As we objectify descriptions, and apply the hypothesis of spatial isotropy, we stop referring directly to chromatic visual sensation. In this article we discuss how physics becomes color-blind by changing the descriptive framework from quark space , to a new, almost-Euclidean space called . In , particles are rotating. Their electric and magnetic axes turn around the polar axis. And in we can construct a Cartesian coordinate system as follows.
The following basis vectors are used to make general descriptions; the axis of the abscissa is directed by , the axis of the ordinate from and the polar-axis by Then, any position vector in can be expressed in terms of , and , its Cartesian coordinates as
The new space is closer than to the ordinary space of everyday experience. And we use a Euclidean metric to determine vector norms. But some other details such as spatial continuity have to be assessed before we could say that is a Euclidean space. So instead we call an atomic space because it is constructed for the presentation and analysis of atomic models.
A Linear Coordinate
Let the -axis be centered on the proton inside an atom of hydrogen . When in its ground-state, the proton is at rest in any inertial frame of reference. And it is extremely stable. So it is a good place to start constructing a coordinate system. The location of on the -axis is specified by the numeric value . This position is called the spatial-origin. So by definition, this special hydrogen atom is always located at the spatial-origin. We use the same spatial-origin in other coordinate-systems to be discussed next. So the special atom is called to distinguish it from other hydrogen atoms. Both and use the same basis vector So we could use this -coordinate to parameterize the one dimensional space discussed earlier.
A Cartesian Plane
As descriptions are objectified, we stop referring directly to sensations. This is done partly by shifting the focus to particles that are larger than quarks. For example, we next use two atoms to define a two-dimensional space, the Cartesian plane.
Let us combine an atom of oxygen with the hydrogen atom shown above, to make a hydroxide anion, The description is again centered on the proton inside The -axis is defined in by sensation, but for more conventional details are required. So in this Cartesian coordinate system, the direction of the unit vector is chosen to align with the O–H chemical bond called , and ultimately fixed by the material presence of atomic oxygen. The -axis is also chosen to be orthogonal to The position of oxygen is then described by the coordinates and where notes the distance between hydrogen and oxygen atoms. The key detail about this arrangement is that it involves two atoms. So is measurable and can meet the definition for being a length. Indeed is observed1Computational Chemistry Comparison and Benchmark Database Edited by Russell D. Johnson III, National Institute of Standards and Technology, Standard Reference Database Number 101, Release 18, Department of Commerce USA, October 2016. to be 96.4 ± 0.1 (pm). This coordinate system uses the atom of hydrogen to furnish a descriptive context for the atom of oxygen, and also for any other atoms that may be included. So is functioning as a frame of reference.
A Three-Dimensional Cartesian Coordinate System
Next we use three atoms to make a three-dimensional Cartesian system. Let us combine another atom of hydrogen with the hydroxide anion to make a molecule of water. The -axis, -axis and spatial-origin are as before, but the -axis still needs to be established. We choose it to be orthogonal to both the and -axes, and in the same plane as the chemical bonds in water. There are two possible orientations, identified by The number is called the handedness of the coordinate system. The water bonds are called They make an angle of with each other. So for example, the position of the new hydrogen atom might be given by the coordinates The material presence of three atoms ensures that the three-dimensional framework is scientifically well-founded: Lengths and angles can actually be measured. In fact they are reported2Ibid. to be
Thus a physical three-dimensional coordinate system is established in principle. And, it may be extended to include other atoms just by making more measurements. Different atoms are assigned different coordinates, that algebraically represent different geometric positions. This water-based coordinate system is not very practical. But it demonstrates that we can finally put aside some concerns about Pauli’s exclusion principle. From now on, when considering a bundle of particles, we assume that Pauli’s principle is satisfied if they all have different Cartesian coordinates.
|1||Computational Chemistry Comparison and Benchmark Database Edited by Russell D. Johnson III, National Institute of Standards and Technology, Standard Reference Database Number 101, Release 18, Department of Commerce USA, October 2016.|