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Disambiguation: there is an unrelated notion of a Hamilton or Hamiltonian operator also called nabla in vector analysis?.
Given a Poisson manifold and a vector field , a Hamiltonian for is a smooth function such that is the derivation corresponding to .
Conversely, one says that is the Hamiltonian vector field of .
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.
The simplest, so-called “natural”, Hamiltonian (function) of a dynamical system is the sum of the kinetic and potential energy:
Knowing only as a function on phase space (so as a function of position and momentum ), we can derive other quantities as functions on phase space. In particular, we have: * velocity, , * force, .
Setting and , we derive the equations of motion in Hamiltonian mechanics.
The quantum mechanics of a point particle in the Schrödinger picture is encoded in a Hilbert space bundle with connection over the real line – the worldline – of the particle.
For the fiber is the space of quantum states of the system, at given parameter time . Since this bundle is necessarily trivializable, we imagine fixing a trivialization . Then the flat connection on the bundle is canonically a 1-form on with values in linear operators on .
The component 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 is constant as a function on , this parallel transport assigns to the path from to in the map
If instead does depend on – called the case of time-dependent quantum mechanics – then the full formula for parallel transport applies, which is given by the path-ordered exponential?
In the physics literature this path-ordered exponential is known as the Dyson formula .
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 via
exp
provided the Hamiltonian is time-independent.
Hamiltonian | Legendre transform | Lagrangian |
---|---|---|
Lagrangian correspondence | prequantization | prequantized Lagrangian correspondence |
higher and integrated Kostant-Souriau extensions:
(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for -principal ∞-connection)
(extension are listed for sufficiently connected )
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.