Linear Operator
Action of linear operators on each basis vector gives the action on any arbitrary vector.
Definition
A linear operator, \(L\), on \(\mathcal{V}\) is a linear transformation \(L:\mathcal{V}\to \mathcal{V}\).
In other words, a linear operator is a linear transformation that maps a vector space \(\mathcal{V}\) onto itself.
Special Operators
Assume \(A\), \(B\) are linear. Let \(|a\rangle\in\mathcal{V}\)
- \(A=0\) means \(A|a\rangle = 0|a\rangle=|0\rangle = `0'\)
- \(A=B\) implies \(A|a\rangle = B|a\rangle\)
- \(C=A+B\) means \(C|a\rangle = A|a\rangle + B|a\rangle\)
- \(D=AB\) means \(D|a\rangle = AB|a\rangle = A(B|a\rangle)\)
- \(A^n|a\rangle = A\cdot A\cdots A|a\rangle\)
- Identity: \(I=\mathbb{I}=E\) means \(I|a\rangle=|a\rangle\)
Knowing \(A^n\) allows us to define (almost) arbitrary functions of an operator from a power series, ’Functions’ Taylor series.
Functions of Operators
Example
\(e^A = \sum_{n=0}^\infty \frac{A^n}{n!}\), \(A^0=I\).
\(\sin A, \cos A, \sqrt{A}, \ln A, \det A\)
Must show convergence of series.
Example in QM: \(U(t) = \exp\left(-\frac{i\mathcal{H}t}{\hbar}\right)\)
Representation of Operators
Suppose we have a basis \({|i\rangle}\). \(A|j\rangle\in\mathcal{V}\), so for every \(j\) we can expand \(A|j\rangle\) in our basis: \(A|j\rangle=\sum_{i=1}^nA^i_j|i\rangle\). \({A_j^i}\) is a set of \(n^2\) numbers completely characterize \(A\) but in a basis-dependent way.
So if, \(|\alpha\rangle = A|a\rangle = \sum_{i=0}^n\alpha^i|i\rangle\), \(|a\rangle=\sum_{i=0}^n a^i|i\rangle\).
\(\alpha^i = \sum_{j=0}^nA_j^ia^j = A_j^ia^j\) (in Einstein notation). In the matrix representation, \(i\) is the row and \(j\) is the column in \(A_j^i\). (I.e. the top index is the row and the bottom is the column)