Hi all
Recently started my degree in chemistry, and we spent a considerable amount of time looking at molecular orbital theory, including the Schrodinger wave equation. We were not taught any detailed maths behind this; the level of detail we were taught is:
HΨ = EΨ
We were told that H is a Hamiltonian operator, Ψ is the wavefunction (which we studied in more detail w/r/t orbitals), and E is the energy of the wavefunction. E is an eigenvalue of H, and Ψ is an eigenfunction of H. This I've accepted and read around a bit, and am happy with.
In preparation for my January exams I've looked at last year's exam paper (for which answers are unavailable). One question is along these lines:
HΨ = EΨ
a. Identify the eigenvalue and eigenfunction, and explain what this means.
Having discussed this with fellow students, some older students assigned to tutor us, and my lecturer, I'm no closer to grasping the second part of this question. I was assured today by my lecturer that eigenvalues/functions, in themselves, have no physical significance. I had the notion that the presence of eigenvalues/functions might mean that the values input are a valid solution to the Schrodinger equation, but I was told that there are other prerequisites for this which we will study next semester (our current course is a "catch up" of sorts for first year students).
I've been told that the question isn't going to come up again, but I'm intrigued nonetheless - what is the significance of the eigenvalue and eigenfunction in this equation?
Thanks for any help
Kaeroll
QUOTE (Kaeroll+Dec 8 2006, 03:10 PM)
What is the significance of the eigenvalue and eigenfunction in this equation?
By AWT the Schrodinger equation describes the wave spreading in aether foam, where the mass density of foam is proportional to energy density (compare the increasing of soap foam density during shaking). By such way, each the energy wave is making the environment more dense, which in turn affects the wave spreading. The simple DHTML applet (for MSIE browser) or 2D Java applet illustrates such behavior.
For Schrodinger equation the time independent solution exist only and if only the energy comes in certain values of energy, which are called the "eigenvalues" and the solution corresponding to each eigenvalue is the "eigenfunction".
The word "eigenvalue" comes from the German "Eigenwert" which means proper or characteristic value. "Eigenfunction" is from "Eigenfunction" meaning "proper or characteristic function". The physical meaning of eigenfunction is quite simple by AWT: because each the wave makes the aether foam more dense, it has an tendency to autofocus by it. If the energy level is sufficient, it leads to the time stable (independent) solution, where the wave resonates with the wave of it's own energy density inside the resulting dense blob as the standing wave packet.
If the energy density increases or decreases, the Aether blob collapses or expands accordingly, so it becomes to small/large to fulfill the interference condition of fixed number of standing waves. As the result, the solution becomes unstable and the wave packet collapses or expands to fulfill the resonance condition again. By such way the standing wave keeps its internal energy density on the eigenvalue solution automatically. But it means at the same time, the standing wave packet cannot exchange arbitrary amount of energy with it's neighborhood - just the certain quanta of energy in short pulses of light, so called photons.
From such behavior the name of quantum theory follows. By such way, the AWT explains the quantum mechanic by the Newtonian wave mechanic of elastic foam.
By AWT the Schrodinger equation describes the wave spreading in aether foam, where the mass density of foam is proportional to energy density (compare the increasing of soap foam density during shaking). By such way, each the energy wave is making the environment more dense, which in turn affects the wave spreading. The simple DHTML applet (for MSIE browser) or 2D Java applet illustrates such behavior.
For Schrodinger equation the time independent solution exist only and if only the energy comes in certain values of energy, which are called the "eigenvalues" and the solution corresponding to each eigenvalue is the "eigenfunction".
The word "eigenvalue" comes from the German "Eigenwert" which means proper or characteristic value. "Eigenfunction" is from "Eigenfunction" meaning "proper or characteristic function". The physical meaning of eigenfunction is quite simple by AWT: because each the wave makes the aether foam more dense, it has an tendency to autofocus by it. If the energy level is sufficient, it leads to the time stable (independent) solution, where the wave resonates with the wave of it's own energy density inside the resulting dense blob as the standing wave packet.
If the energy density increases or decreases, the Aether blob collapses or expands accordingly, so it becomes to small/large to fulfill the interference condition of fixed number of standing waves. As the result, the solution becomes unstable and the wave packet collapses or expands to fulfill the resonance condition again. By such way the standing wave keeps its internal energy density on the eigenvalue solution automatically. But it means at the same time, the standing wave packet cannot exchange arbitrary amount of energy with it's neighborhood - just the certain quanta of energy in short pulses of light, so called photons.
