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AnssiH · 05-30-2009

Okay, time to start dissecting the this thread!

Quote:

Originally Posted by Doctordick

...I took the collection of ontological elements standing behind any explanation to be “unknowns” and then attempted to set down the relationships those unknowns had to obey: the result was the derivation of my fundamental equation. The presentation of that proof may be found here; where the following relationship is both defined and derived

$\left\{\sum_i \vec{\alpha}_i \cdot \vec{\nabla}_i + \sum_{i \neq j}\beta_{ij}\delta(\vec x_i - \vec x_j)\right\}\vec{\Psi} = K\frac{\partial}{\partial t}\vec{\Psi} = iKm\vec{\Psi}$

I have already given a specific proof that Schrödinger's equation is an approximate solution to my fundamental equation...

...and expanded that proof to show that a three dimensional representation of any explanation may be achieved via a three dimensional representation of that self same equation: i.e., it leads to a three dimensional form of Schrödinger equation.

...I also showed that the Dirac delta function could be used as a rational representation of any rules that universe might have. This step provided another subtle alteration in the original numbers which was somewhat unexpected. Notice that multiplication of each of the original numbers representing any observation by any specific constant also has utterly no consequences within this model: i.e., the model is scale invariant.

...if the data belonging to a given observation could be divided into two (or more) sets having negligible influence on one another, those sets could be examined independently of one another: i.e., these collections would end up being constrained by exactly the same relationship which constrained the original universe. This is to say that these subsets (or “objects”) could be analyzed as a universes unto themselves

...there is a subtle problem here: the fundamental equation was constrained (see appendix 3 of the original proof) to be valid only in the rest frame of the universe. The central issue here is that the two collections of elemental entities either have significant influence on one another or they do not. If they do not have any significant influence on one another, the constraint that the equation is only valid in the rest frame of “the universe” cannot be a valid constraint as either object may be considered to be a universe unto itself: i.e., the rest frame of one collection of elemental entities may not be the same as the rest frame of the other. The solution to this problem lies with the scaling of the geometry between the two systems: there must exist a consistent way of converting a solution in one system to a solution in the other independent of any influence between the two.

Up to this point I can only say "check".

Quote:

Now, I have already shown that a given solution in the rest frame is easily transformed to a solution where the frame of reference is no longer at rest. Such a transformation is simply obtained via multiplication of $\vec{\Psi}$ by the simple function

$\prod_{j=1}^n e^{i\frac{Px_j}{n\hbar}}$ .

This change in $\vec{\Psi}$ will simply add P/n to the momentum in the x direction of every elemental entity in the universe (the universe consisting of the elemental entities which make up that independent object). In other words, the transformation simply adds P to the momentum of the object and thus the object is no longer at rest in the rest frame used to solve for $\vec{\Psi}$ . Thus it is that we can always transform a solution in the rest frame of one object to a solution in the rest frame of the other (note that the transformation also requires a change in energy which is just as easily obtained).

Yeah, I had to scratch my head with that, simply because I can't immediately see what happens there, due to my unfamiliarity with these math tricks...

I asked about it in a PM, and DD noticed it was missing the $\hbar$ at the denominator, so that's now fixed in the OP. The following quote is from a PM:

Quote:

Originally Posted by Doctordick

...just as the function $e^{\frac{-iqt}{K\sqrt{2}}}$ shifted the energy of by a constant factor $\frac{-iq}{K\sqrt{2}}$ (as a direct consequence of the differential with respect to “t”), the function $e^{i\frac{Px_j}{n\hbar}}$ shifts the momentum in the x direction of the jth entity by the constant factor $i\frac{P}{n\hbar}$ (as a direct consequence of the differential with respect to x_j). Both of these effects are a consequence of the product rule of differentiation. The product indicates addition of $i\frac{P}{n\hbar}$ for every value of j (all n entities) so the sum of all n terms (times $i\hbar$ will be “P”. Thus it follows that the change in $\Psi$ (changed to $\Phi$ times the product) produces an equation where the product can be divided out and thus creates a new function $\Phi$ which gives exactly the same probability distribution except for the fact that the momentum of the universe in the x direction is no longer zero but turns out to be “P”. In is no more than a common trick done in quantum mechanics.

I am getting old and careless. I will leave it to you to check the algebra.

Yup, so if I understand this part correctly, here goes... To make things simpler for myself, I just concentrated on a partial derivative of a single x. I.e:

$\frac{\partial}{\partial x} \vec{\Psi} = \frac{\partial}{\partial x} \vec{\Phi} e^{i\frac{Px}{n\hbar}}$

The right hand side can be written:

$\left\{ \frac{\partial}{\partial x} \vec{\Phi} \right\} e^{i\frac{Px}{n\hbar}} + \vec{\Phi} \frac{\partial}{\partial x} e^{i\frac{Px}{n\hbar}}$

$=$

$\left\{ \frac{\partial}{\partial x} \vec{\Phi} \right\} e^{i\frac{Px}{n\hbar}} + \vec{\Phi} i\frac{P}{n\hbar} e^{i\frac{Px}{n\hbar}}$

And at that point the e's could be factored out, leaving us with:

$\left\{ \frac{\partial}{\partial x} + i\frac{P}{n\hbar} \right\} \vec{\Phi}$

I hope that's all valid, and judging from your comment "the function $e^{i\frac{Px_j}{n\hbar}}$ shifts the momentum in the x direction of the jth entity by the constant factor $i\frac{P}{n\hbar}$ (as a direct consequence of the differential with respect to x_j)" I think I am on the right track with this. That operation applied to each element would seem to do exactly what you are saying it would.

I'll try and continue from here tomorrow...

-Anssi

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