Pauli’s Exclusion Principle

We have shown throughout the Imagineer’s Chronicles and its companion book "The Reality of the Fourth *Spatial* Dimension" there would many theoretical advantages to defining the universe in terms of four *spatial* dimensions instead of four dimensional space-time.

One is that it would allow for the derivation of Pauli’s exclusion principle or the fact that no two identical fermions such as electrons can have the same four quantum numbers simultaneously by extrapolating the laws of classical physics in a three-dimensional environment to a fourth *spatial* dimension.

There are four quantum numbers.

The first, designated by the letter "n", and it describes the electron shell, or energy level.  Its value ranges from 1 to "n", where "n" is the shell containing the outermost electron of that atom.  The second, or ℓ quantum number, describes the subshell (0 = s orbital, 1 = p orbital, 2 = d orbital, 3 = f orbital, etc.).  Its value can range from 0 to n − 1.  The third, or m_ℓ, describes the specific orbital within a subshell.  Finally fourth, quantum number with the designator "s" describes the spin of the electron within that orbital.  An electron can have a spin of ±½; m_s will be either, corresponding with "spin" and "opposite spin."  Each electron in any individual orbital must have different spins; therefore, an orbital never contains more than two electrons.

Pauli’s exclusion principle is considered one of the most important principles in physics because if electrons could occupy the same quantum state they would all congregate in a single point corresponding to the lowest-energy state.  If this occurred atoms would have no volume.  However, Pauli’s exclusion principle tells us that each additional electron added to an atom must occupy higher energy level with respect to the lower-energy electrons the atom originally contained.  Therefore, they occupy an extended volume rather than a volume less one dimensional point.

In the article "Defining potential and kinetic energy?" Nov. 26, 2007 it was shown that one can define all forms of energy in terms of a curvature or displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

This curvature is analogous to the space-time curvature "The General Theory of Relativity" assumes is responsible for gravitational energy.  However it differs in that it allows for the bi-directional movement of the "surface" of three-dimensional space with respect to a four *spatial* dimension.  This assumption is based on observations of our three-dimensional environment which indicate that one can move in two directions upward or downwards and backward or forward in a spatial dimension whereas they indicate that one can only move in one direction, forward in a time or a space-time dimension.

As mentioned earlier using the concept of four *spatial* dimensions instead of four dimensional space has many theoretical advantages.

One is that as the article "Why is energy/mass quantized?" Oct. 4, 2007 showed that one can derive its quantum mechanical properties by extrapolating the laws governing resonance in a three-dimensional environment to the bi-directional spatial displacements caused by a matter wave moving on a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension.

There are four conditions required for resonance to occur in a classical Newtonian environment, an object, or substance with a natural frequency, a forcing function at the same frequency as the natural frequency, the lack of a damping frequency and the ability for the substance to oscillate spatial.

The existence of four *spatial* dimensions would give the "surface" of three-dimensional space (the substance) the ability to oscillate spatially between a third and fourth *spatial* dimensions thereby fulfilling one of the requirements for classical resonance to occur.

These oscillations would be caused by an event such as the decay of a subatomic particle or the shifting of an electron in an atomic orbital.  This would force the "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension to oscillate with the frequency associated with the energy of that event.

Therefore, these bi-directional oscillations in a "surface" of a three dimensional space would meet the requirements mentioned above for the formation of a resonant system or "structure" in space.

Observations of a three-dimensional environment show the energy associated with resonant system can only take on the incremental or discreet values associated with a fundamental or a harmonic of the fundamental frequency of its environment.

Similarly the energy associated with resonant systems in four *spatial* dimensions could only take on the incremental or discreet values associated a fundamental or a harmonic of the fundamental frequency of its environment.

These resonant systems in four *spatial* dimensions are responsible for the incremental or discreet energies associated with quantum mechanical systems.

As mentioned earlier the article "Defining potential and kinetic energy?" showed all forms of energy including the angular momentum of particles can be defined in terms of a displacement in a "surface* of three-dimensional space manifold with respect to a fourth *spatial* dimension. 

Using your right-hand: Curl your fingers into a half-circle around the wire, they point in the direction of the magnetic field, B Point your thumb in the direction of the conventional current

In three-dimensional space the orientation of the angular momentum of a particle is determined by the right hand rule which says that its  angular momentum or energy of one with a counter clockwise spin would be directed "upward" with respect to the two-dimensional plane in which it is spinning while one with a clockwise spin would be "downwardly" directed. 

Therefore one can derive the fourth or spin quantum number in terms of the direction a "surface" of three-dimensional space is displaced with respect to a fourth *spatial* dimension.  For example if one defines energy of a electron with a spin of -1/2 in terms of a downward directed displacement one would define a +1/2 spin as an upwardly directed one. .

Using this concept one can theoretical derive Pauli’s Exclusion Principle or the reason why only two particle of opposite spins can occupy a quantum orbital by extrapolating the laws of a three-dimensional environment  to a fourth *spatial* dimension

In three-dimensional space the frequency or energy of a resonant system is defined by the vibrating medium and the boundaries of its environment.

For example the resonant frequency or energy of a stationary or standing wave generated when a violin string plucked is determined by its shape of the instrument sounding box and the length of its strings.

Similarly the resonant system that defines energy of atom orbitals defined in the article "Why is energy/mass quantized?" would be defined by the size and shape of its orbital.

This means that atomic orbital will have a unique energy of the standing wave associated with its resonant frequency.

The reason no two identical fermions such as electrons can fill the same energy level or have the same four quantum numbers simultaneously can be understood comparing how those quantum numbers or orbitals are filled to the filling of a bucket of water.

There a two ways to fill a bucket.  One is by pushing it down and allowing the water to flow over its edge or by using a cup to raise it to the level of the buckets rim.

Similarly there would be two ways fill an atomic orbital according to the concepts presented in the article "Defining potential and kinetic energy?".  One would be by forcing the "surface" of three-dimensional space "downward" with respect to a fourth *spatial* while the other would be raise it up to the energy level associated with an electron in that orbital.

However the energy required by each method will not be identical for the same reason that it requires slightly less energy to fill a bucket of water by pushing it down below its surface than using a cup to fill it.

This explains why two elections in the same atomic orbital have different quantum numbers.

However it also explains why no two quantum particles can have the same quantum number because observations of water show that there is a direct relationship between the magnitude of a displacement in its surface to the magnitude of the energy resisting that displacement. 

Similarly the magnitude of a displacement in a "surface" of a three-dimensional space manifold with respect to a fourth *spatial* dimension caused by two quantum particles with similar quantum numbers would greater than that caused by a single one.  Therefore, two electrons in the same orbital will repel each other and seek the lower energy state associated with a different spin quantum number.

This shows how one can derive of Pauli’s exclusion principle or the fact that no two identical fermions such as electrons can have the same four quantum numbers simultaneously by extrapolating the laws of classical physics in a three-dimensional environment to a four *spatial* dimension.

Later Jeff

Copyright Jeffrey O’Callaghan 2011

	The Imagineer’s Chronicles Vol 2 — 2011  Book $9.24 Ebook $5.97		The Imagineer’s Chronicles 2007 thru 2010 Book $15.33 Ebook $5.97
		The Reality of the Fourth Spatial Dimension   Book  $8.99   Ebook $5.97

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