Linear Operator

Action of linear operators on each basis vector gives the action on any arbitrary vector.

Definition

A linear operator, $L$ , on $V$ is a linear transformation $L : V \to V$ .

In other words, a linear operator is a linear transformation that maps a vector space $V$ onto itself.

Special Operators

Assume $A$ , $B$ are linear. Let $| a ⟩ \in V$

$A = 0$ means $A | a ⟩ = 0 | a ⟩ = | 0 ⟩ = ‘ 0^{'}$
$A = B$ implies $A | a ⟩ = B | a ⟩$
$C = A + B$ means $C | a ⟩ = A | a ⟩ + B | a ⟩$
$D = A B$ means $D | a ⟩ = A B | a ⟩ = A (B | a ⟩)$
$A^{n} | a ⟩ = A \cdot A \dots A | a ⟩$
Identity: $I = I = E$ means $I | a ⟩ = | a ⟩$

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: $U (t) = \exp (- \frac{i H t}{ℏ})$

Representation of Operators

Suppose we have a basis $| i ⟩$ . $A | j ⟩ \in V$ , so for every $j$ we can expand $A | j ⟩$ in our basis: $A | j ⟩ = \sum_{i = 1}^{n} A_{j}^{i} | i ⟩$ . $A_{j}^{i}$ is a set of $n^{2}$ numbers completely characterize $A$ but in a basis-dependent way.

So if, $| α ⟩ = A | a ⟩ = \sum_{i = 0}^{n} α^{i} | i ⟩$ , $| a ⟩ = \sum_{i = 0}^{n} a^{i} | i ⟩$ .

$α^{i} = \sum_{j = 0}^{n} A_{j}^{i} a^{j} = A_{j}^{i} a^{j}$ (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)

Linear Operator

Definition

Special Operators

Functions of Operators

Example

Representation of Operators

Classes of Operators

Reitz Representation Theorem

Outer Products

Projection Operators