a plane on a surface.

 

We see 'colors' arrive at our location; we assume they are evolving from a TV (a television is a good example of a tangent vector bundle on a manifold), which is 'outside' our location where we view.

 

The expectation of seeing colors on a real device that projects it, is congruent with being located near a TV.

 

[i keep using "null" because it corresponds to the empty set {}' This is the lambda function, or punctuation in a logic - null punctures logic. This is because the empty set is always empty = infinitely empty = full of null sets of nulls which are all infinitely empty.

 

We can extend null because any null-thing is an empty thing; a null-curve has no 'time' in it because time is null-colored or no-time; this 'freezes' the time-logic so we can see space; we see space because it isn't colored when we see time.

 

It also represents 'potential' in physics; potential energy is the 'fixed' term, kinetic energy 'varies in time' or nullspace. Alternately we see 'energy vary' in space when we null-time. We are not 'allowed' to do both; we need null because it's the 'fixed color' of everything - time, space, logic, we would not be able to compare anything with anything else otherwise.]

 

 

Everything we 'see' is actually a TV, projecting color at us; this extends to everything we hear, smell/taste, and 'touch'. The last is a pressure 'operator' that informs our location. We either 'see' events evolving and projecting 'to' us, or we evolve from our location to them, or project 'our location'.

 

These views are both physically and mathematically equivalent, and also describe a complete (sigma-finite) algebraic space, in which we permute color over a 'brain' or the extensions called 'eyes', that see, etc.

I've edited or amended some of DD's OP, in light of my grasp of 'reality' or 'time', which is a linear frame of reference; the Euclidean plane is also the 'event' plane, or, events construct one for us (or we do this) again either view is unimportant, or trivially true.

 

Thus:

By definition, [imath]\vec{\Psi}[/imath] is a mathematical representation of our expectations. Those expectations are the result of a ... real-time(ity). The explanation itself is a epistemological construct ... ["is" a] free explanation of the past. ... you do in fact (in [math] real-tim(e^{(i|t\sigma_y)} [/math]) have expectations.

 

There are two facts extant here: first, a function (a method of obtaining one's expectations from a given set of known elements: i.e., [imath]\vec{\Psi}[/imath]) exists and that function must be a solution to my fundamental equation.

 

It is very important here to remember that [imath]\vec{\Psi}[/imath] is a mathematical representation of our expectations and is not necessarily a correct representation of the future.

 

What I am trying to point out is that our expectations are never necessarily correct ...; what is ...the known past .. consistent with those expectations, [is] not the future.

 

The future is a totally unknown [, a nullspace, or 'the empty set {}']. Our only defense of our expectations is that the volume of information which goes to make up the past is far far in excess of the next “present” (from our perspective)

 

the equation of interest [to our expectations of deriving it,] is

[math]\left\{\sum_i \vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}\beta_{ij}\delta(x_i -x_j)\delta(\tau_i - \tau_j) \right\}\vec{\Psi} = K\frac{\partial}{\partial t}\vec{\Psi}.[/math]

 

This expression is quite analogous to a differential equation describing the evolution of a many body system which, as anyone competent in physics knows, is not an easy thing to solve. [This ties to uncertainty, what we can actually 'know' about future events]

 

 

What we would like to do is to reduce the number of arguments to something which can be handled: i.e., we want to know the nature of the equations which must be obeyed by a subset of those variables. In an interest towards accomplishing that result, my first step is to divide the problem into two sets of variables: set number one will be the set referring to our “valid” ontological elements (together with the associated tau indices) [for which another definition is 'all points tangent to my location, that send information about the world to me,] and set number [is] all the remaining arguments.

 

[set #1, by induction, must be finite and set #2 can be infinitely possible -> possibly infinite which ties to the probability of seeing any event.]

 

 

So that, if look, then if see, then event -sequence; it cannot happen 'the other way around', we can't go back to check that we saw "reality', instead of 'something strange, or "weird" looking.

[Doctordick]

Hi Anssi, we seem to have hatched a “Troll”. There always seem to be people who just enjoy despoiling their environment. As my parents always told me, “it takes all kinds to make a world!” One could add that barking dogs are barking dogs; I have always suspected they actually think they are talking to us.

 

But, back to your note.

It can't be an intelligence issue, I think it's just a communication difficulty; I think there must be a way to explain this also to those who have never seen any problems with naive realism.

I think “communication” is the most difficult problem confronting us. One can see the universe as “trying to communicate with us”; we just can't seem to pick up on what it is saying.

Also I can of course see your commentary is an epistemological explanation of relativistic time behaviour, and not a suggestion of aether ontology.

Thank you. You are the first person to comprehend the significance of that issue. Actually, though much is made of Einstein's theory eliminating aether ontology, it always struck me that talking about the “properties” of “space-time” (such as it being a “foam” on a fine scale and/or mass being due to a distortion of space-time) makes it essentially an aether theory. But, of course, no one seems to understand that perspective.

