Quirky Quantum Concepts by Eric L. Michelsen

Author:Eric L. Michelsen
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

4.3.1 Transforming Vector Components to a New Basis

We describe a general transformation matrix in two complementary ways. We start with an N-dimensional vector expressed in the orthonormal basis b i :

(4.6)

It is important to note that the previous is a vector equation and is true in any basis. The components of in the b-basis are a i , i = 1,…,N. Furthermore, the a i are given by:

How would we convert the components a i of into a new basis, n j ? That is:

The inner product on the right, for each c j , can be evaluated in any basis we choose. In particular, it can be evaluated in our old basis, the b-basis. In matrix notation, we have:

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