% Lecture notes for Physical Mathematics I
% Ewan Stewart
% KAIST
% 2012/4/13

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\begin{document}

\section{Bases and components}

Hilbert spaces can be finite, countably infinite, or uncountably infinite dimensional.
In the first two cases there will be a discrete set of basis vectors, while in the latter case the basis vectors are usually described in terms of continuous parameters.

\subsection{Discrete bases}

A set of \keyword{basis vectors} $\ket\alpha$ is chosen to be \keyword{orthonormal}
\begin{equation}
\braket\alpha\beta = \delta_{\alpha\beta}
\end{equation}
and \keyword{complete}
\begin{equation}
\sum_\alpha \ket\alpha \bra\alpha = 1
\end{equation}
Then an arbitrary vector $\ket\phi$ can be expressed in \keyword{components} as
\begin{equation}
\ket\phi = \sum_\alpha \phi_\alpha \ket\alpha
\quad \textrm{where} \quad \phi_\alpha = \braket\alpha\phi
\end{equation}
Similarly for covectors
\begin{equation}
\bra\phi = \sum_\alpha \phi_\alpha^* \bra\alpha
\quad \textrm{where} \quad \phi_\alpha^* = \braket\phi\alpha
\end{equation}
and linear operators
\begin{equation}
A = \sum_{\alpha,\beta} A_{\alpha\beta} \ket\alpha \bra\beta
\quad \textrm{where} \quad A_{\alpha\beta} = \bra\alpha A \ket\beta
\end{equation}
and their Hermitian conjugates\footnote{Note that $A^\dagger_{\alpha\beta} \equiv (A^\dagger)_{\alpha\beta} \neq (A_{\alpha\beta})^\dagger = (A_{\alpha\beta})^* \equiv A_{\alpha\beta}^*$.}
\begin{equation}
A^\dagger = \sum_{\alpha,\beta} A^\dagger_{\alpha\beta} \ket\alpha \bra\beta
\quad \textrm{where} \quad A^\dagger_{\alpha\beta} = A_{\beta\alpha}^*
\end{equation}
The contraction of a covector with a vector can be expressed as
\begin{equation}
\braket\phi\psi = \sum_\alpha \phi_\alpha^* \psi_\alpha
\end{equation}
and a linear operator acting on a vector as
\begin{equation}
A \ket\phi = \sum_{\alpha,\beta} A_{\alpha\beta} \phi_\beta \ket\alpha
\end{equation}

\subsection{Continuous bases}

When the basis vector label $\alpha$ is a continuous variable, orthonormality becomes
\begin{equation}
\braket\alpha\beta = \frac{1}{\fn{g}{\beta}} \fn{\delta}{\alpha,\beta}
\end{equation}
and completeness becomes
\begin{equation}
\int \d\alpha \fn{g}{\alpha} \ket\alpha \bra\alpha = 1
\end{equation}
where $\fn{g}{\alpha} > 0$ corresponds to the freedom to reparameterize the label $\alpha$.
Usually we take $\fn{g}{\alpha} = 1$.
The delta function $\fn{\delta}{\alpha,\beta}$ is defined in Section~\ref{delta}.
An arbitrary vector, covector, linear operator or contraction can be expressed in components as
\begin{equation}
\ket\phi = \int \d\alpha \fn{g}{\alpha} \fn{\phi}{\alpha} \ket\alpha
\quad \textrm{where} \quad \fn{\phi}{\alpha} = \braket\alpha\phi
\end{equation}
\begin{equation}
\bra\phi = \int \d\alpha \fn{g}{\alpha} \fn{\phi^*}{\alpha} \bra\alpha
\quad \textrm{where} \quad \fn{\phi^*}{\alpha} = \braket\phi\alpha
\end{equation}
\begin{equation}
A = \int \d\alpha \d\beta \fn{g}{\alpha} \fn{g}{\beta} \fn{A}{\alpha,\beta} \ket\alpha \bra\beta
\quad \textrm{where} \quad \fn{A}{\alpha,\beta} = \bra\alpha A \ket\beta
\end{equation}
\begin{equation}
\braket\phi\psi = \int \d\alpha \fn{g}{\alpha} \fn{\phi^*}{\alpha} \fn{\psi}{\alpha}
\end{equation}

\subsection{Delta function}
\label{delta}

The \keyword{delta function} is defined by
\begin{equation}
\int \d\beta \fn{\delta}{\alpha,\beta} \fn{f}{\beta} = \fn{f}{\alpha}
\end{equation}
which is the continuum analogue of $\sum_\beta \delta_{\alpha\beta} f_\beta = f_\alpha$.
Its representation depends on the Hilbert space, but is usually given by
\begin{equation}
\fn{\delta}{\alpha,\beta} = \fn{\delta}{\beta-\alpha}
\end{equation}
or some generalization thereof \footnote{See Homework~9.}, where
\begin{equation}
\fn{\delta}{\alpha} = \left\{
\begin{array}{ccc}
\infty & \textrm{if} & \alpha = 0
\\
0 & \textrm{if} & \alpha \neq 0
\end{array}
\right.
\end{equation}
with the \emph{simple} divergence at $\alpha = 0$ normalized by
\begin{equation}
\int \d\alpha \fn{\delta}{\alpha} = 1
\end{equation}