A lot of the objections that I've seen, don't seem to be very thoughtful to me, and often times just plain odd. I guess the problem there also is that people just don't or can't give it the time to understand exactly how the perspective differs from whatever idea they have in their head about relativistic time relationships. Like that Modest' post, while I thought it reflects some desire to really understand what you are saying, it did also look like a first reaction commentary...

My single biggest problem is the simple fact that I am answering a question which has apparently never even occurred to anyone except you and I. Notice that your knowledge of math and physics (the most significant tools used in my presentation) is almost non-existent compared to Modest or Erasmus (and/or many others who I have attempted to reach) and yet you have utterly no problem whatsoever understanding the intention of my work. This fact alone should be sufficient to point out the nature of my problem: the issue I am discussing is simply beyond the comprehension of practically everyone. In fact I would go so far as to say that, if any of them with a decent knowledge of math and physics actually were to begin to comprehend the issue we are discussing, their realization that I am absolutely correct would follow almost immediately. I could quit posting and go back to bed.

Well you are probably right... That's a bit unfortunate. I wonder if Pyrotex had the chops to easily follow the math/logic itself... He's got physics background and he seems somewhat properly aligned philosophically to understand the discussion...

My point exactly; he simply does not understand the question. To quote you:

I guess it's characteristic to internet forum discussions that people use relatively little effort to try and comprehend what is being said, and mainly only comprehend and respond to the tidbits they already knew. You know, whatever "sounds valid" from the get-go. And whatever sounds invalid is never thought over. Certainly it would take longer to "think it over" (and understand the perspective of the other party) than most people are willing to spend on their contributions.

 

[imath]\cdots[/imath]

 

Certainly it seems many people don't see over their defined entities, and conclude that ontologically reality is a set of things that move from one place to another, and we are simply trying to figure out what those things are...

Not just the internet forum discussion; it is a characteristic of most all educated people – they spent a lot of hard time learning those “tidbits they already know” and they really don't want to consider the possibility they were erroneously convinced.

(Just a tiny typo there btw, the [imath]\vec{\Psi}[/imath] is missing _r)

Just fixed it. I am very impressed by your ability to spot those things; it means you are carefully looking and that is an important issue.

Right. The only part of that that I don't understand is, how and why those terms amount to a constant rather than "0".

Think about what they represent. Essentially, they amount to the probability that element i and j are in the same place in the universe and (since in the end, since [imath]x_1[/imath] has not been integrated over) how that probability varies when [imath]x_1[/imath] changes (how that probability behaves as a function of [imath]x_1).[/imath] Clearly from the assumptions we have made, it doesn't and what do we call something which doesn't change? Don't we call things which don't change constants?

Yeah, I also get the impression that this is starting to make sense to me :D

But "phew", there are so many potential pitfalls here! Like I said before, I so feel like I'm walking on a mine field. And most steps I decide to try out are straight towards a mine!

The kind of mathematics we are dealing with is not usually seen on the introductory level and your abilities are impressive. You are just not used to dealing with such things; just like that constant thing I just explained to you. I know you are kicking yourself for that oversight but don't feel bad. Everything in the universe is simple when it's obvious to you but it will never be obvious until you look at it from the right perspective.

 

So we arrive at the final step!

I.e. find out what the probability distribution of x (of the element of interest), when tau is allowed to be anything at all?

 

[imath]\cdots[/imath]

 

So I suppose this issue is similar to how the time derivative was removed earlier. I reviewed posts #54 - #60 where Bombadil helped me with it.

To a great extent, yes! But there is a simpler way to look at it (simple if you understood the mathematical nature of modern quantum mechanics). What we have here is two significant variables, the coordinate measure [imath]\tau[/imath] and momentum in the tau direction, defined to be the expectation value of [imath] \frac{\partial}{\partial \tau}[/imath] (multiplied by a constant which is not really important here) when the wave function [imath]\vec{\Phi}(x,\tau,t)[/imath] is known. The simple relationship of great value here is the Heisenberg “uncertainty principle”.

[math]\Delta x\Delta p \ge \frac{1}{2}\hbar[/math]

 

If the momentum in the tau direction [imath](p_\tau[/imath] represented by [imath]-i\hbar\frac{\partial}{\partial \tau})[/imath] is known exactly, the the uncertainty in position [imath]\tau[/imath] is infinite. This is exactly the case we want to be fact. In essence, a fixed known momentum corresponds exactly to the circumstance we have proposed: i.e., actual position in the tau direction is then unknowable. Thus I come to the conclusion that the operator [imath]\frac{\partial}{\partial \tau}[/imath] can be replaced with that constant “iq” and [imath]\frac{\partial^2}{\partial \tau^2},[/imath] which is defined to be [imath]\frac{\partial}{\partial \tau} \frac{\partial}{\partial \tau},[/imath] can thus certainly be replaced with [imath]-q^2[/imath].

