The Gram-Schmidt process is a mathematical technique used in linear algebra to convert a set of linearly independent vectors into a set of orthonormal vectors. This process is commonly used in applications involving vector spaces, such as in quantum mechanics.
In quantum mechanics, it is often necessary to work with orthonormal basis vectors to accurately describe the state of a quantum system. The Gram-Schmidt process allows for the creation of such basis vectors from a set of linearly independent vectors, making it a crucial tool in quantum mechanics.
Sure! In the 2nd edition of QM Sankar, example 1.3.2 walks through the Gram-Schmidt process using a set of three linearly independent vectors in three-dimensional space. The process involves finding the orthogonal projections of each vector onto the span of the previous vectors and then normalizing them to create the orthonormal basis vectors.
In example 1.3.2, the Gram-Schmidt process is used to create an orthonormal basis for a three-dimensional vector space. This basis can then be used for various calculations and applications in quantum mechanics.
One limitation of the Gram-Schmidt process is that it can only be applied to a set of linearly independent vectors. If the initial set of vectors is not linearly independent, the process will fail to produce an orthonormal basis. Additionally, the process can be computationally intensive for large sets of vectors.