Vector model of the atom

In physics, in particular quantum mechanics, the vector model of the atom is a model of the atom in terms of angular momentum. It can be considered as the extension of the Rutherford-Bohr-Sommerfeld atom model to multi-electron atoms. In physics, in particular quantum mechanics, the vector model of the atom is a model of the atom in terms of angular momentum. It can be considered as the extension of the Rutherford-Bohr-Sommerfeld atom model to multi-electron atoms. The model is a convenient representation of the angular momenta of the electrons in the atom. Angular momentum is always split into orbital L, spin S and total J: Given that in quantum mechanics, angular momentum is quantized and there is an uncertainty relation for the components of each vector, the representation turns out to be quite simple (although the background mathematics is quite complex). Geometrically it is a discrete set of right-circular cones, without the circular base, in which the axes of all the cones are lined up onto a common axis, conventionally the z-axis for three-dimensional Cartesian coordinates. Following is the background to this construction. The commutator implies that for each of L, S, and J, only one component of any angular momentum vector can be measured at any instant of time; at the same the other two are indeterminate. The commutator of any two angular momentum operators (corresponding to component directions) is non-zero. Following is a summary of the relevant mathematics in constructing the vector model. The commutation relations are (using the Einstein summation convention):