Operations in Bra-Ket Notation #
This article describes how to handle bra-ket notation that appears in quantum mechanics. It focuses on the possible operations rather than providing a detailed explanation of the notation itself.
1. Representation of Functions #
$$ \begin{align} \psi(\mathbf{r}) &= \langle\mathbf{r}|\psi\rangle \label{e1}\\ \psi(\mathbf{p}) &= \langle\mathbf{p}|\psi\rangle \label{e2} \end{align} $$
$|\psi\rangle$: Description of a state without using a specific basis1
$\psi(\mathbf{r})$: State description in position representation
$\psi(\mathbf{p})$: State description in momentum representation
2. Hermitian Conjugate #
The Hermitian conjugate of an operator $\hat{A}$ is expressed as $\hat{A}^{\dagger}$, using $\dagger$ to denote Hermitian conjugation.
The complex conjugate of $\hat{A}$ is expressed as $\hat{A}^{\ast}$, using $\ ^\ast$ to denote complex conjugation.
Applying Hermitian conjugation affects operators, bras, and kets as follows:
$$ \begin{align} {|{\psi}\rangle}^{\dagger} = \langle{\psi}| \label{e3}\\ {\langle{\psi}|}^{\dagger} = |{\psi}\rangle \label{e4} \end{align} $$
$$ \begin{align} \label{e5} \bigl(\hat{A}^{\dagger}\bigr)^{\dagger} = \hat{A} \end{align} $$
$$ \begin{align} \label{e6} \bigl(\hat{B}\hat{A}\bigr)^{\dagger} = \hat{A}^{\dagger}\hat{B}^{\dagger} \end{align} $$
$$ \begin{align} \label{e7} \bigl(\hat{A}|\psi\rangle\bigr)^{\dagger} = \langle{\psi}|\hat{A}^{\dagger} \end{align} $$
$$ \begin{align} \label{e8} \bigl(c_1 | \psi_1 \rangle + c_2 | \psi_2 \rangle\bigr)^{\dagger} = c_1^{\ast}\langle\psi_1| + c_2^{\ast}\langle\psi_2| \end{align} $$
$$ \begin{align} \label{e9} \bigr(|\psi\rangle\langle\phi|\bigr)^{\dagger} = |\phi\rangle\langle\psi| \end{align} $$
$$ \begin{align} \label{e10} \bigl(\langle\phi|\hat{A}|\psi\rangle\bigr)^{\dagger} &= |\psi\rangle^{\dagger} \hat{A}^{\dagger}\langle\phi| ^{\dagger} = \langle\psi|\hat{A}^{\dagger}|\phi\rangle \end{align} $$
$$ \begin{align} \label{e11} \bigl( \langle\phi|\psi\rangle \bigr)^{\dagger} = \langle\psi|\phi\rangle \end{align} $$
$c_1, c_2$: Scalars
If the inverse operator $\hat{A}^{-1}$ of $\hat{A}$ exists, the following relation holds:
$$ \begin{align} \Bigl(\hat{A}^{-1}\Bigr)^{\dagger} = \Bigl(\hat{A}^{\dagger}\Bigr)^{-1} \end{align} $$
3. Insertion of the Identity Operator #
If the states $|\phi\rangle$ form a complete set (or as a condition for completeness), the identity operator $\hat{I}$ can be expressed as:
$$ \begin{align} \hat{I} = \int d\phi |\phi\rangle\langle\phi| \end{align} $$
The identity operator does not affect operations, akin to the multiplicative identity “1,” and can be inserted anywhere in an expression.
Considering a set of states $|x\rangle$ that form a complete basis, for instance, when examining the inner product of $\phi$ and $\psi$, various transformations are possible as shown below:
$$ \begin{align} \langle\phi|\hat{A}|\psi\rangle &= \langle\phi|\hat{A}\hat{I}|\psi\rangle \\ &=\langle\phi|\hat{A}\left(\int dx |x\rangle\langle x| \right) |\psi\rangle \\ &=\int dx \langle\phi|\hat{A}|x\rangle \langle x|\psi\rangle \\ \end{align} $$
Similarly, it can also be added before $\hat{A}$ as:
$$ \begin{align} \langle\phi|\hat{A}|\psi\rangle &= \langle\phi|\hat{I}\hat{A}\hat{I}|\psi\rangle \\ &=\langle\phi|\left(\int dx’ |x’\rangle\langle x’| \right) \hat{A}\left(\int dx |x\rangle\langle x| \right) |\psi\rangle \\ &=\int dx’ \int dx \langle\phi|x’\rangle \langle x’|\hat{A}|x\rangle \langle x|\psi\rangle \end{align} $$
If $x$ represents position, $\langle x|\psi\rangle = \psi(x)$ represents the position representation of the state $|\psi\rangle$, so:
$$ \begin{align} \langle\phi|\hat{A}|\psi\rangle \equiv \int dx’ \int dx \phi^{*}(x’) A(x’,x) \psi(x) \end{align} $$
where $A(x’,x) \equiv \langle x’|\hat{A}|x\rangle$ has been defined.