In this post, I will explain how to built a 4-D torus that would tell why a matter wave wavelength depends on its speed (De Broglie wavelength).

1) Take a slinky. Make a mark along the slinky. I cut out an inch of tape paper and attached it to each spiral. Then, I marked one side of the tape paper, this represent "arrows".

2) Make a clockwise turn around the slinky edge and glue the tail arrow with the head arrow. This makes a torus. The twisted path of the arrows is very noticeable.

3) Take another slinky. Do the same as before but a counter-clockwise turn.

Heuristically, these slinkies have to be together. It is too difficult to see anything when they are together, so I draw them separately. You have to arrange them in the way shown in Figure 1. The clockwise-turned slinky is at the right of the Figure. The movement of its arrows is clockwise. The counterclockwise-turned slinky is at the left of the Figure. The movement of its arrows is counterclockwise. This makes that the wavelength left on flatland depends on the object speed.

Figure 1 the two slinkies are arranged as shown. The clockwise turned is at the right of the Figure. This arrangement allows to have a wave that just depends on the speed of the object. This occurs because of the clockwise and counterclockwise movements.

These arrows are electric field vectors that get printed in the plane. The impression will look like this:

Figure 2 electric field vectors imprinted in the plane. The wavelength will depend on the speed of the object and will use the two dimensions of the plane to travel.

On Fig 2 undistinguishables states occur at 0, 180 and 360 degrees. Notice that the printed arrows close a circle when the particle arrive to 180 degrees. Thus, an observer in the plane may judge that one complete round has been made. However, the intersection at 180 degrees is different to the original state at 0 degrees. After another round, the printed pattern is exactly like the state at the origin. This is a characteristic of a spin 1/2 particle.

## Saturday, March 30, 2013

## Saturday, March 23, 2013

### Particle-Wave duality

After I read Mikio Kaku's "Hyperspace", I felt that the quantum weirdness was about the lack of room to explain it. I will explain the particle-wave duality based on this principle -"Lack of Room"

In order to increase the room, we need more dimensions. Lets start with four. To do that, our familiar three-dimensional word has to be flattening out. This produces a plane, which now contains our familiar three-dimensions. This plane exists at present time, all the time. This means that the future is above and the past is under it.

This new time dimension is like the other three ones. But observers in the plane cannot detect anything in it. However, a significant portion of the physical object is there. A physical particle symmetrically exists in this wider room, half of it in the past and the other half in the future. The only part of the particle that an observer can detect is the intersection at the present time. This exercise shows you the need to understanding the intersection of a higher dimensional body, into a lower dimensional space.

To do that, I begun to use the tools suggested in "Hyperspace" and Thomas Banchoff's "Beyond the Third Dimension: Geometry, Computer Graphic and Higher Dimensions" to imagine a four dimensional 4-D object. One of the tools is slicing. You take a 3-D solid and make it travel through a 2-D surface (a plane). This will produce slices of the object in the plane. Then, you make a revolution of every slice. This revolutions will make back a 3-D object. The shape of these 3-D objects will depend on where the slice occurred and how the revolution was performed. The point is that those 3-D objects are the intersections of the 4-D object by a 3-D plane. Thus, the sequence of the intersection of a hypersphere ( 4-D sphere) by a 3-D space will be a 3-D sphere which diameter increases, it reached a maximum and then decreases. This occurs until the sphere disappeared. Figure 1.

Figure 1 the 3-D sphere is sliced by the 2-D plane. This produces a circle that increases in diameter, reaches a maximum and then, decreases until it disappeared from the 2-D plane. This same exercise with a 4-D sphere (hypersphere) and a 3-D plane (normal 3-D space) will produce a 3-D sphere following the same pattern as the circle in the plane.

Now imaging a donut. That donut is only intersected in the middle. Haft of the donut is in the past and the other half is in the future. You can only see the present. If you follow this, you will get just three shapes: 1) when the donut is cut through its equator. Regardless the orientation of the revolution, you will get two concentrical spheres. 2) when the donut is cut perpendicular to its equator. You will get two separated circles. Here you can do two different revolutions of the two circles. One through a symmetrical edge that join them, which will give two separated spheres. Another through an edge perpendicular to the previous one, which will render another donut. You can see this last intersection in http://www.bekkoame.ne.jp/~ishmnn/java/hypertorus.html, see Figure 2.

Figure 2 the intersection of the donut through its equator will give two concentric circles. The revolution of these circles will give two concentric 3-D spheres (bottom of the Figure). The intersection of the donut perpendicular to its equator will give two circles. The revolution performed as depicted in the Figure gives two 3-D spheres or a torus. The structures at the far right of the figure are what have been found for the deuteron (JL Forest et al., 1996).

Now, the reader may ask - Why am I going through all this? Who cares about these higher dimensions? Fair enough. We need evidence indicating that this would be relevant. After a while, I found that the structures just described occur in the atom nucleus and they are shown at the far right of Figure 2 (J.L. Forest et al. 1996 "Femtometer Toroidal Structures in Nuclei") and in the chemical bond ( "Laplacian of the electron density of the urea molecule"), see Figure 3.

