Deriving Schrödinger's Equation from my Fundamental Equation

[AnssiH]

Originally Posted by Doctordick

The issue of what actually exists is central to the subject of ontology. It is clear from the work of many thinkers over the centuries that actually answering such a question is impossible (i.e., solipsism, the idea that nothing actually exists, cannot be disproved). I am saying, maybe some things we think exist actually do exist and some things we think exist don't; the fact that we cannot answer such a question can not be taken as proof it cannot be so. Thus I used the word “valid” for the sole purpose of identifying which case I am talking about because the two parts (though one cannot ever actually identify which is which) must obey subtly different rules. Each and every truly flaw-free explanation must explain everything which actually exists while, on the other hand, it is possible that some ontological elements presumed to exist (by some specific explanation) may only be hypothesized and yet there can still exist a flaw-free explanation which require these elements: the central point being that the fact that an explanation is “flaw-free” is no guarantee that every element presumed in that explanation actually exists. This is an issue often forgotten by many serious scientists.

When I read
I presume you are referring to exactly that issue and I want to be sure that presumption is not in error.

Yes, I believe we are on the same page on this.

Here you are absolutely correct and the issue you bring up is the single most far reaching misrepresentation of reality which is out there blocking perception of what I am talking about. I would put this difficulty squarely in the same category as the presumption that “God exists” used in most medieval arguments prior to modern science; it so blocks their thinking that they are blind to the issues you and I are talking about. People who think along such lines simply can not be reached by any intellectual argument and it is really a waste of time to try.

I say that though even I am occasionally driven to try myself.

Well yeah, I wouldn't like to conclude it's a complete waste of time. Granted, when that issue about "persistent identity" has come up, I've usually had some people respond with "that's obviously nonsense" in the blink of an eye. But, then some people seem to understand the issue more readily. 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.

Hmmm, but I can see more and more clearly the problems you said you've had, with mathematicians, physicians and philosophers all thinking this is not really their department... At least philosophers should be interested, but I guess they are easily scared by all the math, as it looks deceptively like someone is yet again trying to prove a specific ontology with a lot of math, so they decide it is coming from someone who doesn't even understand the problem of ontology.

The “What is spacetime” thread is a perfect example of exactly what I was just talking about. The people posting there have absolutely no comprehension of the inadequacy of their beliefs.
You are absolutely correct, no one is paying the slightest attention to anyone else. It reminds me of a room full of parrots. As I have said many times, thinking is not an easy process and most people would rather avoid it if at all possible. Even Modest, who is clearly a rather rational person, would rather spout off then actually try to understand anything new. I had hoped he was reading some of my stuff but, in his latest post to me (a rather long and involved missive), he made it quite clear that actually thinking about what I said was too much trouble.

I saw it, and thought it didn't display a very good understanding of what you were trying to say. Well, at least he acknowledged in the beginning of the post that he did not have time to properly think about what was being said.

This is an excellent presentation of the situation. I am curious, have you read my thread, “An 'analytical-metaphysical' take on Special Relativity!” And, if you have, did you find it clear?

I've skimmed it through, and I would like to walk through the math after the Schrödinger's Equation bit (I will need help). Nevertheless, from what I understand (and have understood from your earlier commentary of that same issue), it seems entirely reasonable to me. When I first learned what Lorentz' transformation is (just by digging up information from the net), and realized exactly how it is a consequence of the assumption of isotropic speed of light, it was pretty evident that all the essential relationships of relativity could be expressed with framework that implies absolute simultaneity, or any kind of simultaneity one wishes for. Just had no idea how those frameworks would be laid down mathematically.

I actually made a brief comment about your thread, on my reply to Freeztar here:
http://hypography.com/forums/philoso...tml#post259609

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

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...

That would be nice but I am afraid I doubt it quite seriously.

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...

So, now let's get down to the issues central to your confusion!

The real problem seems to be your lack of mathematics. Because of your lack of practice in the field, your mind almost invariably jumps to misinterpretations of the symbolic representations. First of all, your comment,

is erroneous.

Ah, because the result of the dirac delta function is actually infinity there? It is really just the result of the integration over all the possibilities that allows one to say, what I said in the next sentence; "from each integration we get the probability that the value of is the same as the value of .

