# 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)

## Projection Operators

Created: 2023-06-25 Sun 02:26

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