 

As you guessed, this fact is very closely related to the behavior of [imath]e^{iq\tau}[/imath]. The central issue is the nature of wave phenomena itself. We usually picture waves like ocean waves, a sinusoidal variation in amplitude: i.e., the amplitude (height in the case of water waves) is essentially a sine or cosine function of position. It is interesting to note that [imath]e^{ix}=cos x + i sin x[/imath] when x is a real number. Actually it is also true if x is complex but we are not really concerned with that case.

 

You should check out “Computation of [imath]e^z [/imath] for a complex z” which you will find under item #5 in Exponential function.

 

You were actually quite close to the correct result

So focusing on the latter term:

 

[math]

\frac{\partial^2}{\partial \tau^2} e^{iq\tau} = \left\{ iqe^{iq\tau} \right\}^2

[/math]

 

Actually don't really know how all those squares work in the algebraic manipulations...

What you should have done is as follows:

[math]\frac{\partial^2}{\partial \tau^2} e^{iq\tau} =\frac{\partial}{\partial \tau} \frac{\partial}{\partial \tau} e^{iq\tau} = \frac{\partial}{\partial \tau}iq e^{iq\tau}=iq\frac{\partial}{\partial \tau} e^{iq\tau}=(iq)^2e^{iq\tau}=-q^2e^{iq\tau}[/math]

 

Just a simple error: squaring an operator simply means applying that operator twice. When it comes to mathematics, guesses (either educated or uneducated) should not be made.

 

The last step of all is to understand that [imath]A^2-B^2[/imath] can always be factored into (A+B)(A-B). Multiply A(A-B)+B(A-B) out in detail and you should understand that.

 

Have fun -- Dick

[Rade]

...(i.e., solipsism, the idea that nothing actually exists, cannot be disproved)

Dear DD, this is not what is meant by the philosophy of solipsism. If I hold solipsism to be true, I do not hold that "nothing actually exists", I hold that "My mind is the only thing that I know exists." . Just a side-bar correction to a misunderstanding you have of basic definition of terms in philosophy.

 

But, please do continue with your mathematical exposition of how you uniquely derive the Schrödinger's Equation from another Equation. Your approach is of great interest to science because, as we can read in just about any textbook of Quantum Chemistry, such as the 1983 text by Donald A. McQuarrie, p. 78..."We cannot derive the Schrödinger's Equation anymore than we can derive Newton's laws, and Newton's second law, f=ma, in particular". And such position has been known since the dawn of quantum theory, where we read from the 1935 textbook by Linus Pauling and E. Bright Wilson, Jr titled "Introduction to Quantum Mechanics with Applications to Chemistry", p. 52.."...the Schrödinger's Equation, ..., and the interpretation of the wave function are conveniently taken as fundamental postulates, with no derivation from other principles necessary". And further on p. 52 ..."the wave equation of Schrödinger...is not derived from other physical laws nor obtained as a necessary consequence of any experiment; instead, it is assumed to be correct, and the results predicted by it are compared with data from the laboratory".

 

Now, DD, since Linus Pauling received a Nobel Prize on a life of work in chemistry holding to the above worldview that the Schrödinger's Equation is not derived, and here you are claiming that it can so be derived, I for one find this to be of great importance.

 

I do not know if others reading this thread understand the implications of what you here present. In my opinion, if you are correct and you have derived the Schrödinger's Equation from a more Fundamental Equation, then the historical interpretation of the Schrödinger's Equation is incorrect, or at least incomplete, and perhaps you DD then receive a future Nobel Prize ! And here you thought as you stated above that I had no idea what you were trying to present here. Well, I understand completely what you are presenting here, it is what Thomas Kuhn called a paradigm shift which often leads to revolutions in scientific understanding.

 

But I keep coming back to the same problem I have presented to you for four year now, how is it that you not get this revolutionary derivation of the Schrödinger's Equation published ?! I know, I know, you have told me now three times why, but I fear a more fundamental reason that in fact your Fundamental Equation is not so fundamental after all.

 

So, DD, while you continue with your mathematical dialog with AnssiH which is of course important to the thread OP, perhaps others can provide input why the Fundamental Equation that is the basis of your worldview is not so fundamental after all. That is, I think it is time in this thread discussion to open a side-bar discussion to show how your Fundamental Equation can be falsified. For if it cannot be so falsified, then we open the door to the possibility of a scientific revolution.

[AnssiH]

My single biggest problem is the simple fact that I am answering a question which has apparently never even occurred to anyone except you and I. Notice that your knowledge of math and physics (the most significant tools used in my presentation) is almost non-existent compared to Modest or Erasmus (and/or many others who I have attempted to reach) and yet you have utterly no problem whatsoever understanding the intention of my work. This fact alone should be sufficient to point out the nature of my problem: the issue I am discussing is simply beyond the comprehension of practically everyone.

 

Well, hmmm, yeah, I'd say - just so it wouldn't sound like I'd think it's an intelligence issue - that the problem is that people are looking at this from very wrong perspective somehow. History repeats itself somehow... It's sort of like back when nature used to be "intentionally designed and created", as it just didn't make sense that complex systems like the nature could just develop without intentional guidance. Or, when it just didn't make sense that earth could revolve around the sun, or when it just didn't make sense that earth could be round.

