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\begin{document}

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\section*{Hamiltonian}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{hamiltonians}{}\section*{{Hamiltonians}}\label{hamiltonians}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_mechanical_systems}{In mechanical systems}\dotfill \pageref*{in_mechanical_systems} \linebreak
\noindent\hyperlink{in_classical_mechanics}{In classical mechanics}\dotfill \pageref*{in_classical_mechanics} \linebreak
\noindent\hyperlink{in_quantum_mechanics}{In quantum mechanics}\dotfill \pageref*{in_quantum_mechanics} \linebreak
\noindent\hyperlink{physical_meaning_and_relation_to_unitary_transformations}{Physical meaning and relation to unitary transformations}\dotfill \pageref*{physical_meaning_and_relation_to_unitary_transformations} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


Disambiguation: there is an unrelated notion of a Hamilton or Hamiltonian operator also called [[nabla]] in [[vector analysis]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Given a [[Poisson manifold]] $(X, \{-,-\})$ and a [[vector field]] $v \in \Gamma(T X)$, a \textbf{Hamiltonian} for $v$ is a [[smooth function]] $h_v \in C^\infty(X)$ such that $\{h_v,-\}$ is the [[derivation]] corresponding to $v$.

Conversely, one says that $v$ is the [[Hamiltonian vector field]] of $h_v$.

\hypertarget{in_mechanical_systems}{}\subsection*{{In mechanical systems}}\label{in_mechanical_systems}

Given a [[classical mechanical system]] evolving in time, there is a [[symplectic manifold]] (or at least [[Poisson manifold]]) equipped with the [[vector field]] that generates time evolution. Its Hamiltonian is often called \emph{the} Hamiltonian. This is the concept that Hamilton originally considered and which hence gives the name to the general situaiton.

\hypertarget{in_classical_mechanics}{}\subsubsection*{{In classical mechanics}}\label{in_classical_mechanics}

The simplest, so-called ``natural'', Hamiltonian (function) of a [[dynamical system]] is the sum of the kinetic and potential energy:

\begin{displaymath}
H = T + V.
\end{displaymath}
Knowing only $H$ as a function on [[phase space]] (so as a function of [[position]] $q^i$ and [[momentum]] $p_i$), we can derive other quantities as functions on phase space. In particular, we have: * [[velocity]], $v^i = \partial{H}/\partial{p_i}$, * [[force]], $f_i = -\partial{H}/\partial{q^i}$.

Setting $v^i = \mathrm{d}q^i/\mathrm{d}t$ and $f_i = \mathrm{d}p_i/\mathrm{d}t$, we derive the [[equations of motion]] in [[Hamiltonian mechanics]].

\hypertarget{in_quantum_mechanics}{}\subsubsection*{{In quantum mechanics}}\label{in_quantum_mechanics}

The [[quantum mechanics]] of a point [[particle]] in the \emph{[[Schrödinger picture]]} is encoded in a [[Hilbert space]] [[bundle]] $\mathcal{H} \to \mathbb{R}$ [[connection on a bundle|with connection]] $\nabla$ over the real line -- the \emph{worldline} -- of the particle.

For $t \in \mathbb{R}$ the fiber $\mathcal{H}_t$ is the \textbf{space of [[quantum state]]s} of the system, at given parameter time $t$. Since this bundle is necessarily trivializable, we imagine fixing a trivialization $\mathcal{H} \simeq \mathcal{H}_0 \times \mathbb{R}$. Then the flat connection on the bundle is canonically a [[differential form|1-form]] on $\mathbb{R}$ with values in [[linear operator]]s on $\mathbf{H}$.

\begin{displaymath}
A = H \;d t \in \Omega^1(\mathbb{R}, End(\mathcal{H}))
  \,.
\end{displaymath}
The component $H \in End(\mathcal{H})$ of this canonical 1-form is the \textbf{Hamilton(ian) operator} (or the \emph{quantum Hamiltonian}) of the system.

Its [[parallel transport]] is the \textbf{time evolution} of quantum states. If $H$ is constant as a function on $\mathbb{R}$, this parallel transport assigns to the path $\gamma$ from $t_1$ to $t_2$ in $\mathbb{R}$ the map

\begin{displaymath}
U :
  (t_1 \stackrel{\gamma}{\to} t_2) 
   \mapsto
  (\mathcal{H}_{t_1} 
  \stackrel{exp\left(-\frac{i}{\hbar}H (t_2-t_1)\right)}{\to} \mathcal{H}_{t_2})
  \,.
\end{displaymath}
If instead $H$ does depend on $t$ -- called the case of \emph{time-dependent quantum mechanics} -- then the full formula for parallel transport applies, which is given by the [[path-ordered exponential]]

\begin{displaymath}
U :
  (t_1 \stackrel{\gamma}{\to} t_2) \mapsto
  (\mathcal{H}_{t_1} \stackrel{P exp \left(-\frac{i}{\hbar}\int_{t_1}^{t_2}H  d t\right)}{\to} \mathcal{H}_{t_2})
  \,.
\end{displaymath}
In the physics literature this path-ordered exponential is known as the \textbf{[[Dyson formula]]} .

\hypertarget{physical_meaning_and_relation_to_unitary_transformations}{}\paragraph*{{Physical meaning and relation to unitary transformations}}\label{physical_meaning_and_relation_to_unitary_transformations}

The [[eigenvalue]]s of the Hamiltonian operator for a closed quantum system are exactly the energy eigenvalues of that system. Thus the Hamiltonian is interpreted as being an ``energy'' operator. Conservation of energy occurs when the Hamiltonian is time-independent.

Transformations and evolutions in standard quantum mechanics are represented via [[unitary operator]]s where a time evolving unitary is related to the Hamiltonian $H$ via

$U(0,t) =$exp$\left(-\frac{i}{\hbar}H t\right),$

provided the Hamiltonian is time-independent.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[quadratic Hamiltonian]]

\end{itemize}
[[!include Hamiltonian and Lagrangian -- table]]

\begin{itemize}%
\item [[Hamilton's equations]]


\item [[quantum operator]]


\item [[Hamiltonian action]]


\item [[Hamiltonian form]]


\item [[propagator]]



\end{itemize}
[[!include geometric quantization extensions - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

Named after [[William Rowan Hamilton]].

[[!redirects Hamiltonian]] [[!redirects hamiltonian]] [[!redirects Hamiltonians]] [[!redirects hamiltonians]] [[!redirects Hamiltonian function]] [[!redirects hamiltonian function]] [[!redirects Hamiltonian functions]] [[!redirects hamiltonian functions]] [[!redirects Hamiltonian operator]] [[!redirects hamiltonian operator]] [[!redirects Hamiltonian operators]] [[!redirects hamiltonian operators]] [[!redirects Hamilton operator]] [[!redirects hamilton operator]] [[!redirects Hamilton operators]] [[!redirects hamilton operators]]



\end{document}