?

The fact that you include the line, “in the original input arguments”, leads me to think that you do not understand the integration operation itself. Integration sums the function (in this case, over all possible values of the arguments being integrated over (in our case of interest, all arguments except .

When you perform that sum over (where k is neither i nor j) the Dirac delta function (which only depends upon and ) can be factored out.

You mean, when I perform an integration pass over (i.e. some argument not in the Dirac delta function)...? So the Dirac delta function simply doesn't play a role at all at the other integration passes... When I was writing my previous post, that possibility crossed my mind and I actually wrote it down too... but for some reason that I can't remember, I decided it can't be the correct interpretation and scratched it before posting

(Just a tiny typo there btw, the is missing _r)

In essence, all you are integrating over is which amounts to the sum of all possibilities times the probability of each of those possibilities which (under the definition of probabilities) is one. Only when you go to integrate over either or does the Dirac delta function play a roll. In that specific case only (when you are either integrating over or ) the Dirac delta function yields zero anytime these two arguments are different and infinity when they are the same.

You must remember that you are not, at that moment, integrating over both of them; you are integrating over only one or the other. Note that, due to the fact that the probability that , where one or the other is some fixed value is only one possibility out of an infinite number of possibilities (that one fixed value is the one you are not integrating over). Thus the answer is “infinity times zero” ordinarily undefined; however, in this case by definition of the Dirac delta function it is simply the remainder of the function (i.e., don't include the Dirac delta function) evaluated when the argument being integrated over is equal to the one not being integrated over.

Right, I think I understand that now.

Essentially, when summed over all possibilities (for the argument being integrated over) the answer is, the probability that the argument being integrated over is exactly equal to the argument not being integrated over no matter what in the universe that second argument might be.

Right, that's, at a conceptual level, exactly what I was expecting to find, but didn't know how to interpret the math that way exactly. But, I think I am starting to understand how it works mathematically as well. E.g. after having performed the integration pass over the , we'd be looking at a function that still includes the , and hence it's value is still affecting the "result" we get from the integration of . Hmm, that's probably a bit bad way to put it, but, essentially that allows us to say "the answer is, the probability that the argument being integrated over is exactly equal to the argument not being integrated over no matter what in the universe that second argument might be

Now, the Dirac delta function has been integrated over and is no longer there! When you go to integrate over each and every argument yet to be integrated over (including the other argument in the Dirac delta function we just integrated over) we are back to a situation where the integral is over what amounts to the sum of all possibilities for the argument being integrated over. The net result is, “one” times the probability that and are the same “period”.

Right, conceptually exactly what I was expecting to find.

Remember, we have not integrated over so the net result of the integration to this point is still a function of ; however, we are presuming the value of this function is essentially not really dependent upon the value of of interest to us. Consider the probability an element some where in a neighboring galaxy is in the same place as another some other hypothetical element in that galaxy, what impact do you think that should have upon the experiment we are doing in our laboratory. My point being, only when hypothetical elements are in the same place as the element we are interested in do we expect the behavior of to depend upon that occurrence. So the great majority of those terms essentially amount to a background constant.

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

The only term of serious importance (in the deduction of Schrödinger's equation) are those terms arising from the Dirac delta function and and they will be identical.

Right.

...at this point, I hope I have cleared up your confusion a bit. The real problem here is that you are not familiar with integral calculus and to go straight into implied results of an integration over an infinite number of variables is a bit much to expect you to understand, but, from your comments, I get the impression that the whole thing is beginning to make sense to you.

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

Anyway, at this point, I went back to where this conundum with g(x) started; I reviewed my post #85 to make sure I still remembered how we got there. I did, so, where we stand with the OP is exactly here:

Originally Posted by Doctordick

...we can write the differential equation to be solved as

At this point we must turn to analysis of the impact of our axis, a pure creation of our own imagination and not a characteristic of the actual data defining the collection of referenced elements we need to explain. Since we are interested in the implied probability distribution of x, we must (in the final analysis) integrate over the probability distribution of tau.

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

Since tau is a complete fabrication of our imagination, the final certainly cannot depend upon tau. It follows directly from this observation that the dependence of on tau must (at worst) be of the form . It follows directly from this observation that the differential equation can be written.

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.

