Notes I

Linear operators

A finite-dimensional vector space with \(\text{dim}(V)=n\) is a vector space \(V\) for which there exists some finite set of vectors \(U=\{u_{1},u_{2},\ldots,u_{n}\}\) that spans the vector space, or \(\text{span}(U)=V\). An important notation to understand in quantum mechanics is the braket notation. With this notation, we denote a vector as a so-called “ket”, represented as \(|v_{i}\rangle\in V\), where \(V\) is a vector space. The “bra” portion of the braket notation is represented as \(\langle v_{i}|\), and we will explain its significance later.

In quantum mechanics, we commonly refer to matrices as “linear operators”. A linear operator is a function \(A: V \rightarrow W\) from a vector space \(V\) to a vector space \(W\) that is linear in its inputs, namely

where \(A\) is a linear operator. However, to simplify notation, instead of tediously representing an action of some linear operator \(A\) on a vector with \(A(|v_{i}\rangle)\), we can write it as \(A|v_{i}\rangle\). There are two trivial linear operators. For some vector space \(V\), \(I_{V}\) is the identity linear operator, and \(I_{V}|v_{i}\rangle=|v_{i}\rangle\) for all vectors \(|v_{i}\rangle\in V\). Also, \(0\) is the zero vector, and \(0|v_{i}\rangle=0\) for all vectors \(|v_{i}\rangle\in V\). A linear operator \(A\) is equivalent to a matrix representation. To see how this fits into the bigger picture, if we have some vector space \(V_{m}\) and another vector space \(V_{n}\), then the \(m\times n\) linear operator \(A\) with entries \(A_{ij}\) takes a vector from \(V_{m}\) and maps it to a vector in \(V_{n}\). In other words,

(1 June, 2023)