From such behavior the name of quantum theory follows. By such way, the AWT explains the quantum mechanic by the Newtonian wave mechanic of elastic foam.
QUOTE (Kaeroll+Dec 8 2006, 12:10 PM)
HΨ = EΨ
We were told that H is a Hamiltonian operator, Ψ is the wavefunction (which we studied in more detail w/r/t orbitals), and E is the energy of the wavefunction. E is an eigenvalue of H, and Ψ is an eigenfunction of H. This I've accepted and read around a bit, and am happy with.
In finite vector spaces, operators (O) are nothing more than square matricies. They operate on vectors (V) by matrix multiplication.
O V = ?
But in the theory of square matricies (linear algebra) for every operator there are predicted to be non-zero vectors, called eigenvectors, where the effect of the operator is identical to multiplying by a scalar constant (λ). For interesting cases, there are many eigenvectors.
O V_1 = λ_1 V_1
O V_2 = λ_2 V_2
O V_3 = λ_3 V_3
...
Naturally, you can determine what the eigenvalues/eigenvectors are by solving the equation ( O - λ I ) V = 0 -- but this is hard to do with pen and paper if the number of dimensions is high.
If λ_i = λ_j = λ_ij then the corresponding eigenvectors are called degenerate, and any linear combination of eigenvectors has the same eigenvalue.
O ( a V_i + b V_j ) = λ_ij ( a V_i + b V_j )
But the most important thing about eigenvectors is that they form an orthogonal basis. We can normalize the inner product of each vector so that the inner product <V_1,V_1> = 1. This is the best basis (an orthonormal basis) for expressing the vector if we care about expressing the operator:
Since V = a_1 V_1 + a_2 V_2 + a_3 V_3 + ...
then O V = a_1 λ_1 V_1 + a_2 λ_2 V_2 + a_3 λ_3 V_3 + ...
and <V,OV> = a_1 λ_1 + a_2 λ_2 + a_3 λ_3 + ...
Creating an orthonormal basis is called "diagonalizing" the operator, because the matrix is a diagonal one. When two operators commute AB - BA = [A,B] = 0 -- which says that you can find an orthonormal basis which diagonalizes both A and B.
--
Now Hilbert space is a special type of vector space where the number of dimensions is considered to be infinite. But luckily, all the math I have shown here so far is still good.
H is clearly an operator and you can think of it as a matrix.
δΨ/δx can be thought of as a linear operator (δ/δx) operating on a vector/function of x (Ψ). An eigenfunction is therefore analogous to an eigenvector. E, an energy, is just a number (scalar), so
H Ψ = E Ψ
is clearly analogous to
O V = λ V
Thus, for this eigenfunction,
<Ψ, H Ψ> = <Ψ, E Ψ> = E <Ψ, Ψ>
-- and <Ψ, Ψ> is usually normalized to 1.
so H Ψ = E Ψ says <Ψ, H Ψ> = E which says Ψ is one, perhaps the only one, of the possible wave equations which has exactly the energy E.
We were told that H is a Hamiltonian operator, Ψ is the wavefunction (which we studied in more detail w/r/t orbitals), and E is the energy of the wavefunction. E is an eigenvalue of H, and Ψ is an eigenfunction of H. This I've accepted and read around a bit, and am happy with.
In finite vector spaces, operators (O) are nothing more than square matricies. They operate on vectors (V) by matrix multiplication.
O V = ?
But in the theory of square matricies (linear algebra) for every operator there are predicted to be non-zero vectors, called eigenvectors, where the effect of the operator is identical to multiplying by a scalar constant (λ). For interesting cases, there are many eigenvectors.
O V_1 = λ_1 V_1
O V_2 = λ_2 V_2
O V_3 = λ_3 V_3
...
Naturally, you can determine what the eigenvalues/eigenvectors are by solving the equation ( O - λ I ) V = 0 -- but this is hard to do with pen and paper if the number of dimensions is high.
If λ_i = λ_j = λ_ij then the corresponding eigenvectors are called degenerate, and any linear combination of eigenvectors has the same eigenvalue.
