A Space for Atomic Models of Molecules

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 $\mathbb{Q}$, to a new, almost-Euclidean space called $\mathbb{E}$. In $\mathbb{E}$, particles are spinning. Their electric and magnetic axes rotate around the $z$-axis. And in $\mathbb{E}$ we can construct a Cartesian coordinate system as follows.

The algebraic vector spaceXlink.png $\mathbb{E}$ is a set of position vectors $\, \overline{r}$ for some collection of atoms $\, \mathbf{A}$1, $\mathbf{A}$2, $\mathbf{A}$3$\mathbf{A}$N. This mathematical construction is generically written as

$\mathbb{E} = \left\{ \overline{r}^{ 1 }, \ \overline{r}^{2}, \ \overline{r}^{3} \ldots \ \overline{r}^{\, i} \ldots \ \overline{r}^{\, N} \right\}$

The following basis vectorsXlink.png are used to make general descriptions; the axis of the abscissa is directed by $\hat{x} \equiv (1, 0, 0)$, the axis of the ordinate from $\hat{y} \equiv (0, 1, 0)$ and the polar-axis by $\hat{z} \equiv (0, 0, 1)$. Then, any position vector in $\mathbb{E}$ can be expressed in terms of $x$, $y$ and $z$, its Cartesian coordinates as

$\overline{r} = x \hat{x} + y \hat{y} + z \hat{z}$

The new space $\mathbb{E}$ is closer to the ordinary space of everyday experience than $\mathbb{Q}$. But some details about continuity and metrics have to be discussed before we can say that $\mathbb{E}$ is Euclidean.

A one-dimensional coordinate system centered on an atom of hydrogen.
A one-dimensional coordinate system centered on an atom of hydrogen.

A Linear Coordinate

Let the $z$-coordinate be centered on the proton inside an atom of hydrogen, $\mathrm{H}$. 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 $\mathrm{H}$ on the $z$-axis is specified by the numeric value $z=0$. 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 H0 to distinguish it from other hydrogen atoms. Both $\mathbb{Q}$ and $\mathbb{E}$ use the same basis vector $\hat{z}=(0,0,1)$. So we could use this $z$-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.

A two-dimensional Cartesian coordinate system defined by the two atoms that form a hydroxide ion. The red ball represents oxygen.
A two-dimensional Cartesian coordinate system defined by the two atoms that form a hydroxide ion. The red ball represents oxygen.

Let us combine an atom of oxygen with the hydrogen atom $\mathrm{H}_{0}$ shown above, to make a hydroxide anion, $\mathrm{OH}^{–}$. The description is again centered on the proton inside $\mathrm{H}_{0}$. The $x$-axis is defined in principle by sensation, but in $\mathbb{E}$ more detail is required. So in this Cartesian coordinate system, the direction of the unit vector $\hat{x}=(1,0,0) \,$ is chosen to align with the O–H chemical bond called $\mathbb{B} \sf{(2)}$, and ultimately fixed by the material presence of atomic oxygen. The $x$-axis is also chosen to be orthogonal to $\hat{z}$. The position of oxygen is then described by the coordinates $z = 0$ and $x = \ell$ where $\ell$ notes the distance between hydrogen and oxygen atoms. The key detail about this arrangement is that it involves two atoms. So $\ell$ can meet the definition for being a length, and be measured. Indeed $\ell$ is observed1 to be $96.4 ± 0.1 \ \sf{(pm)}$. This coordinate system uses the atom of hydrogen to furnish a descriptive context for the atom of oxygen, and also any other atoms that may be included. So $\mathrm{H}_{\sf{0}}$ 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 $\mathrm{OH}^{–}$ to make ${\mathrm{H}} _{\sf{2}} {\mathrm{O}}$, a molecule of water. The $z$-axis, $x$-axis and spatial-origin are as before, but the $y$-axis still needs to be established. We choose it to be orthogonal to both the $x$ and $z$-axes, and in the same plane as the chemical bonds in water. There are two possible orientations, identified by $\varpi = ±1$. The number $\varpi$ is called the parity of the coordinate system. The bonds in water are called $\mathbb{B} \sf{(aqueous)}$. They make an angle of $\vartheta$ with each other. So for example, the position of the new hydrogen atom is given by the coordinates

A three-dimensional Cartesian coordinate system defined by the three atoms that make a molecule of water.
A three-dimensional Cartesian coordinate system defined by the three atoms that make a molecule of water.

$\begin{align} \overline{r} ^{\, \mathrm{H}} = \ell \left( 1-\cos\vartheta , \ \varpi\sin\vartheta, \ 0 \right) \end{align}$

The material presence of three atoms ensures that the three-dimensional framework is scientifically well-founded. Lengths and angles can actually be measured. Indeed they are reported2 to be

$\ell = 95.8 ± 0.1 \ \sf{(pm)}$ and

$\vartheta=104.4776 ± 0.0019 \ \sf{(degrees)}$

Thus a physical three-dimensional coordinate system is established in principle. And, it may be extended indefinitely 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.

Right.png Next step: Euclidean space.
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