 

Kuhn talked about that, the problem is people try to understand new information in terms of their old paradigms, and, it continues to surprise me how common that mistake is. Don't get me wrong, I know I make that mistake a lot; I know because I've often spotted myself doing it. After spending enough time trying to understand what someone is really saying, when something hasn't made the slightest sense to me.

 

Whoever's reading this and wondering what's the paradigm shift needed here, then try and stop thinking about knowledge from an ontological perspective, i.e. "what reality is really like", and start thinking about epistemological perspective, i.e. "what sort of knowledge can we have about reality". Or "when we are building our conception of reality, what are we working with?"

 

In fact I would go so far as to say that, if any of them with a decent knowledge of math and physics actually were to begin to comprehend the issue we are discussing, their realization that I am absolutely correct would follow almost immediately. I could quit posting and go back to bed.

 

I think you are right; certainly doesn't seem like an overwhelming amount of logic that we are dealing with here. Also, even with my lacking knowledge of physics, I am very acutely aware of the ontological complications of different physics interpretations (QM & relativity and also the incoherence between the two), and I do see the significance of this work as a coherent (and quite satisfying) explanation for why reality seems so eluding at the very limits of physics.

 

Not just the internet forum discussion; it is a characteristic of most all educated people – they spent a lot of hard time learning those “tidbits they already know” and they really don't want to consider the possibility they were erroneously convinced.

 

Yeah, that very much sounds like what Kuhn has warned us about. I guess it's natural. It tends to follow that paradigm shifts are brought in by the new generations as they are not so deeply invested into the old ways of thinking.

 

Just fixed it. I am very impressed by your ability to spot those things; it means you are carefully looking and that is an important issue.

 

It's just out of necessity, and the more familiar I'll get with the math, the less likely I am to spot small errors like that as, I'll have very specific expectations as to what I'm looking at and I stop looking at it so carefully.

 

I guess that's part of the problem that people have when they are trying to understand you; after the first skimming, they have certain expectations about what they think you are talking about, and they stop hearing the important details. I have certainly been quite amazed by some objections, like the other party just seemed to read a completely different post than what I read :D

 

Oh I shouldn't laugh, it's really quite unfortunate circumstance :(

 

Think about what they represent. Essentially, they amount to the probability that element i and j are in the same place in the universe and (since in the end, since [imath]x_1[/imath] has not been integrated over) how that probability varies when [imath]x_1[/imath] changes (how that probability behaves as a function of [imath]x_1).[/imath] Clearly from the assumptions we have made, it doesn't and what do we call something which doesn't change? Don't we call things which don't change constants?

 

Yeah, I got confused over trying to understand what would such "constant" be "operating on" then. But, now I suppose you just mean that it all amounts to something that will not have any effect on our element of interest. That much I had understood. Communication failure, "check" :D

 

So we arrive at the final step!

To a great extent, yes! But there is a simpler way to look at it (simple if you understood the mathematical nature of modern quantum mechanics). What we have here is two significant variables, the coordinate measure [imath]\tau[/imath] and momentum in the tau direction, defined to be the expectation value of [imath] \frac{\partial}{\partial \tau}[/imath] (multiplied by a constant which is not really important here) when the wave function [imath]\vec{\Phi}(x,\tau,t)[/imath] is known. The simple relationship of great value here is the Heisenberg “uncertainty principle”.

[math]\Delta x\Delta p \ge \frac{1}{2}\hbar[/math]

 

My understanding of the uncertainty principle is incredibly shallow. If I've understood it at all correctly, it is somehow a consequence of quantum elements being described as wave-like entities, including the measurement devices that affect the situation at hand. Have not really put in the time to really understand that well. Should I? (Right now, I can very easily take the principle as valid on faith - within your framework too as it boils down to our probabilistic expectations)

 

If the momentum in the tau direction [imath](p_\tau[/imath] represented by [imath]-i\hbar\frac{\partial}{\partial \tau})[/imath] is known exactly, the the uncertainty in position [imath]\tau[/imath] is infinite. This is exactly the case we want to be fact. In essence, a fixed known momentum corresponds exactly to the circumstance we have proposed: i.e., actual position in the tau direction is then unknowable. Thus I come to the conclusion that the operator [imath]\frac{\partial}{\partial \tau}[/imath] can be replaced with that constant “iq” and [imath]\frac{\partial^2}{\partial \tau^2},[/imath] which is defined to be [imath]\frac{\partial}{\partial \tau} \frac{\partial}{\partial \tau},[/imath] can thus certainly be replaced with [imath]-q^2[/imath].

 

Ahha...

