PX262 - G3 - matrix representation

taking a dynamical variable, $\hat{Q}$ , with an eigenstate, $ψ$ , with eigenvalue, $q :$

\hat{Q} ψ = q ψ

expanding $ψ$ in terms of some complete orthonormal set of functions, $ϕ_{n}$ which may not necessarily be eigenfunctions of $\hat{Q} :$

ψ = \sum_{n} a_{n} ϕ_{n}

applying $\hat{Q} :$

\sum_{n} a_{n} \hat{Q} ϕ_{n} = q \sum_{n} a_{n} ϕ_{n}

multiplying it by the $ϕ_{m}^{*}$ and integrating over all space:

\begin{matrix} \int \sum a_{n} ϕ_{m}^{*} \hat{Q} ϕ_{n} d τ = q \int \sum_{n} a_{n} ϕ_{m}^{*} ϕ_{n} d τ = q a_{m} \\ ∴ \sum_{n} Q_{m n} a_{n} = q a_{m} \end{matrix}

where,
$$Q_{mn}= \int \phi_{m}^{*} \hat Q \phi_{n},d\tau$$

expressing as matrices:

(\begin{matrix} Q_{11} & Q_{12} & \dots \\ Q_{21} & Q_{22} & \dots \\ ⋮ & ⋮ & ⋱ \end{matrix}) (\begin{matrix} a_{1} \\ a_{2} \\ \dots \end{matrix}) = q (\begin{matrix} a_{1} \\ a_{2} \\ \dots \end{matrix})

matrices still obey the same commutation relations
replacing complex conjugate by hermitian conjugate, $A^{†} = A^{T *}$
matrix representation is helpful for:
- working with spin
- numerical calculations