nLab
Hamiltonian

In classical mechanics

The simplest, so-called “natural”, Hamiltonian 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 state?s 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 operator?s 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 operator 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.