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%%% Primary Title: Hamiltonian operator
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%%% Owner: bci1
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\begin{document}
\textbf{Definition 0.1}
The {Hamiltonian operator} \textbf{H} introduced in \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} (\htmladdnormallink{QM}{http://planetphysics.us/encyclopedia/FTNIR.html}) by Schr\"odinger (and thus sometimes also called the \emph{\htmladdnormallink{Schr\"odinger operator}{http://planetphysics.us/encyclopedia/Hamiltonian2.html}}) on the \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $L^2(\Rset^n)$ is given by the action:
\[
\psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n),
\]
The \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} defined above $[-\nabla^2 +V(x)]$ , for a potential \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $V(x)$ specified as the real-valued function $V\colon \Rset^n \to \Rset$ is called the {\em Hamiltonian operator}, \textbf{H}, and only very rarely the {\em Schr\"odinger operator}.
\subsection{Schr\"odinger formulation of QM}
The \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} conservation (quantum) law written with the operator \textbf{H} as the
Schr\"odinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical \htmladdnormallink{computation}{http://planetphysics.us/encyclopedia/LQG2.html} device in quantum mechanics of \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} with a finite number of degrees of freedom. There is also, however, the alternative approach in the Heisenberg picture, or formulation, in which the \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} and other \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} are time-dependent whereas the state vectors $\psi$ are time-independent, which reverses the time dependences betwen operators and state vectors from the more popular Schr\"odinger formulation. Other formulations of \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} occur in
quantum field theories (\htmladdnormallink{QFT}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}), such as \htmladdnormallink{QED}{http://planetphysics.us/encyclopedia/HotFusion.html} (\htmladdnormallink{quantum electrodynamics}{http://planetphysics.us/encyclopedia/QED.html}) and \htmladdnormallink{QCD}{http://planetphysics.us/encyclopedia/HotFusion.html} (quantum chromodynamics).
\subsection{Heisenberg formulation of QM}
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 Schr\"odinger'' formulation because the Lorentz invariance from \htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} is also encountered in the Heisenberg picture,
and also because there is a `correspondence' between the \htmladdnormallink{commutator}{http://planetphysics.us/encyclopedia/Commutator.html} of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of \htmladdnormallink{classical mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}. If the state vector $\psi$, or
$\left| \psi \right\rangle$ does not change with time as in the Heisenberg picture, then the `equation of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}' of a (quantum) observable operator is :
\[
\frac{d}{dt} A_{quantum} = (i\hbar)^{-1}[A,H] + \left(\frac {\partial A}{\partial t}\right)_{classical}
\]
\end{document}