But, I was not able to apply that information here, apparently.

Focusing on:

Suppose I can write that down as:

So focusing on the latter term:

Actually don't really know how all those squares work in the algebraic manipulations... A lot of uneducated guesses in the above, and I really don't know how to proceed from that point on... :I (And I have a distinct feeling that I already went very wrong somewhere )

-Anssi

[Boof-head]

Doc, do you mind if I demonstrate why watching TV means you have to be tangent to bundle map B over V?
with colors, of course;

you have to be in the neighborhood of a surface S, with color in it. These colors are because of stripes (in CRT ones) or a matrix of pixels.

There is a Euclidean plane aligned transverse to your line of sight; there is a vector space V.

this is all you need, the 'machine' is polarizing map M w/azimuthal angle generator z, that polarizes the vectors (the ones you see are projected normal to your xy plane)

stripes over S are 3-colored; sections across the stripes divide them equally and this is the submanifold or color map B(s);
z extends V a vector map or tangent bundle over E;
a brightness function b takes z to B

This B is a subgroup of the algebra of states (sigma-finite);
Colors over C are the complex phase-space,

The phase-space is then, for each color c (as above)



so we can derive the appropriate Hamiltonian for z and b mixing.

if T then if V then TV, the tangent bundle's phase-space.
We need a map that takes abc to rgb (M) in the space; this is the Hamiltonian H(b,s) in a time-dependent form (or we could forget about colors and use electrons insted)
it's all in there; you simply need to align this with a time-dependent transfer function, which is time-independent when you switch off the TV.

Here we can claim that the function "taking color to" is "taking it from" M or B, so that S = (M|B), for the surface; this is the 'subspace in the machine' - we are seeing a map when we watch TV (or, are we really watching electrons?)

[Boof-head]

...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 . 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 , 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:

Originally Posted by Doctordick

By definition, 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 ) 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., ) exists and that function must be a solution to my fundamental equation.

It is very important here to remember that 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

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.

Originally Posted by AnssiH

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.

Originally Posted by AnssiH

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.

Originally Posted by AnssiH

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.

Originally Posted by AnssiH

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:

Originally Posted by AnssiH

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.



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.

Originally Posted by AnssiH

(Just a tiny typo there btw, the 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.

Originally Posted by AnssiH

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 has not been integrated over) how that probability varies when changes (how that probability behaves as a function of 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?

Originally Posted by AnssiH

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

Originally Posted by AnssiH

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



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 and momentum in the tau direction, defined to be the expectation value of (multiplied by a constant which is not really important here) when the wave function is known. The simple relationship of great value here is the Heisenberg “uncertainty principle”.

If the momentum in the tau direction represented by is known exactly, the the uncertainty in position 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 can be replaced with that constant “iq” and which is defined to be can thus certainly be replaced with .

As you guessed, this fact is very closely related to the behavior of . 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 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 for a complex z” which you will find under item #5 in Exponential function.

You were actually quite close to the correct result

Originally Posted by AnssiH

So focusing on the latter term:

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

What you should have done is as follows:

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 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]

Originally Posted by Doctordick

...(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]

Originally Posted by Doctordick

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

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 has not been integrated over) how that probability varies when changes (how that probability behaves as a function of 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"

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 and momentum in the tau direction, defined to be the expectation value of (multiplied by a constant which is not really important here) when the wave function is known. The simple relationship of great value here is the Heisenberg “uncertainty principle”.

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 represented by is known exactly, the the uncertainty in position 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 can be replaced with that constant “iq” and which is defined to be can thus certainly be replaced with .

Ahha...

As you guessed, this fact is very closely related to the behavior of . 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 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 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:

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 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:

And I figured out the bit. I guess it is related to what happens in the next step in the OP...

Notice that, if the term is moved to the right side of the equal sign, we may factor that side and obtain,

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

Moving the term to the right side:

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

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 , but don't know how the step before is analogous to , 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]

Originally Posted by Rade

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 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 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]

Originally Posted by AnssiH

Moving the term to the right side:

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

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 , but don't know how the step before is analogous to , 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 : 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:

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

which is actually the same thing since i² is “-1” and i is defined to be . Now, since we have the necessary negative sign, the term can be factored and the result is:

Originally Posted by AnssiH

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