Adjoint

An adjoint is basically a Conjugate transpose but for an operator (whereas a conjugate transpose can only be applied to a matrix)

Given operator $\hat{A}$ , it’s adjoint is the operator $\hat{A}^+$ such that $\forall\left\lvert i \right\rangle,\left\lvert j \right\rangle$

\begin{gather*} \left\langle i \right\rvert\hat{A}^+\left\lvert j \right\rangle=(\left\langle j \right\rvert\hat{A}\left\lvert i \right\rangle)^*\\ (\hat{A}^+)_{ij}=\hat{A}^*_{ji} \end{gather*}

Where, for any operator $\hat{O}$ , a matrix Element $\left\langle i \right\rvert\hat{O}\left\lvert j \right\rangle$ is just the $(i,j)$ -th element of the matrix representing $\hat{O}$ in basis $\{\left\lvert i \right\rangle\}$

Note that $\left\lvert i \right\rangle$ and $\left\lvert j \right\rangle$ are arbitrary Basis State

where $A^*$ is the complex conjugation of $A$

properties:

\begin{gather*} (\hat{A}^+)^+=\hat{A}\\ (\hat{A}+\hat{B})^+=\hat{A}^++\hat{B}^+\\ (\hat{A}\hat{B})^+=\hat{B}^+\hat{A}^+ \end{gather*}

Example:

Consider operator

\begin{gather*} \hat{A}=\left\lvert 1 \right\rangle\left\langle 2 \right\rvert+2i\left\lvert 2 \right\rangle\left\langle 1 \right\rvert+3\left\lvert 2 \right\rangle\left\langle 2 \right\rvert\\ \Rightarrow\text{adjoint of $\hat{A}$}=\left\lvert 2 \right\rangle\left\langle 1 \right\rvert-2i\left\lvert 1 \right\rangle\left\langle 2 \right\rvert+3\left\lvert 2 \right\rangle\left\langle 2 \right\rvert \end{gather*}

So w.r.t. to basis $\{\left\lvert 1 \right\rangle,\left\lvert 2 \right\rangle\}$ , $\hat{A}$ is

\begin{gather*} \hat{A}=\begin{pmatrix} 0 & 1 \\ 2i & 3 \end{pmatrix} \end{gather*}

The conjugate transpose of $\hat{A}=\hat{A}^+$ which is

\begin{gather*} \hat{A}^+=\begin{pmatrix} 0 & -2i \\ 1 & 3 \end{pmatrix} \end{gather*}

Which, once we pick an orthonormal basis, the matrix of $\hat{A}^+$ equals the conjugate transpose of the matrix of $\hat{A}$

\begin{gather*} \hat{A}^+=\text{adjoint of $\hat{A}$} \end{gather*}