PlanetPhysics/Hamiltonian Operator 3

Definition 0.1 The {Hamiltonian operator} H introduced in [[../QuantumParadox/|quantum mechanics]] ([[../FTNIR/|QM]]) by Schr\"odinger (and thus sometimes also called the [[../Hamiltonian2/|Schr\"odinger operator]]) on the [[../NormInducedByInnerProduct/|Hilbert space]] Failed to parse (unknown function "\Rset"): {\displaystyle L^2(\Rset^n)} is given by the action:

Failed to parse (unknown function "\Rset"): {\displaystyle \psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n), }

The [[../QuantumSpinNetworkFunctor2/|operator]] defined above $[-{\mathrm{\nabla }}^2+V(x)]$ , for a potential [[../Bijective/|function]] $V(x)$ specified as the real-valued function Failed to parse (unknown function "\Rset"): {\displaystyle V\colon \Rset^n \to \Rset} is called the Hamiltonian operator , H , and only very rarely the Schr\"odinger operator .

Schr\"odinger formulation of QM

The [[../CosmologicalConstant/|energy]] conservation (quantum) law written with the operator H as the Schr\"odinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical [[../LQG2/|computation]] device in quantum mechanics of [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|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 [[../QuantumSpinNetworkFunctor2/|observable]] and other [[../QuantumOperatorAlgebra4/|operators]] 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 [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum theories]] occur in quantum field theories ([[../QuantumOperatorAlgebra5/|QFT]]), such as [[../HotFusion/|QED]] ([[../QED/|quantum electrodynamics]]) and [[../HotFusion/|QCD]] (quantum chromodynamics).

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 [[../SR/|general relativity]] is also encountered in the Heisenberg picture, and also because there is a `correspondence' between the [[../Commutator/|commutator]] of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of [[../MathematicalFoundationsOfQuantumTheories/|classical mechanics]]. If the state vector $\psi$, or $|\psi ⟩$ does not change with time as in the Heisenberg picture, then the `equation of [[../CosmologicalConstant/|motion]]' of a (quantum) observable operator is :

$\frac{d}{dt}{A}_{quantum}=(i\hslash {)}^{-1}[A,H]+{\left(\frac{\partial A}{\partial t}\right)}_{classical}$