+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Physics +--{: .hide} [[!include physicscontents]] =-- =-- =-- # Hamiltonians * table of contents {: toc} Disambiguation: there is an unrelated notion of a Hamilton or Hamiltonian operator also called [[nabla]] in [[vector analysis]]. ## Definition Given a [[Poisson manifold]] $(X, \{-,-\})$ and a [[vector field]] $v \in \Gamma(T X) $, a **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 $h_v$. ## 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 _the_ Hamiltonian. This is the concept that Hamilton originally considered and which hence gives the name to the general situaiton. ### In classical mechanics The simplest, so-called "natural", Hamiltonian (function) of a [[dynamical system]] is the sum of the kinetic and potential energy: \[ H = T + V. \] 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]]. ### In quantum mechanics The [[quantum mechanics]] of a point [[particle]] in the _[[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 _worldline_ -- of the particle. For $t \in \mathbb{R}$ the fiber $\mathcal{H}_t$ is the **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 $\mathbf{H}$. $$ A = H \;d t \in \Omega^1(\mathbb{R}, End(\mathcal{H})) \,. $$ The component $H \in End(\mathcal{H})$ of this canonical 1-form is the **Hamilton(ian) operator** (or the *quantum Hamiltonian*) of the system. Its [[parallel transport]] is the **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 $$ U : (t_1 \stackrel{\gamma}{\to} t_2) \mapsto (\mathcal{H}_{t_1} \stackrel{exp\left(-\frac{i}{\hbar}H (t_2-t_1)\right)}{\to} \mathcal{H}_{t_2}) \,. $$ If instead $H$ does depend on $t$ -- called the case of _time-dependent quantum mechanics_ -- then the full formula for parallel transport applies, which is given by the [[path-ordered exponential]] $$ U : (t_1 \stackrel{\gamma}{\to} t_2) \mapsto (\mathcal{H}_{t_1} \stackrel{P exp \left(-\frac{i}{\hbar}\int_{t_1}^{t_2}H d t\right)}{\to} \mathcal{H}_{t_2}) \,. $$ In the physics literature this path-ordered exponential is known as the **[[Dyson formula]]** . #### 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. ## Related concepts * [[quadratic Hamiltonian]] [[!include Hamiltonian and Lagrangian -- table]] * [[Hamilton's equations]] * [[quantum operator]] * [[Hamiltonian action]] * [[Hamiltonian form]] * [[propagator]] [[!include geometric quantization extensions - table]] ## 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]]