Quaternions Sample Clauses

Quaternions. Xxx X. X. Xxxxxxxx introduced quaternions as a generalization of complex numbers. Walking with his wife along the Royal Canal in Dublin, the defining equations of quaternions i2 = j2 = k2 = ijk = −1 (1.46) came to him in a burst of inspiration. In his excitement he carved them into stone at the Xxxxxxxx Xxxxxx [00]. The story does not elaborate on what his wife was doing meanwhile. One of the consequences of the defining equation (1.46) is that the basic quaternions i, j, k do not commute. There are different representations of the algebraic structure of quaternions, the most common being in terms of xxx Xxxxx matrices (1.2) (see below). Xxxxxxxx spent much of the rest of his life trying to realize the useful- ness and beauty of complex numbers in his quaternions. There are strong reasons why that cannot work17, and thus he was not very successful. So why do we want to use quaternions? For us, the main reason, perhaps, is bookkeeping. The Hamiltonian in a basis which is a direct product of a real space state vector and a two dimensional spin state vector, has a natural decomposition into blocks of 2 × 2 matrices, which can then be thought of as a single quaternion. Instead of taking the Hamiltonian to be a 2N × 2N complex matrix, one can consider it to be an N × N matrix of quaternions. What does one gain by doing this? Mainly an economic way of expressing symmetry relations and performing calculations18.