nLab
Hamiltonian

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 $v\in \Gamma (TX)$, a Hamiltonian for $v$ is a smooth function $h_v\in C^{\mathrm{\infty }}(X)$ 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:

(1)$$H=T+V.$$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 $\mathscr{H}\to ℝ$ with connection $\nabla$ over the real line – the worldline – of the particle.

For $t\in ℝ$ the fiber ${\mathscr{H}}_t$ is the space of quantum states of the system, at given parameter time $t$. Since this bundle is necessarily trivializable, we imagine fixing a trivialization $\mathscr{H}\simeq {\mathscr{H}}_0\times ℝ$. Then the flat connection on the bundle is canonically a 1-form on $ℝ$ with values in linear operators on $H$.

$$A=Hdt\in {\Omega }^1(ℝ,\mathrm{End}(\mathscr{H})).$$A = H \;d t \in \Omega^1(\mathbb{R}, End(\mathcal{H})) \,.

The component $H\in \mathrm{End}(\mathscr{H})$ 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 $ℝ$, this parallel transport assigns to the path $\gamma$ from $t_1$ to $t_2$ in $ℝ$ the map

$$U:(t_1\stackrel{\gamma }{\to }{t}_2)\mapsto ({\mathscr{H}}_{t_1}\stackrel{\exp (-\frac{i}{\hslash }H(t_2-t_1))}{\to }{\mathscr{H}}_{t_2}).$$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?

$$U:(t_1\stackrel{\gamma }{\to }{t}_2)\mapsto ({\mathscr{H}}_{t_1}\stackrel{P\exp (-\frac{i}{\hslash }{\int }_{t_1}^{t_2}Hdt)}{\to }{\mathscr{H}}_{t_2}).$$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 operators where a time evolving unitary is related to the Hamiltonian $H$ via

$U(0,t)=$exp$(-\frac{i}{\hslash }Ht),$

provided the Hamiltonian is time-independent.

Related concepts

• quantum operator