\subsection{Eigenbases}

Due to their orthogonality and completeness, the eigenspaces of a Hermitian or unitary operator, or the intersections of the eigenspaces of a set of commuting Hermitian or unitary operators, provide natural choices of vectors for a basis.
In particular, a complete set of commuting Hermitian or unitary operators $A,\ldots,B$, for which all the intersections of the eigenspaces $\hilbert{A}_\alpha \cap \ldots \cap \hilbert{B}_\beta$ are zero or one dimensional, provides a unique \footnote{Up to signs and ordering.} orthonormal basis labelled by the eigenvalues of the intersecting eigenspaces
\begin{equation}
\ket{\alpha,\ldots,\beta} \in \hilbert{A}_\alpha \cap \ldots \cap \hilbert{B}_\beta
\end{equation}
See Figure~\ref{eigenbasis}.

\begin{figure}[h]
\centering
\begin{tikzpicture}
\matrix{
\draw[red] (-1,0,0) -- (0,0,0);
\fill[red!50,opacity=0.8] (0,-1,-1) -- (0,-1,1) -- (0,1,1) -- (0,1,-1) node[below right,opacity=1]{$\hilbert{A}_{\alpha_2}$} -- cycle;
\draw[red] (0,0,0) -- (1,0,0) node[below]{$\hilbert{A}_{\alpha_1}$};
&
\node{$\cap$};
&
\draw[blue] (0,-1,0) -- (0,0,0);
\fill[blue!50,opacity=0.8] (-1,0,-1) -- (-1,0,1) -- (1,0,1) -- (1,0,-1) node[above left,opacity=1]{$\hilbert{B}_{\beta_2}$} -- cycle;
\draw[blue] (0,0,0) -- (0,1,0) node[left]{$\hilbert{B}_{\beta_1}$};
&
\node{$\to$};
&
% background
\fill[red!50,opacity=0.8] (0,-1,-1) -- (0,-1,1) -- (0,0,1) -- (0,0,-1) -- cycle;
\fill[blue!50,opacity=0.8] (-1,0,-1) -- (-1,0,1) -- (0,0,1) -- (0,0,-1) -- cycle;
\draw[red] (-1,0,0) -- (0,0,0);
\draw[blue] (0,-1,0) -- (0,0,0);
% foreground
\fill[red!50,opacity=0.8] (0,1,-1) -- (0,1,1) -- (0,0,1) -- (0,0,-1) -- cycle;
\fill[blue!50,opacity=0.8] (1,0,-1) -- (1,0,1) -- (0,0,1) -- (0,0,-1) -- cycle;
\draw[red] (0,0,0) -- (1,0,0);
\draw[blue] (0,0,0) -- (0,1,0);
&
\node{$\to$};
&
\draw[red!50!blue!50] (-1,0,0) -- (1,0,0) node[below]{$\ket{{\color{red} \alpha_1},{\color{blue} \beta_2}}$};
\draw[red!50!blue!50] (0,-1,0) -- (0,1,0) node[left]{$\ket{{\color{red} \alpha_2},{\color{blue} \beta_1}}$};
\draw[red!50!blue!50] (0,0,-1) -- (0,0,1) node[below left]{$\ket{{\color{red} \alpha_2},{\color{blue} \beta_2}}$};
\\
\node{eigenspaces of $A$};
&
\node{};
&
\node{eigenspaces of $B$};
&
\node{};
&
\node{\begin{tabular}{c} intersections \\ of eigenspaces \end{tabular}};
&
\node{};
&
\node{\begin{tabular}{c} orthonormal \\ basis \end{tabular}};
\\
};
\end{tikzpicture}
\caption{ \label{eigenbasis}
The intersections of the eigenspaces of a complete set of commuting Hermitian or unitary operators provide a unique orthonormal basis labelled by eigenvalues.
}
\end{figure}

In this basis the operators have the simple diagonalized form
\begin{eqnarray}
A & = & \smash{\sum_{\alpha,\ldots,\beta}} \alpha P_{\hilbert{A}_\alpha \cap \ldots \cap \hilbert{B}_\beta}
\eqnarrayrelation{=}
\smash{\sum_{\alpha,\ldots,\beta}} \alpha \ket{\alpha,\ldots,\beta} \bra{\alpha,\ldots,\beta}
\nonumber \\
& \vdots &
\\ \nonumber
B & = & \vphantom{\sum_{\alpha,\ldots,\beta}} \smash{\sum_{\alpha,\ldots,\beta}} \beta P_{\hilbert{A}_\alpha \cap \ldots \cap \hilbert{B}_\beta}
\eqnarrayrelation{=}
\smash{\sum_{\alpha,\ldots,\beta}} \beta \ket{\alpha,\ldots,\beta} \bra{\alpha,\ldots,\beta}
\end{eqnarray}
with the trivial action on the basis vectors
\begin{eqnarray}
A \ket{\alpha,\ldots,\beta} & = & \alpha \ket{\alpha,\ldots,\beta}
\nonumber \\
& \vdots &
\\ \nonumber
B \ket{\alpha,\ldots,\beta} & = & \beta \ket{\alpha,\ldots,\beta}
\end{eqnarray}

A simple physical example is given by the operators $\hat{p}_x$, $\hat{p}_y$, $\hat{p}_z$, representing the $x$, $y$, $z$ components of a particle's momentum.
They form a complete set of commuting Hermitian operators and their mutual eigenvectors $\ket{p_x,p_y,p_z}$, representing a state with momentum $(p_x,p_y,p_z)$, can be used as a basis for the particle's Hilbert space.

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