 

As you guessed, this fact is very closely related to the behavior of [imath]e^{iq\tau}[/imath]. The central issue is the nature of wave phenomena itself. We usually picture waves like ocean waves, a sinusoidal variation in amplitude: i.e., the amplitude (height in the case of water waves) is essentially a sine or cosine function of position. It is interesting to note that [imath]e^{ix}=cos x + i sin x[/imath] when x is a real number. Actually it is also true if x is complex but we are not really concerned with that case.

 

You should check out “Computation of [imath]e^z [/imath] for a complex z” which you will find under item #5 in Exponential function.

 

Hmmm, okay, not really sure if this is relevant, or just additional factoid, so only skimmed it through for now.

 

You were actually quite close to the correct result

What you should have done is as follows:

[math]\frac{\partial^2}{\partial \tau^2} e^{iq\tau} =\frac{\partial}{\partial \tau} \frac{\partial}{\partial \tau} e^{iq\tau} = \frac{\partial}{\partial \tau}iq e^{iq\tau}=iq\frac{\partial}{\partial \tau} e^{iq\tau}=(iq)^2e^{iq\tau}=-q^2e^{iq\tau}[/math]

 

Just a simple error: squaring an operator simply means applying that operator twice. When it comes to mathematics, guesses (either educated or uneducated) should not be made.

 

The last step of all is to understand that [imath]A^2-B^2[/imath] can always be factored into (A+B)(A-B). Multiply A(A-B)+B(A-B) out in detail and you should understand that.

 

Alright! So with that, I understand how you got to:

 

[math] \left\{\frac{\partial^2}{\partial x^2} - q^2 + G(x)\right\}\vec{\Phi}(x,t)= 2K^2\frac{\partial^2}{\partial t^2}\vec{\Phi}(x,t)[/math]

 

And I figured out the [imath]A^2-B^2 = (A+B)(A-B)[/imath] bit. I guess it is related to what happens in the next step in the OP...

 

Notice that, if the term [imath]q^2[/imath] is moved to the right side of the equal sign, we may factor that side and obtain,

[math] \left\{\frac{\partial^2}{\partial x^2} + G(x)\right\}\vec{\Phi}(x,t)=\left\{\sqrt{2}K\frac{\partial}{\partial t}- iq\right\}\left\{\sqrt{2}K\frac{\partial}{\partial t}+iq\right\}\vec{\Phi}(x,t).[/math]

 

...but I still wasn't able to understand the route to that stage. Here's my attempt:

 

Moving the term [imath]q^2[/imath] to the right side:

 

[math] \left\{\frac{\partial^2}{\partial x^2} + G(x)\right\}\vec{\Phi}(x,t) = \left\{ 2K^2\frac{\partial^2}{\partial t^2} + q^2 \right\} \vec{\Phi}(x,t)[/math]

 

Focusing on the right side, I see instead of squaring you are applying the operator twice. But, what I would have written down is:

 

[math]\left\{\sqrt{2}K\frac{\partial}{\partial t} + q \right\}\left\{\sqrt{2}K\frac{\partial}{\partial t}+ q \right\}\vec{\Phi}(x,t)

[/math]

 

So, I understand why there's the square root, but I don't understand to get the -iq and +iq in there like you have them.

 

I do understand how your result is analogous to [imath](A+B)(A-B)[/imath], but don't know how the step before is analogous to [imath]A^2-B^2[/imath], as I changed the sign of q when I moved it from one side of the equation to the other :I

 

-Anssi "waiting for the explanation and getting ready to kick himself"

[AnssiH]

But, please do continue with your mathematical exposition of how you uniquely derive the Schrödinger's Equation from another Equation. Your approach is of great interest to science because, as we can read in just about any textbook of Quantum Chemistry, such as the 1983 text by Donald A. McQuarrie, p. 78..."We cannot derive the Schrödinger's Equation anymore than we can derive Newton's laws, and Newton's second law, f=ma, in particular". And such position has been known since the dawn of quantum theory, where we read from the 1935 textbook by Linus Pauling and E. Bright Wilson, Jr titled "Introduction to Quantum Mechanics with Applications to Chemistry", p. 52.."...the Schrödinger's Equation, ..., and the interpretation of the wave function are conveniently taken as fundamental postulates, with no derivation from other principles necessary". And further on p. 52 ..."the wave equation of Schrödinger...is not derived from other physical laws nor obtained as a necessary consequence of any experiment; instead, it is assumed to be correct, and the results predicted by it are compared with data from the laboratory".

 

Now, DD, since Linus Pauling received a Nobel Prize on a life of work in chemistry holding to the above worldview that the Schrödinger's Equation is not derived, and here you are claiming that it can so be derived, I for one find this to be of great importance.

 

Not only Schrödinger's Equation, but the derivation of any relationships that are commonly seen as "laws of nature", is quite significant as it ties those laws as epistemological of nature (instead of actually ontological), and it explains exactly how.

 

...it is what Thomas Kuhn called a paradigm shift which often leads to revolutions in scientific understanding.

 

But I keep coming back to the same problem I have presented to you for four year now, how is it that you not get this revolutionary derivation of the Schrödinger's Equation published ?!