O ( a V_i + b V_j ) = λ_ij ( a V_i + b V_j )
But the most important thing about eigenvectors is that they form an orthogonal basis. We can normalize the inner product of each vector so that the inner product <V_1,V_1> = 1. This is the best basis (an orthonormal basis) for expressing the vector if we care about expressing the operator:
Since V = a_1 V_1 + a_2 V_2 + a_3 V_3 + ...
then O V = a_1 λ_1 V_1 + a_2 λ_2 V_2 + a_3 λ_3 V_3 + ...
and <V,OV> = a_1 λ_1 + a_2 λ_2 + a_3 λ_3 + ...
Creating an orthonormal basis is called "diagonalizing" the operator, because the matrix is a diagonal one. When two operators commute AB - BA = [A,B] = 0 -- which says that you can find an orthonormal basis which diagonalizes both A and B.
--
Now Hilbert space is a special type of vector space where the number of dimensions is considered to be infinite. But luckily, all the math I have shown here so far is still good.
H is clearly an operator and you can think of it as a matrix.
δΨ/δx can be thought of as a linear operator (δ/δx) operating on a vector/function of x (Ψ). An eigenfunction is therefore analogous to an eigenvector. E, an energy, is just a number (scalar), so
H Ψ = E Ψ
is clearly analogous to
O V = λ V
Thus, for this eigenfunction,
<Ψ, H Ψ> = <Ψ, E Ψ> = E <Ψ, Ψ>
-- and <Ψ, Ψ> is usually normalized to 1.
so H Ψ = E Ψ says <Ψ, H Ψ> = E which says Ψ is one, perhaps the only one, of the possible wave equations which has exactly the energy E.
QUOTE (rpenner+Dec 8 2006, 10:42 PM)
... in the theory of square matricies (linear algebra) for every operator there are predicted to be non-zero vectors, called eigenvectors, where the effect of the operator is identical to multiplying by a scalar constant...creating an orthonormal basis is called "diagonalizing" the operator, because the matrix is a diagonal one.....
This is nice, but I'm not very sure, if it really helps to understand the physical meaning of operators and eigenvalues of quantum wave.
By AWT such behavior is the result of composition of the wave undulations into rotational motion along surface of bubbles, forming the aether foam.
This is nice, but I'm not very sure, if it really helps to understand the physical meaning of operators and eigenvalues of quantum wave.
By AWT such behavior is the result of composition of the wave undulations into rotational motion along surface of bubbles, forming the aether foam.
Zephir, this is the question for which the thread was created. How does anything you posted answer this question?
QUOTE (Kaeroll+Dec 8 2006, 12:10 PM)
One question is along these lines:
HΨ = EΨ
a. Identify the eigenvalue and eigenfunction, and explain what this means.
what is the significance of the eigenvalue and eigenfunction in this equation?
Specifically, your use of the terms "AWT", "aether", "foam", "mass density", "energy density", "autofocus", "sufficient energy level", "resonates", "standing wave packet", "energy density", "Aether blob", "interference condition of fixed number of standing waves", "solution becomes unstable", "arbitrary amount of energy", "neighborhood", "photons", "Newtonian wave mechanic", "elastic", "rotational motion", and "surface of bubbles" indicate your complete disregard of the question which is a question about nonrelatavistic quantum mechanics.
"AWT" is your term for your theory of everything from which we are still waiting for your deveration of your first numerical result.
"aether" is your term for a medium which pervades all space-time and according to you may be 6-dimensional (but you are not very clear on this point)
"foam" is how you claim to explain the aether of being composed of material which obeys Newtonian physics despite the fact that your wave equation is non-Newtonian
"mass density" cannot apply, because as described, the nonrelatavistic function Ψ describes a single point particle
"energy density" cannot apply for a similar reason.
"autofocus" is a mechanism which is logically falsified. Wave solutions do not autofocus with time unless the medium is dispersive, but your wave equation is nondispersive.
"sufficient energy level" is irrelevent, as the question involves a Hamiltonian which is unidentified and an energy level which is unidentified, and the question is given by a professor of quantum mechanics not AWT.
"resonates" does not apply, as there is nothing in the question which could resonate.
"standing wave packet" also does not apply -- you seem to think that the wave quation given is trapped in a box, which is not the stated case.