Figure 3 laplacian of the electron density for urea molecule. Core attractors are observed in all the atoms; oxygen, carbon and nitrogen. It is not observed for hydrogen. The hydrogen bond shows a two- separated sphere structure. The lone pair on the nitrogen consists of two toruses, part of these two toruses are fused on the side of the nitrogen opposed to the carbon atom.

How does this have anything to do with the particle-wave paradox? I think it is obvious that we are trapped in a 3-D plane, trying to observe a 4-D torus, which is impossible. The evidence suggests that the deuteron and the electron are 4-D toruses. This is the particle!

To have more room opens the possibility of a 4-D object (the particle) to leaving a 3-D wave when it moves through the 3-D plane! This 3-D wave is described by the wave equation.

Furthermore, now we can conceive a non-moving particle, i.e. when the wave function collapses.

Is the paradox resolved?

In the next Post, I will describe the 3-D wave.

In order to increase the room, we need more dimensions. Lets start with four. To do that, our familiar three-dimensional word has to be flattening out. This produces a plane, which now contains our familiar three-dimensions. This plane exists at present time, all the time. This means that the future is above and the past is under it.

This new time dimension is like the other three ones. But observers in the plane cannot detect anything in it. However, a significant portion of the physical object is there. A physical particle symmetrically exists in this wider room, half of it in the past and the other half in the future. The only part of the particle that an observer can detect is the intersection at the present time. This exercise shows you the need to understanding the intersection of a higher dimensional body, into a lower dimensional space.

To do that, I begun to use the tools suggested in "Hyperspace" and Thomas Banchoff's "Beyond the Third Dimension: Geometry, Computer Graphic and Higher Dimensions" to imagine a four dimensional 4-D object. One of the tools is slicing. You take a 3-D solid and make it travel through a 2-D surface (a plane). This will produce slices of the object in the plane. Then, you make a revolution of every slice. This revolutions will make back a 3-D object. The shape of these 3-D objects will depend on where the slice occurred and how the revolution was performed. The point is that those 3-D objects are the intersections of the 4-D object by a 3-D plane. Thus, the sequence of the intersection of a hypersphere ( 4-D sphere) by a 3-D space will be a 3-D sphere which diameter increases, it reached a maximum and then decreases. This occurs until the sphere disappeared. Figure 1.

Figure 1 the 3-D sphere is sliced by the 2-D plane. This produces a circle that increases in diameter, reaches a maximum and then, decreases until it disappeared from the 2-D plane. This same exercise with a 4-D sphere (hypersphere) and a 3-D plane (normal 3-D space) will produce a 3-D sphere following the same pattern as the circle in the plane.

Now imaging a donut. That donut is only intersected in the middle. Haft of the donut is in the past and the other half is in the future. You can only see the present. If you follow this, you will get just three shapes: 1) when the donut is cut through its equator. Regardless the orientation of the revolution, you will get two concentrical spheres. 2) when the donut is cut perpendicular to its equator. You will get two separated circles. Here you can do two different revolutions of the two circles. One through a symmetrical edge that join them, which will give two separated spheres. Another through an edge perpendicular to the previous one, which will render another donut. You can see this last intersection in http://www.bekkoame.ne.jp/~ishmnn/java/hypertorus.html, see Figure 2.

Figure 2 the intersection of the donut through its equator will give two concentric circles. The revolution of these circles will give two concentric 3-D spheres (bottom of the Figure). The intersection of the donut perpendicular to its equator will give two circles. The revolution performed as depicted in the Figure gives two 3-D spheres or a torus. The structures at the far right of the figure are what have been found for the deuteron (JL Forest et al., 1996).

Now, the reader may ask - Why am I going through all this? Who cares about these higher dimensions? Fair enough. We need evidence indicating that this would be relevant. After a while, I found that the structures just described occur in the atom nucleus and they are shown at the far right of Figure 2 (J.L. Forest et al. 1996 "Femtometer Toroidal Structures in Nuclei") and in the chemical bond ( "Laplacian of the electron density of the urea molecule"), see Figure 3.

Figure 3 laplacian of the electron density for urea molecule. Core attractors are observed in all the atoms; oxygen, carbon and nitrogen. It is not observed for hydrogen. The hydrogen bond shows a two- separated sphere structure. The lone pair on the nitrogen consists of two toruses, part of these two toruses are fused on the side of the nitrogen opposed to the carbon atom.

How does this have anything to do with the particle-wave paradox? I think it is obvious that we are trapped in a 3-D plane, trying to observe a 4-D torus, which is impossible. The evidence suggests that the deuteron and the electron are 4-D toruses. This is the particle!

To have more room opens the possibility of a 4-D object (the particle) to leaving a 3-D wave when it moves through the 3-D plane! This 3-D wave is described by the wave equation.

Furthermore, now we can conceive a non-moving particle, i.e. when the wave function collapses.

Is the paradox resolved?

In the next Post, I will describe the 3-D wave.

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