 

Funny that you brought up Kuhn just as I thought about him too :thumbs_up And I think you sort of answered yourself, isn't that what Kuhn talks about a lot? The blockages that people have towards new paradigms. We talked about that issue in more detail in the few previous posts with DD; it seems like the blockage is exactly in how people tend to see objects in naive realistic ways, and don't comprehend how the identities of objects can't really be taken as "real" (and/or what consequences that fact has). I'm referring to that "persistent identity" dialog.

 

So, DD, while you continue with your mathematical dialog with AnssiH which is of course important to the thread OP, perhaps others can provide input why the Fundamental Equation that is the basis of your worldview is not so fundamental after all. That is, I think it is time in this thread discussion to open a side-bar discussion to show how your Fundamental Equation can be falsified. For if it cannot be so falsified, then we open the door to the possibility of a scientific revolution.

 

The derivation of the fundamental equation was discussed in quite significant detail throughout the "What can we know about reality" thread, and the thread at Physics Forums (there are links at What can we know about reality).

 

The discussion could be condensed a lot though, but, for now let it be said that I can't think of any significant errors in there. I mean, in the starting point of symmetries springing from ignorance to the real meaning of the data.

 

Even if that starting point somehow contained assumptions that can't be taken as valid, it is quite significant thing to derive "laws of physics" from those symmetries, I'd say.

 

Well, whatever you do, please don't start that discussion in this thread :thumbs_do I value the ability to go back to old posts easily, and it helps when there's no great volume of posts about different topics.

 

-Anssi

[Doctordick]

Moving the term [imath]q^2[/imath] to the right side:

 

[math] \left\{\frac{\partial^2}{\partial x^2} + G(x)\right\}\vec{\Phi}(x,t) = \left\{ 2K^2\frac{\partial^2}{\partial t^2} + q^2 \right\} \vec{\Phi}(x,t)[/math]

 

Focusing on the right side, I see instead of squaring you are applying the operator twice. But, what I would have written down is:

 

[math]\left\{\sqrt{2}K\frac{\partial}{\partial t} + q \right\}\left\{\sqrt{2}K\frac{\partial}{\partial t}+ q \right\}\vec{\Phi}(x,t)

[/math]

 

So, I understand why there's the square root, but I don't understand to get the -iq and +iq in there like you have them.

I do understand how your result is analogous to [imath](A+B)(A-B)[/imath], but don't know how the step before is analogous to [imath]A^2-B^2[/imath], as I changed the sign of q when I moved it from one side of the equation to the other :I

 

-Anssi "waiting for the explanation and getting ready to kick himself"

Don't be so hard on yourself. I know you are going to kick yourself but please don't take it as evidence of incompetence; rather, you are just not used to working with mathematics.

 

Please note that (A+B)(A+B) is equal to A(A+B)+B(A+B) which becomes [imath]A^2+2AB+B^2[/imath]: i.e., we have that troublesome 2AB term. The negative sign in (A+B)(A-B) is very important as it eliminates that middle term. The term:

[math]\left\{ 2K^2\frac{\partial^2}{\partial t^2} + q^2 \right\}[/math]

 

(which is correct) does not posses that important negative sign and thus (as written) does not factor. That is why it was changed to

[math]\left\{ 2K^2\frac{\partial^2}{\partial t^2} -(i q)^2 \right\}[/math]

 

which is actually the same thing since i² is “-1” and i is defined to be [imath]\sqrt{-1}[/imath]. Now, since we have the necessary negative sign, the term can be factored and the result is:

[math]\left\{\sqrt{2}K\frac{\partial}{\partial t} -iq \right\}\left\{\sqrt{2}K\frac{\partial}{\partial t}+ iq \right\}[/math]

 

Well, whatever you do, please don't start that discussion in this thread :) I value the ability to go back to old posts easily, and it helps when there's no great volume of posts about different topics.

I agree with you one hundred percent. If I had a web site I think I could be persuaded to rewrite my original opus. I think I could make it considerably clearer now after being dragged through the all the garbage on this forum. I know I wouldn't put it the same way given another chance and perhaps it would be good to have a succinct correct presentation to refer people to.

 

Have fun -- Dick

[Rade]

Dear AnssiH and DoctorDick,

 

Clearly neither of you find any fault with the assumptions used to derive the Fundamental Equation, but for you to suggest that discussion of these assumptions is not of importance on this thread (or has somehow, somewhere been discussed on other threads) does not hold water for me. The thread has the title, "Deriving Schrödinger's Equation from my Fundamental Equation". If the assumptions of the Fundamental Equation are false, the claimed derivation is false.

 

So, please do continue with your mathematical dialog here--as I stated, it is very important. But I wish to open a side discussion on the assumptions of the Fundamental Equation--if not here, OK, then I request one of you begin a new thread called, "Assumptions of the Fundamental Equation of DoctorDick".