"Aether blob" by definition, you can't have an aether blob, or the aether would have density variations, which could be detected by shining light through, if you ascribe to a 19th century theory that light requires a medium in which to travel.
"interference condition of fixed number of standing waves" wrong for the same reason as "standing wave packet" and because there is no stated "interference condition"
"solution becomes unstable" you are confusing your math terms here.
"arbitrary amount of energy" This is a worksheet problem. everything about the amount of energy is arbitrary and unstated.
"neighborhood" You can't have a neighborhood unless it's localized and there is no indication that this wave describes a "particle in a box"
"photons" a photon is not constrained to "arbitrary amount of energy" but is part of a electromagnetic continum. You seem fixated on the "particle in a box"
"Newtonian wave mechanic" Neither in quantum mechanics nor the stated principles of your private theory of AWT is the Newtonian theory of waves relevant. Your own AWT uses a Lorentz-invariant d'Alembert operator.
"elastic" Your own AWT tells use nothing of the elastic modulus of the "aether foam"
"rotational motion" Your own AWT, by the two stated equations which you claim predict everything, has no math in it to describe rotation, nor a fixed number of dimensions in which to rotate.
"surface of bubbles" This is bad math, as the 4-d space-time cannot be the boundary of 6-d aether foam.
HΨ = EΨ
a. Identify the eigenvalue and eigenfunction, and explain what this means.
what is the significance of the eigenvalue and eigenfunction in this equation?
Specifically, your use of the terms "AWT", "aether", "foam", "mass density", "energy density", "autofocus", "sufficient energy level", "resonates", "standing wave packet", "energy density", "Aether blob", "interference condition of fixed number of standing waves", "solution becomes unstable", "arbitrary amount of energy", "neighborhood", "photons", "Newtonian wave mechanic", "elastic", "rotational motion", and "surface of bubbles" indicate your complete disregard of the question which is a question about nonrelatavistic quantum mechanics.
"AWT" is your term for your theory of everything from which we are still waiting for your deveration of your first numerical result.
"aether" is your term for a medium which pervades all space-time and according to you may be 6-dimensional (but you are not very clear on this point)
"foam" is how you claim to explain the aether of being composed of material which obeys Newtonian physics despite the fact that your wave equation is non-Newtonian
"mass density" cannot apply, because as described, the nonrelatavistic function Ψ describes a single point particle
"energy density" cannot apply for a similar reason.
"autofocus" is a mechanism which is logically falsified. Wave solutions do not autofocus with time unless the medium is dispersive, but your wave equation is nondispersive.
"sufficient energy level" is irrelevent, as the question involves a Hamiltonian which is unidentified and an energy level which is unidentified, and the question is given by a professor of quantum mechanics not AWT.
"resonates" does not apply, as there is nothing in the question which could resonate.
"standing wave packet" also does not apply -- you seem to think that the wave quation given is trapped in a box, which is not the stated case.
"Aether blob" by definition, you can't have an aether blob, or the aether would have density variations, which could be detected by shining light through, if you ascribe to a 19th century theory that light requires a medium in which to travel.
"interference condition of fixed number of standing waves" wrong for the same reason as "standing wave packet" and because there is no stated "interference condition"
"solution becomes unstable" you are confusing your math terms here.
"arbitrary amount of energy" This is a worksheet problem. everything about the amount of energy is arbitrary and unstated.
"neighborhood" You can't have a neighborhood unless it's localized and there is no indication that this wave describes a "particle in a box"
"photons" a photon is not constrained to "arbitrary amount of energy" but is part of a electromagnetic continum. You seem fixated on the "particle in a box"
"Newtonian wave mechanic" Neither in quantum mechanics nor the stated principles of your private theory of AWT is the Newtonian theory of waves relevant. Your own AWT uses a Lorentz-invariant d'Alembert operator.
"elastic" Your own AWT tells use nothing of the elastic modulus of the "aether foam"
"rotational motion" Your own AWT, by the two stated equations which you claim predict everything, has no math in it to describe rotation, nor a fixed number of dimensions in which to rotate.
"surface of bubbles" This is bad math, as the 4-d space-time cannot be the boundary of 6-d aether foam.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
...indicate your complete disregard of the question which is a question about nonrelatavistic quantum mechanics...
