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## Weird dreams

I woke up today having a dream still in my head from the night, but it was a strange one. I was expanding out the Dirac notation representation of an operator in matrix form, but the symbols in the kets were elaborate pictures of Disney princesses that I was drawing with forestry scenery in the background, including little bears. At the point that I woke up from the dream, I noticed that I’d gotten the proportion of the bears wrong in one of the pictures, and they looked like they were ready to eat one of the princess characters.

## Guts

As a side effect of this weird dream I actually started thinking about matrix element representation of operators.

When forming the matrix element of an operator using Dirac notation the elements are of the form \( \bra{\textrm{row}} A \ket{\textrm{column}} \). I’ve gotten that mixed up a couple of times, so I thought it would be helpful to write this out explicitly for a \( 2 \times 2 \) operator representation for clarity.

To start, consider a change of basis for a single matrix element from basis \( \setlr{\ket{q}, \ket{r} } \), to basis \( \setlr{\ket{a}, \ket{b} } \)

\begin{equation}\label{eqn:operatorMatrixElement:20}

\begin{aligned}

\bra{q} A \ket{r}

&=

\braket{q}{a} \bra{a} A \ket{r}

+

\braket{q}{b} \bra{b} A \ket{r} \\

&=

\braket{q}{a} \bra{a} A \ket{a}\braket{a}{r}

+ \braket{q}{a} \bra{a} A \ket{b}\braket{b}{r} \\

&+ \braket{q}{b} \bra{b} A \ket{a}\braket{a}{r}

+ \braket{q}{b} \bra{b} A \ket{b}\braket{b}{r} \\

&=

\braket{q}{a}

\begin{bmatrix}

\bra{a} A \ket{a} & \bra{a} A \ket{b}

\end{bmatrix}

\begin{bmatrix}

\braket{a}{r} \\

\braket{b}{r}

\end{bmatrix}

+

\braket{q}{b}

\begin{bmatrix}

\bra{b} A \ket{a} & \bra{b} A \ket{b}

\end{bmatrix}

\begin{bmatrix}

\braket{a}{r} \\

\braket{b}{r}

\end{bmatrix} \\

&=

\begin{bmatrix}

\braket{q}{a} &

\braket{q}{b}

\end{bmatrix}

\begin{bmatrix}

\bra{a} A \ket{a} & \bra{a} A \ket{b} \\

\bra{b} A \ket{a} & \bra{b} A \ket{b}

\end{bmatrix}

\begin{bmatrix}

\braket{a}{r} \\

\braket{b}{r}

\end{bmatrix}.

\end{aligned}

\end{equation}

Suppose the matrix representation of \( \ket{q}, \ket{r} \) are respectively

\begin{equation}\label{eqn:operatorMatrixElement:40}

\begin{aligned}

\ket{q} &\sim

\begin{bmatrix}

\braket{a}{q} \\

\braket{b}{q} \\

\end{bmatrix} \\

\ket{r} &\sim

\begin{bmatrix}

\braket{a}{r} \\

\braket{b}{r} \\

\end{bmatrix} \\

\end{aligned},

\end{equation}

then

\begin{equation}\label{eqn:operatorMatrixElement:60}

\bra{q} \sim

{\begin{bmatrix}

\braket{a}{q} \\

\braket{b}{q} \\

\end{bmatrix}}^\dagger

=

\begin{bmatrix}

\braket{q}{a} &

\braket{q}{b}

\end{bmatrix}.

\end{equation}

The matrix element is then

\begin{equation}\label{eqn:operatorMatrixElement:80}

\bra{q} A \ket{r}

\sim

\bra{q}

\begin{bmatrix}

\bra{a} A \ket{a} & \bra{a} A \ket{b} \\

\bra{b} A \ket{a} & \bra{b} A \ket{b}

\end{bmatrix}

\ket{r},

\end{equation}

and the corresponding matrix representation of the operator is

\begin{equation}\label{eqn:operatorMatrixElement:100}

A \sim

\begin{bmatrix}

\bra{a} A \ket{a} & \bra{a} A \ket{b} \\

\bra{b} A \ket{a} & \bra{b} A \ket{b}

\end{bmatrix}.

\end{equation}

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