- to assess a two-body mechanical system in an isotropic three-dimensional space requires that both bodies be at least as large as atoms
- but if we make extra assumptions about angular momenta, and limit what sort of interactions are allowed, then we can limit considerations to just a two-dimensional problem
- we can make two-dimensional models of electrons and protons by associating a magnetic field with the proton
- so the smallest two-body, two-dimensional problem we can consider in an isotropic space is an electron orbiting a proton, i.e. the Bohr model

Bohr Balmer

Ajat basket, Penan people. Borneo 20th century, 20 (cm) diameter by 35 (cm) height. Hornbill motif. From the Teo Family collection, Kuching. Photograph by D Dunlop. |

One common way of dealing with this issue to to assume that a sub-atomic particle has been absorbed by an atom that *does* have a well-defined position. Then both particles are supposedly in the same place. Another possibility is to conjecture additional fields to align a particle's orientation. Such presumptions are codified in various three-dimensional arrangements that assign quarks to sub-orbital events by convention. These designs are called **sub-atomic** particle models. For example, here are some ways of representing protons and electrons in a three-dimensional space.

Electron |

$\Large{ k }$ | $\large{ \delta _{\hat{m}} }$ | $\large{ \delta _{\hat{e}} }$ | $\large{ \delta _{\theta} }$ | $\large{ \sf{P}_{\it{k}} }$ |

1 | +1 | 0 | +1 | $\,$ |

2 | 0 | -1 | +1 | $\bar{\sf{u}} \ \ \bar{\sf{b}}\sf{t} \ \ \mathrm{2}\bar{\sf{g}}$ |

3 | -1 | 0 | +1 | $\,$ |

4 | 0 | +1 | +1 | $\bar{\sf{u}} \ \ \bar{\sf{s}}\sf{c} \ \ \mathrm{2}\sf{e}$ |

5 | +1 | 0 | -1 | $\,$ |

6 | 0 | -1 | -1 | $\bar{\sf{u}} \ \ \bar{\sf{b}}\sf{t} \ \ \mathrm{2}\bar{\sf{g}}$ |

7 | -1 | 0 | -1 | $\,$ |

8 | 0 | +1 | -1 | $\bar{\sf{u}} \ \ \bar{\sf{s}}\sf{c} \ \ \mathrm{2}\sf{e}$ |

Proton |

$\Large{ k }$ | $\large{ \delta _{\hat{m}} }$ | $\large{ \delta _{\hat{e}} }$ | $\large{ \delta _{\theta} }$ | $\large{ \sf{P}_{\it{k}} }$ |

1 | +1 | 0 | +1 | $\sf{d} \ \ \bar{\sf{t}} \sf{b}$ |

2 | 0 | -1 | +1 | $\,$ |

3 | -1 | 0 | +1 | $\sf{d} \ \ \bar{\sf{t}} \sf{b}$ |

4 | 0 | +1 | +1 | $\,$ |

5 | +1 | 0 | -1 | $\sf{d} \ \ \bar{\sf{t}} \sf{b}$ |

6 | 0 | -1 | -1 | $\,$ |

7 | -1 | 0 | -1 | $\sf{d} \ \ \bar{\sf{t}} \sf{b}$ |

8 | 0 | +1 | -1 | $\,$ |

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Bohr Model of Hydrogen |