[Doctordick]

Clearly neither of you find any fault with the assumptions used to derive the Fundamental Equation, but for you to suggest that discussion of these assumptions is not of importance on this thread (or has somehow, somewhere been discussed on other threads) does not hold water for me.

Well I am sorry that you feel the position “does not hold water” but I am afraid I have to side with Anssi here. As you well know, I am of the opinion that you have no understanding of what I have been talking about and consider you somewhat of a troll with regard to that issue. If you truly want to discuss the validity of my fundamental equation, a thread for that purpose already exists and, if you wish to complain about any of the steps in that derivation, I (and I think Anssi also) would be happy to answer any rational points you wished to make on that thread.

 

The thread you need to read begins with the following post.

This is a thread started to discuss a serious problem deeply embedded in the whole fabric of philosophical thought. idsoftwaresteve has referred this problem through the metaphor of a map of reality from which we can direct our thoughts and I suggested that the real problem is coming up with a method of drawing such a map when we don't know what we are talking about.

Go back and read that thread in detail and post comments the moment you find fault with any logical step (please insert a quote from whatever specific post you find questionable so that I may understand your complaint). Make a rational presentation and I will respond. Make irrational comments having nothing to do with the derivation and I will go back to ignoring you.

 

I hope you understand what I just said -- Dick

[Rade]

Thank you DoctorDick--I will read your thread and post my questions there. Sorry for the interruption.

 

Edit:

 

Well, DD, I have read your first post in the thread you mention and would like to make a point here, since it is related to this thread, about this comment you made:

 

Comment of DoctorDick on thread mentioned above....If one holds that only logical thoughts are rational, then scientific progress becomes impossible since any deductions must be based on things presumed to be valid without reason (those axioms one starts with) and that is certainly irrational.....

 

So, you claim above: (1) non-logical thoughts can be rational and are necessary for scientific progress, (2) it is irrational thinking to deduce from fundamental axioms, and (3) the community of scientists that now hold that the Schrödinger Equation is logically valid as an axiom and use it to solve problems are irrational thinkers !

 

Well, these are very revolutionary claims indeed and I strongly suggest that others with interest in this thread read the above thread for it does introduce an aspect of the philosophy of DoctorDick not apparent here. But DoctorDick, do you really believe there has been no scientific progress in chemistry and physics in the past 70 years by those that have applied the Schrödinger Equation as a axiom and have logically and rationally concluded that it is not required for the Schrödinger Equation to be derived from any equation more fundamental--such as the one you discuss in this thread ?

[Qfwfq]

But, please do continue with your mathematical exposition of how you uniquely derive the Schrödinger's Equation from another Equation. Your approach is of great interest to science because, as we can read in just about any textbook of Quantum Chemistry, such as the 1983 text by Donald A. McQuarrie, p. 78..."We cannot derive the Schrödinger's Equation anymore than we can derive Newton's laws, and Newton's second law, f=ma, in particular". And such position has been known since the dawn of quantum theory, where we read from the 1935 textbook by Linus Pauling and E. Bright Wilson, Jr titled "Introduction to Quantum Mechanics with Applications to Chemistry", p. 52.."...the Schrödinger's Equation, ..., and the interpretation of the wave function are conveniently taken as fundamental postulates, with no derivation from other principles necessary". And further on p. 52 ..."the wave equation of Schrödinger...is not derived from other physical laws nor obtained as a necessary consequence of any experiment; instead, it is assumed to be correct, and the results predicted by it are compared with data from the laboratory".

I do not agree with what those guys say. I can actually understand a chemist getting away with such a point of view but it isn't my point of view as a physics graduate and it is even less Dick's.

 

I can't afford the time to properly follow and judge Dick's work but I do say Rade that your objections are of no avail. I think you should distinguish between the ability to effectively use something (such as the Schrödinger equation) and being able to claim that it could only be a primitive axiom.

 

BTW does McQuarrie not know that [imath]f=ma[/imath] can actually be viewed as a consequence of the Schrödinger equation for the simple case of scalar particle in a potential? Actually this is just because it can be derived from the corresponding classical Hamiltonian.

[Rade]

To Qfwfq,

 

Thank you, this is very interesting report. If you would have time, could you please give here a summary of your understanding of how the Schrödinger's Equation is derived from another equation or set of assumptions and is not taken to be an axiom, as I presented in a few posts above. Now, Linus Pauling is believed to be one of the most intelligent scientists of the 20Th Century, and understood quantum theory well, so I do find it of great interest that you claim Pauling does not know what he is talking about concerning the Schrödinger's Equation. Is it then your understanding that the Fundamental Equation of DoctorDick is the only valid way that the Schrödinger's Equation can be derived, or do you know of other published derivations ? A textbook citation that presents how the Schrödinger's Equation can be derived from some prior set of assumptions or equation for all possible solutions would be greatly appreciated and help with just about all of my questions.

===

 

Edit:

 

Well, a search of internet resulted in this peer reviewed published derivation of the Schrödinger's Equation:

 

http://arxiv.org/PS_cache/physics/pdf/0610/0610121v1.pdf

 

So, looks like we have something that we can compare to the approach taken here by DoctorDick using his Fundamental Equation.