The answer to the question concerning the "eigenvalues" and "eigenfunction" of Schrodinger wave solution can be very simple: the eigenvalues are simply the certain values of energy (density), which the time independent solution of the Schrodinger equation must follow. And the "eigenfunction" is the solution of Schrodinger equation corresponding such energy (density) levels.
This is all, what the mainstream physic (even the very qualified one) can say to this problem. Simply because the Schrodinger equation follows directly from quantum mechanic postulates. The matrix operator calculus, you've presented has nothing to add to such answer. This is just a one of many formalisms, by which the Schrodinger equation can be solved. I can present for example the finite difference scheme, which has diagonal matrix representation, too, because it leads to the sparse system of simple equations, which is solvable by the common methods of linear algebra. We can notify the Dirac statistic formalism of Feynman integral path formalism as well.
But whole the pile of math equations DOESN'T EVEN EXPLAIN the real physical meaning of the "eigenvalues" and "eigenfunction". It just describes the way, how this math model can be solved. No less, no more. The relevant physical interpretation is, what is missing in quantum and relativity theory.
And this is what, the whole contemporary physic is about. You can live with this easily, it's your decision - but please, don't ask it for the others. They have right to understand the physical meaning of this math. Don't forget, we are living in the 21 century, not in medieval times. The knowledge is not based on the belief into hidden beauty of math equations - but the understanding of reality.
The answer to the question concerning the "eigenvalues" and "eigenfunction" of Schrodinger wave solution can be very simple: the eigenvalues are simply the certain values of energy (density), which the time independent solution of the Schrodinger equation must follow. And the "eigenfunction" is the solution of Schrodinger equation corresponding such energy (density) levels.
This is all, what the mainstream physic (even the very qualified one) can say to this problem. Simply because the Schrodinger equation follows directly from quantum mechanic postulates. The matrix operator calculus, you've presented has nothing to add to such answer. This is just a one of many formalisms, by which the Schrodinger equation can be solved. I can present for example the finite difference scheme, which has diagonal matrix representation, too, because it leads to the sparse system of simple equations, which is solvable by the common methods of linear algebra. We can notify the Dirac statistic formalism of Feynman integral path formalism as well.
But whole the pile of math equations DOESN'T EVEN EXPLAIN the real physical meaning of the "eigenvalues" and "eigenfunction". It just describes the way, how this math model can be solved. No less, no more. The relevant physical interpretation is, what is missing in quantum and relativity theory.
And this is what, the whole contemporary physic is about. You can live with this easily, it's your decision - but please, don't ask it for the others. They have right to understand the physical meaning of this math. Don't forget, we are living in the 21 century, not in medieval times. The knowledge is not based on the belief into hidden beauty of math equations - but the understanding of reality.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"AWT" is your term for your theory of everything from which we are still waiting for your derivation of your first numerical result
I'm not payed for some derivations. What I'll obtain in doing this? The common people will not understand me either.
I'm not payed for some derivations. What I'll obtain in doing this? The common people will not understand me either.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"aether" is your term for a medium which pervades all space-time and according to you may be 6-dimensional...
The Aether is the ancient concept, not mine. And I'm not saying, the Aether is 6-dimensional, the Aether has no fixed number of dimensions, being infinitely recursive.
The Aether is the ancient concept, not mine. And I'm not saying, the Aether is 6-dimensional, the Aether has no fixed number of dimensions, being infinitely recursive.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"foam" is how you claim to explain the aether of being composed of material which obeys Newtonian physics despite the fact that your wave equation is non-Newtonian...
The double pendulum doesn't fulfill the harmonic wave solution, but it doesn't means, it violates the Newtonian physic. The system of mutually connected strings cannot be described by the wave equation at the global level, although each part of it follows the wave equation.
The double pendulum doesn't fulfill the harmonic wave solution, but it doesn't means, it violates the Newtonian physic. The system of mutually connected strings cannot be described by the wave equation at the global level, although each part of it follows the wave equation.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"mass density" cannot apply, because as described, the nonrelatavistic function Ψ describes a single point particle...
Single point function? LOL..
Single point function? LOL..
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"autofocus" is a mechanism which is logically falsified. Wave solutions do not autofocus with time unless the medium is dispersive, but your wave equation is nondispersive....
Wave equation of foam is the nonlinear function of it's own solution.
Wave equation of foam is the nonlinear function of it's own solution.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"sufficient energy level" is irrelevant, as the question ... is given by a professor of quantum mechanics.....
