$ \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\abs}[2][]{\left\lvert#2\right\rvert_{\text{#1}}} \newcommand{\ket}[1]{\left\lvert#1 \right.\rangle} \newcommand{\bra}[1]{\langle\left. #1\right\rvert} \newcommand{\braket}[1]{\langle #1 \rangle} \newcommand{\dd}{\text{d}} \newcommand{\dv}[2]{\frac{\dd #1}{\dd #2}} \newcommand{\pdv}[2]{\frac{\partial}{\partial #1}} $
FullPage
Overview
Notations
Concepts
An operator is a mathematical object that acts on a ket and turns it into a new ket. See Postuate 2.

Notations

Operators tend to be matrices. Decomposed into its eigenvectors, it is written $$\begin{align*} A=\sum_{i}\alpha_i|a_i\rangle \langle a_i|, \end{align*}$$ and its terms are $$\begin{align*} A_{ij}=\braket{\phi_i| A |\phi_j}, \end{align*}$$ where $\braket{\phi_i|A|\phi_j}$ is the probability of measuring the eigenstate of $\bra{\phi_i}$ on $A\ket{\phi_j}$, often given experimentally (?).

Concepts

The eigenvectors of an operator forms a basis for the Hilbert space, and so by definition of basis, every quantum state can be represented as a superposition of the eigenvectors of the operator.
The eigenvalues of the operator is the possible results of the measurement, where the associate eigenvector is the quantum state that has the measurement of the eigenvalue.
Hilbert space is defined by operators.
We sometimes (when? I think for measureable observables?) require that an operator acts correctly on both a bra and a ket. That is, if $A\ket{\alpha}=\ket{\beta}$, then we would like an associated operator such that $\bra{\alpha}A^{\dagger}=\bra{\beta}$. This is the Hermitian conjungate.
Operators can be writtin in terms of its eigenvector basis. This is the spectral decomposition of an operator.
An operator is a mathematical object that acts on a ket and turns it into a new ket. See Postuate 2.

Notations

Operators tend to be matrices. Decomposed into its eigenvectors, it is written $$\begin{align*} A=\sum_{i}\alpha_i|a_i\rangle \langle a_i|, \end{align*}$$ and its terms are $$\begin{align*} A_{ij}=\braket{\phi_i| A |\phi_j}, \end{align*}$$ where $\braket{\phi_i|A|\phi_j}$ is the probability of measuring the eigenstate of $\bra{\phi_i}$ on $A\ket{\phi_j}$, often given experimentally (?).

Concepts

The eigenvectors of an operator forms a basis for the Hilbert space, and so by definition of basis, every quantum state can be represented as a superposition of the eigenvectors of the operator.
The eigenvalues of the operator is the possible results of the measurement, where the associate eigenvector is the quantum state that has the measurement of the eigenvalue.
Hilbert space is defined by operators.
We sometimes (when? I think for measureable observables?) require that an operator acts correctly on both a bra and a ket. That is, if $A\ket{\alpha}=\ket{\beta}$, then we would like an associated operator such that $\bra{\alpha}A^{\dagger}=\bra{\beta}$. This is the Hermitian conjungate.
Operators can be writtin in terms of its eigenvector basis. This is the spectral decomposition of an operator.
FullPage
Overview
Notations
Concepts
FullPage
Overview
Notations
Concepts
An operator is a mathematical object that acts on a ket and turns it into a new ket. See Postuate 2.

Notations

Operators tend to be matrices. Decomposed into its eigenvectors, it is written $$\begin{align*} A=\sum_{i}\alpha_i|a_i\rangle \langle a_i|, \end{align*}$$ and its terms are $$\begin{align*} A_{ij}=\braket{\phi_i| A |\phi_j}, \end{align*}$$ where $\braket{\phi_i|A|\phi_j}$ is the probability of measuring the eigenstate of $\bra{\phi_i}$ on $A\ket{\phi_j}$, often given experimentally (?).

Concepts

The eigenvectors of an operator forms a basis for the Hilbert space, and so by definition of basis, every quantum state can be represented as a superposition of the eigenvectors of the operator.
The eigenvalues of the operator is the possible results of the measurement, where the associate eigenvector is the quantum state that has the measurement of the eigenvalue.
Hilbert space is defined by operators.
We sometimes (when? I think for measureable observables?) require that an operator acts correctly on both a bra and a ket. That is, if $A\ket{\alpha}=\ket{\beta}$, then we would like an associated operator such that $\bra{\alpha}A^{\dagger}=\bra{\beta}$. This is the Hermitian conjungate.
Operators can be writtin in terms of its eigenvector basis. This is the spectral decomposition of an operator.
An operator is a mathematical object that acts on a ket and turns it into a new ket. See Postuate 2.

Notations

Operators tend to be matrices. Decomposed into its eigenvectors, it is written $$\begin{align*} A=\sum_{i}\alpha_i|a_i\rangle \langle a_i|, \end{align*}$$ and its terms are $$\begin{align*} A_{ij}=\braket{\phi_i| A |\phi_j}, \end{align*}$$ where $\braket{\phi_i|A|\phi_j}$ is the probability of measuring the eigenstate of $\bra{\phi_i}$ on $A\ket{\phi_j}$, often given experimentally (?).

Concepts

The eigenvectors of an operator forms a basis for the Hilbert space, and so by definition of basis, every quantum state can be represented as a superposition of the eigenvectors of the operator.
The eigenvalues of the operator is the possible results of the measurement, where the associate eigenvector is the quantum state that has the measurement of the eigenvalue.
Hilbert space is defined by operators.
We sometimes (when? I think for measureable observables?) require that an operator acts correctly on both a bra and a ket. That is, if $A\ket{\alpha}=\ket{\beta}$, then we would like an associated operator such that $\bra{\alpha}A^{\dagger}=\bra{\beta}$. This is the Hermitian conjungate.
Operators can be writtin in terms of its eigenvector basis. This is the spectral decomposition of an operator.
FullPage
Overview
Notations
Concepts