Geometric Interpretation Of Schrodinger's

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GEOMETRICAL STUDY OF SCHROEDINGER'S FORMULA

[tex]\vec J = \mathfrak{Re}[\Psi^* (\vec r) (\frac{-i \hbar}{m} \vec \nabla) \Psi (\vec r) ][/tex]

If we take a look on previous expression, we could continue with the importance of complex numbers.

The complex numbers are very important to represent points or vectors in plane, and can be expressed this way:

a = b·x+c·y

If we choose the real axis numbers like X, and imaginaries like Y:

a = b + c·i

But we could have been done this:

a = b·i + c

If we take Geometric Algebras (GA), more exactly the even sub group of GA, we can represent vectors like part quaternions, and rotations through quaternion operators.

If we take an element of volume (area in draw):

Clipboard01.0.jpg


And take a volume in a posterior time:

[tex] V_t+dt = \left( dx + \frac {\partial u}{\partial x} dx \cdot dt \right) \left( dy + \frac {\partial v}{\partial y} dy \cdot dt \right) [/tex]

And the relative change of volume is:

[tex] \frac {d \left( \frac {V_t_+_d_t - V_t} {V_t} \right) } {dt} = \frac {dV_R}{dt} = \frac {\partial u}{\partial x} + \frac {\partial v}{\partial y} [/tex]

We have that:

[tex] \frac {\partial V_R}{\partial t} = \frac {\partial u}{\partial x} + \frac {\partial v}{\partial y} + \frac {\partial w}{\partial z} = \nabla \cdot \vec {v} [/tex]

The dilatation of volume can be directly expressed with velocity gradient in space:

A volume element can be:

TRANSLATED:

Clipboard05.6.jpg


DEFORMATED:
Clipboard06.jpg


ROTATED
Clipboard07.jpg


or HAVE CUTTING EFFORTS

Clipboard08.jpg


If we analyze the angles [tex]\theta_1[/tex] y [tex]\theta_2[/tex], it can easily verify geometrically that:

[tex] TAN~~ d \theta_1 \simeq d \theta_1 \simeq \frac {\left( \frac {\partial v} {\partial x} dx \cdot dt \right) }{dx} [/tex]

And:
[tex] TAN~~ d \theta_2 \simeq d \theta_2 \simeq \frac {\left( \frac {- \partial u} {\partial y} dy \cdot dt \right) }{dy} [/tex]

And we have:

[tex] \dot {\theta_1} = \frac {d \theta_1}{dt} = \frac {\partial v}{\partial x} [/tex]

and

[tex] \dot {\theta_2} = \frac {d \theta_2}{dt} = - \frac {\partial u}{\partial y} [/tex]

And angular velocity in Z axis will be:

[tex] \omega_z = \frac {1}{2} \left( \frac {\partial v}{\partial x} - \frac {\partial u}{\partial y} \right) = \frac {1}{2} \dot {\theta_1} + \dot {\theta_2} [/tex]

Now, if we take this fraction of the Schrödinger's:

[tex] \nabla \left[ \frac {i \hbar} {2m} \left( \psi \nabla \psi^*} - \psi^* \nabla \psi \right) \right] [/tex]

Or much clear:

[tex] \nabla \left[ \frac {1} {2} \left( \psi \vec v \psi^*} - \psi^* \vec v \psi \right) \right] [/tex]


Does anybody likes to play in discovering what it means?

-----------
References:

- Quaternion and Rotation Sequences, Jack B. Kuipers
- Vectors, tensors and the Basic Equations of Fluid Mechanics, Rutherford Aris.
- Vibraciones y ondas. A. P. French.
- Lectures On Clifford (Geometric) Algebras. Rafal Ablamowicz et. al.
- Mecánica de Fluidos. Victor L. Streeter et. al.
- De Natura Visibilium et Invisibilium. R. Aparicio. Ed. Elaleph.
- Siglo XXI: La Física que nos espera. R. Aparicio. Ed. Elaleph.[/quote]
 
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Answers and Replies

  • #3
Tom Mattson
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Plat00n,

I am having trouble figuring out what you are trying to do here, and it is mostly because I can't see the images you linked to. img tags don't work outside of the General Discussion Forum. If you want the images to be viewable here you'll need to attach them to a post.

Thanks,
 
  • #4
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Tom Mattson said:
Plat00n,
I am having trouble figuring out what you are trying to do here, and it is mostly because I can't see the images you linked to. img tags don't work outside of the General Discussion Forum. If you want the images to be viewable here you'll need to attach them to a post.
Thanks,

You can see it in the blogspot, that I have used to link it. Here is:

http://denaturavisibiliumetinvisibilium.blogspot.com/

Thanks.
 
  • #5
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Links

Now images can be seen.
 
  • #6
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Anything to say?

Does anybody has any interesting oppinion about this? It's all wrong? All ok? ???
 
  • #7
nrqed
Science Advisor
Homework Helper
Gold Member
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Tom Mattson said:
Plat00n,

I am having trouble figuring out what you are trying to do here, and it is mostly because I can't see the images you linked to. img tags don't work outside of the General Discussion Forum. If you want the images to be viewable here you'll need to attach them to a post.

Thanks,

Hi Tom....

I saw the interesting thread on your class on differential forms (using Bachman`s book). I would like to discuss differential forms at a very elementary level... I hope you can help clarifying some points. I will post later this weekend in the diff geometry board. I hope you will be around to help me understand!

Regards,

Pat
 
  • #8
jtbell
Mentor
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Plott029 said:
Does anybody has any interesting oppinion about this?

Perhaps more people who might be interested in discussing this would have seen it if you had posted it in the Quantum Physics forum instead of here (one of the homework-help forums). I saw this thread for the first time today.

You could try posting a link to this thread in the Quantum Physics forum, or maybe the moderators would be willing to move this thread over there. I think they wouldn't like it if you re-post your entire first post there.
 
  • #9
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Thanks for your information.
 
  • #10
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Endorsers and referees are wanted to write the first of 3 papers on Geometric (Clifford) Algebras, concretely the Even Sub Group of Clifford Algebras and his appllication with the quantum mechanics.
 
  • #11
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Quaternions

Does anybody thinks that include a wave function in quaternions, not in complex numbers, can be a way to discover "hidden variables" that Einstein was looking for?

Can be that EPR and Bell inequalities are really bad interpretations due to the lack of better numbers?

Is the Schrödinger equation an amputated equation? Can be derived from another much generic?
 

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