HUH?
HUH?
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
..."resonates" does not apply, as there is nothing in the question which could resonate.......
Says who?
Says who?
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
..."standing wave packet" also does not apply -- you seem to think that the wave equation given is trapped in a box, which is not the stated case......
The wave is shaking the aether foam, it makes the foam more dense at the place of the wave, as the result, the wave undulates in its own density blob like particle in the box.
The wave is shaking the aether foam, it makes the foam more dense at the place of the wave, as the result, the wave undulates in its own density blob like particle in the box.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
.."Aether blob" by definition, you can't have an aether blob, or the aether would have density variations, which could be detected by shining light throughl.....
We are calling these density variations a quantum fluctuations and photons.
We are calling these density variations a quantum fluctuations and photons.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
..."interference condition of fixed number of standing waves" wrong for the same reason as "standing wave packet" and because there is no stated "interference condition".....
It's obvious, you cannot even imagine the wave spreading through foam, but it's not my problem. I'm just collecting the evidence, how dumb even quite clever people can be, whenever the discussion goes outside the scope of the usual paradigm. Why do you think, I'm wasting my time with you? Everything was explained here by many times.
It's obvious, you cannot even imagine the wave spreading through foam, but it's not my problem. I'm just collecting the evidence, how dumb even quite clever people can be, whenever the discussion goes outside the scope of the usual paradigm. Why do you think, I'm wasting my time with you? Everything was explained here by many times.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
..."solution becomes unstable" you are confusing your math terms here.......
Prove it.
Prove it.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
..."neighborhood" You can't have a neighborhood unless it's localized and there is no indication that this wave describes a "particle in a box"......
Nope, it describes the quantum wave packet, i.e. the wave confined by it's own energy density.
Nope, it describes the quantum wave packet, i.e. the wave confined by it's own energy density.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
.."photons" a photon is not constrained to "arbitrary amount of energy" but is part of a electromagnetic continuum. You seem fixated on the "particle in a box".."......
I'm not saying, the "photon is constrained to arbitrary amount of energy". Concerning the "particle in the box", it's just you who is using such concept all the time, not me...
You're just projecting your own mental problem into my way of thinking. I never used the words "particle in the box" in my post.
I'm not saying, the "photon is constrained to arbitrary amount of energy". Concerning the "particle in the box", it's just you who is using such concept all the time, not me...
You're just projecting your own mental problem into my way of thinking. I never used the words "particle in the box" in my post.QUOTE (rpenner+Dec 9 2006, 01:19 AM)
...Neither in quantum mechanics nor the stated principles of your private theory of AWT is the Newtonian theory of waves relevant. Your own AWT uses a Lorentz-invariant d'Alembert operator..."......
Prove it.
Prove it.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"elastic" Your own AWT tells use nothing of the elastic modulus of the "aether foam"...
How the wave equation can exist in nonelastic medium?
How the wave equation can exist in nonelastic medium?
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
..."rotational motion" Your own AWT, by the two stated equations which you claim predict everything, has no math in it to describe rotation, nor a fixed number of dimensions in which to rotate"...
The AWT doesn't assumes some "fixed number of dimensions". The composition of wave undulations in different directions leads to such motion naturally.
The AWT doesn't assumes some "fixed number of dimensions". The composition of wave undulations in different directions leads to such motion naturally.
QUOTE (rpenner+Dec 9 2006, 01:19 AM)
"surface of bubbles" This is bad math, as the 4-d space-time cannot be the boundary of 6-d aether foam....
This is bad argumentation, as I don't say something like this. The spacetime is formed by the foam gradients, not some boundary.
This is bad argumentation, as I don't say something like this. The spacetime is formed by the foam gradients, not some boundary.
Does mass gravity travel from great depths, to influence other things, or is this supposed influence made by one aspect of space, bumping into one another?
PhysOrg scientific forums are totally dedicated to science, physics, and technology. Besides topical forums such as nanotechnology, quantum physics, silicon and III-V technology, applied physics, materials, space and others, you can also join our news and publications discussions. We also provide an off-topic forum category. If you need specific help on a scientific problem or have a question related to physics or technology, visit the PhysOrg Forums. Here you’ll find experts from various fields online every day.
To quit out of "lo-fi" mode and return to the regular forums, please click here.