nLab
Hamiltonian

Context

Physics

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Hamiltonians

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

Definition

Given a Poisson manifold (X,{,})(X, \{-,-\}) and a vector field vΓ(TX)v \in \Gamma(T X) , a Hamiltonian for vv is a smooth function h vC (X)h_v \in C^\infty(X) such that {h v,}\{h_v,-\} is the derivation corresponding to vv.

Conversely, one says that vv is the Hamiltonian vector field of h vh_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 HH as a function on phase space (so as a function of position q iq^i and momentum p ip_i), we can derive other quantities as functions on phase space. In particular, we have: * velocity, v i=H/p iv^i = \partial{H}/\partial{p_i}, * force, f i=H/q if_i = -\partial{H}/\partial{q^i}.

Setting v i=dq i/dtv^i = \mathrm{d}q^i/\mathrm{d}t and f i=dp i/dtf_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} with connection \nabla over the real line – the worldline – of the particle.

For tt \in \mathbb{R} the fiber t\mathcal{H}_t is the space of quantum states of the system, at given parameter time tt. Since this bundle is necessarily trivializable, we imagine fixing a trivialization 0×\mathcal{H} \simeq \mathcal{H}_0 \times \mathbb{R}. Then the flat connection on the bundle is canonically a 1-form on \mathbb{R} with values in linear operators on H\mathbf{H}.

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

The component HEnd()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 HH is constant as a function on \mathbb{R}, this parallel transport assigns to the path γ\gamma from t 1t_1 to t 2t_2 in \mathbb{R} the map

U:(t 1γt 2)( t 1exp(iH(t 2t 1)) 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 HH does depend on tt – 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 1γt 2)( t 1Pexp(i t 1 t 2Hdt) 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 eigenvalues 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 HH via

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

provided the Hamiltonian is time-independent.

\leftarrow \rightarrow

higher and integrated :

( of of for 𝔾\mathbb{G}-)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\infty of (Ω𝔾)(\Omega \mathbb{G})- on XX
1 under
1
2
2
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(extension are listed for sufficiently connected XX)

References

Named after William Rowan Hamilton.

Last revised on March 16, 2015 at 14:14:30. See the history of this page for a list of all contributions to it.