Definition 0.1
The Hamiltonian operator H introduced in quantum mechanics by Schrödinger (and thus sometimes also called the
Schrödinger operator) on the Hilbert space
is given by the action:
The operator
defined above
, for a potential function
specified as the real-valued function
is called the Hamiltonian operator, H, and only very rarely the Schrödinger operator. The energy conservation (quantum) law written with the operator H as the
Schrödinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical computation
device in quantum mechanics of systems
with a finite number of degrees of freedom. There is also, however, the alternative approach in the Heisenberg picture, or formulation, in which the observable
and other operators
are time-dependent whereas the state vectors
are time-independent, which reverses the time dependences betwen operators and state vectors from the more popular Schrödinger formulation. Other formulations of quantum theories
occur in
quantum field theories (QFT), such as QED
(quantum electrodynamics) and QCD
(quantum chromodynamics).
Although the two formulations, or pictures, are unitarily (or mathematically) equivalent, however, sometimes the claim is made that the Heisenberg picture is “more natural and fundamental than the Schrödinger” formulation because the Lorentz invariance from general relativity
is also encountered in the Heisenberg picture,
and also because there is a `correspondence' between the commutator of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of classical mechanics. If the state vector
, or
does not change with time as in the Heisenberg picture, then the `equation of motion'
of a (quantum) observable operator is :
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