# Matrix Representation of Operators and Wavefunctions

## Vector Representation of an eigenstate

For a set of vectors $$\{|1 \rangle, |2 \rangle, ... | \infty \rangle \}$$ that spans the space we are interested in, the arbitrary eigenstate can be decomposed

$| \psi \rangle = \sum_i^n c_i | i \rangle = \begin{pmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{pmatrix} \label{1A}$

The $$\{|1 \rangle, |2 \rangle, ... | \infty \rangle \}$$ constitutes a basis (one of many possible) for the space.

Most of the operators we are discussed are linear so

$\hat{A} | \psi \rangle = \hat{A} \left( \sum_i^n c_i | i \rangle \right) = \sum_i^n c_i \hat{A} | i \rangle \label{2A}$

## Matrix Representation of an Operator

Operators can be expressed as matrices that "operator" on the eigenvector discussed above

$\hat{A} | i \rangle = \sum_i^n A_{ij} | i \rangle \label{3A}$

The number $$A_{ij}$$ is the $$ij^{th}$$ matrix element of $$A$$ in the basis select.

Hermitian Operators

Hermitian operators are operators that satisfy the general formula

$\langle \phi_i | \hat{A} | \phi_j \rangle = \langle \phi_j | \hat{A} | \phi_i \rangle \label{Herm1}$

If that condition is met, then $$\hat{A}$$ is a Hermitian operator. For any operator that generates a real eigenvalue (e.g., observables), then that operator is Hermitian. The Hamiltonian $$\hat{H}$$ meets the condition and a Hermitian operator. Equation \ref{Herm1} can be rewriten as

$A_{ij} =A_{ji}$

where

$A_{ij} = \langle \phi_i | \hat{A} | \phi_j \rangle$

and

$A_{ji} = \langle \phi_j | \hat{A} | \phi_i \rangle$

Therefore, when applied to the Hamiltonian operator

$\boxed{H_{ij}^* =H_{ji}.}$

## Multiplication

We can define an inner product (dot product) of two eigenstates $$| \phi_1 \rangle$$ and $$| \phi_2\ \rangle$$

$\langle \phi_1 | \phi_2 \rangle = \sum_{i=1}^n c_i^* c_j \label{4A}$

which looks like this in vector representations

$\langle \phi_1 | \phi_2 \rangle = ( c_1^* \; c_2^* \; ... c_n^* ) \begin{pmatrix} c_1 \\ c_2 \\ ... \\ c_n \end{pmatrix} \label{5A}$

This form emphasizes the dot product nature of the inner product multiplication.

From equation $$\ref{5A}$$, we can express the bra form of the eigenstate as

$\langle \phi | = \sum_i^n c_i^* \langle i | \label{6A}$

## Completeness Relation

For vectors $$|i \rangle$$ forming an orthonormal basis $$\langle i | j \rangle = \delta_{ij}$$ for all space then

$\sum_i^n | i \rangle \langle j| = 1 \label{7A}$

Those terms in this sum are outer products and are matrices (remember inner products are scalars).

## Diagonal Representation of an operator

$\hat{A} = \sum_i ^n \lambda_i | i \rangle \langle i| \label{8A}$

This is a matrix that has non-zero element everywhere except the diagonal. E.g.,

$\begin{pmatrix} 4 & 0 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -4 \end{pmatrix}$