[lawcat]

Very nicely explained by Dr. Ward.

[Qfwfq]

Is it then your understanding that the Fundamental Equation of DoctorDick is the only valid way that the Schrödinger's Equation can be derived,

I did not mean this at all. It can be viewed as following from the de Broglie-Einstein relations and Hamiltonian formulation of the dynamics. Dick is, however, discussing a totally different matter.

[AnssiH]

Make a rational presentation and I will respond. Make irrational comments having nothing to do with the derivation and I will go back to ignoring you.

...So, you claim above: (1) non-logical thoughts can be rational and are necessary for scientific progress, (2) it is irrational thinking to deduce from fundamental axioms, and (3) the community of scientists that now hold that the Schrödinger Equation is logically valid as an axiom and use it to solve problems are irrational thinkers...

 

:omg:

 

Rade, please... I don't want you to feel like we'd be ignoring important aspects of this issue, and I can assure you that I, and DD, take the logical validity of any part of the derivation very seriously. Unfortunately, your previous posts only tell me that you don't understand the topic sufficiently to be of help with that, I'm sorry.

 

Let's keep this thread as uncluttered as possible and stay on Schrödinger part. You can post to the other thread for questions and concerns about the derivation of the fundamental equation. But if you do...

 

Well, a search of internet resulted in this peer reviewed published derivation of the Schrödinger's Equation:

 

http://arxiv.org/PS_cache/physics/pdf/0610/0610121v1.pdf

 

So, looks like we have something that we can compare to the approach taken here by DoctorDick using his Fundamental Equation.

 

...I also expect you to think for yourself little while before you post. You should know by now that that paper is completely different matter from what we are discussing here. It certainly is immediately obvious to me, to Qfwfq, and I'd hope to anyone else who has been following this for as long as you have. If it's really not obvious to you, think about where the Schrödinger Equation is derived from in the paper you linked to, and where it's derived from in DD's work.

 

Let it be said that if your contributions don't improve, I just don't think it's time well spent trying to explain it to you. Truth be told, after your previous posts, I was not sure if I should respond to you at all anymore because I can't be sure if you are just trolling or what... :shrug: Had I not talked with you before, I'm sure I would not have responded at all.

 

-Anssi

[Rade]

......I also expect you to think for yourself little while before you post....-Anssi

Dear AnssiH and DoctorDick. I am sorry if I upset you two, and I think you will most likely find it hard to believe that I do take thinking seriously. But I can only say that I do not think that I can know what I do not know and, rather than view me as an unthinking troll, view me as a lost sheep, whose thinking needs to be educated. AnssiH, you are the only person that shows any interest at all in what DoctorDick has to say and I greatly respect that you continue to dig deeper into understanding of his philosophy and mathematics. Let me attempt to explain my recent posts:

 

The title of this thread is called: Deriving Schrödinger's Equation from my Fundamental Equation

 

It is a very long thread, now going for many months. I have been reading along, since the beginning, and you will find I did provide a few useful posts in the past.

 

Recently I have been reading some textbooks on quantum chemistry and found the very interesting comments that I posted a few posts ago--that is, the conclusion held by more than just a few scientists that "the Schrödinger's Equation CANNOT BE DERIVED" ?! And this is not a view held by some high school student that happens to post on the internet--it is the view of Linus Pauling who was an EXPERT in quantum theory and perhaps one of the most intelligent scientists of the 20th Century. Well, I find this of interest, since here I have been reading for many months on this thread about how the Schrödinger's Equation can be derived, and then Linus Pauling telling me it cannot. So, why should I accept one option over the other. Either Linus Pauling is incorrect or DoctorDick is incorrect--or perhaps they use different concepts of what it means to 'derive' something such as an equation.

 

Then, I read the post of Qfwfq and he more or less indicates that Linus Pauling is incorrect--that the Schrödinger's Equation can be "derived" (whatever that means to him) and that the approach taken by DoctorDick is valid (at least I think that is what he is saying). Then I find on the internet a paper that I posted that also derives the Schrödinger's Equation--of course not in the same way as DoctorDick.

 

So, while DoctorDick likely has discovered a new way to derive the Schrödinger's Equation, his approach is not the only way to do so. You see, I was under the impression that it was the only way (given statement by Linus Pauling and others). So, I hope this explanation helps you understand why I jumped into this thread discussion at this point in time.

 

Then DoctorDick directed me to another thread, of great interest to me, and I can see where I will have many questions over there--so I will keep those types of questions in that other thread.

 

====

 

Now, I do have another question which seems to be proper here in this thread. Can the Fundamental Equation of DoctorDick also derive the "Hamilton Principle" ? -- or is the Fundamental Equation = Hamilton Principle ? It is not clear to me which takes priority as being more fundamental. See below for nice summary of Hamilton Principle:

 

Hamilton's principle - Wikipedia, the free encyclopedia