Operator

Let T^\hat{T} be an operator.

An operator is a function where

T^:V→V\begin{gather*} \hat{T}:V\rightarrow V \end{gather*}

where V are the Set of all possible Vector in a Hilbert space.

In quantum mechanics, all observed operators are linear which means

T^(a∣u⟩+b∣v⟩)=aT^(∣u⟩)+bT^(∣v⟩)\begin{gather*} \hat{T}(a\left\lvert u \right\rangle+b\left\lvert v \right\rangle)=a\hat{T}(\left\lvert u \right\rangle)+b\hat{T}(\left\lvert v \right\rangle) \end{gather*}

If we were to observe a non-linear operator, then that would cause wacky things.

also they can be represented as Matrix

T^∣ψ⟩=∑mψm′∣m⟩\begin{gather*} \hat{T}\left\lvert \psi \right\rangle=\sum_{m}\psi_m'\left\lvert m \right\rangle \end{gather*}

where

ψm′=∑nTnmψn\psi_{m}'=\sum_nT_{nm}\psi_n [ψ1′ψ2′⋮ψd′]=[T11T12⋯T1dT21T22⋯T2d⋮Td1Td2⋯Tdd][ψ1ψ2⋮ψd]\begin{gather*} \begin{bmatrix} \psi_1' \\ \psi_2' \\ \vdots \\ \psi_d' \end{bmatrix}= \begin{bmatrix} T_{11} & T_{12} & \cdots & T_{1d} \\ T_{21} & T_{22} & \cdots & T_{2d} \\ \vdots \\ T_{d1} & T_{d2} & \cdots & T_{dd} \end{bmatrix}\begin{bmatrix}\psi_1 \\ \psi_2 \\ \vdots \\\psi_d\end{bmatrix} \end{gather*}

where ∣m⟩\left\lvert m \right\rangle is the basis for ∣ψ⟩\left\lvert \psi \right\rangle

Operators must be square.

a∣v⟩∈V;a∈F,∣v⟩∈V\begin{gather*} a\left\lvert v \right\rangle\in V\quad ;\quad a\in \mathbb{F},\left\lvert v \right\rangle\in V \end{gather*}

See also: Continuous Operator