Introduction to Mathematical Physics/Quantum mechanics/Some observables

__toc__ Hamiltonian operators Hamiltonian operator \index{hamiltonian operator} has been introduced as the infinitesimal generator times of the evolution group. Experience, passage methods from classical mechanics to quantum mechanics allow to give its expression for each considered system. Schr\"odinger equation rotation invariance implies that the hamiltonian is a scalar operator (see appendix chapgroupes). Classical energy of a free particle is Its quantum equivalent, the hamiltonian is: Passage relations Quantification rules ([#References references]) indicate how for a physical quantity already defined in classical mechanics, the operator of quantum mechanics has to be defined. In a general way, observable associated to physical quantity defined classically is obtained by replacing in suitably symmetrised\footnote{Indeed, in general the product of two operators does not commute, on the contrary of the scalar product of two vectors or the product of two scalars. For instance the classical quantity where is the classical momentum and the classical position have the equivalent: where is the momentum observable, and the position observable, observables that are introduced in the next part of the present section. For any operator and , the quantity is called the commutator \index{commutator} of and and is in genral different from zero. } expression of , r and p by observables and . Position operator Classical notion of position of a particle leads to associate to a particle a set of three operators (or observables) called position operators\index{position operator} and defined by their action on a function of the orbital Hilbert space: Momentum operator In the same way, to "classical" momentum of a particle is associated a set of three observables . Action of operator is defined by \index{momentum operator}: eqdefmomP Operators and verify commutation relations called canonical commutation relations \index{commutation relations} : where is Kronecker symbol (see appendix secformultens) and where for any operator and , . Operator is called the commutator of and . Kinetic momentum operator A kinetic momentum \index{kinetic moment operator} , is a set of three operators that verify following commutation relations \index{commutation relations}: that is: where is the permutation signature tensor (see appendix secformultens). Operator is called a vector operator (see appendix chapgroupes. Orbital kinetic momentum Operator defined by is a kinetic momentum. It is called orbital kinetic momentum. Let us evaluate (see ([#References references])) commutator: where canonical commutation relations have been used. Changing notations: To orbital kinetic momentum is associated a magnetic moment : Postulates for the electron. We have seen at section secespetat that state space for an electron (a fermion of spin ) is the tensorial product orbital state space and spin state space. One defines an operator called spin operator that acts inside spin state space. It is postulated that this operator is a kinetic momentum and that it appears in the hamiltonian {\it via} a magnetic momentum. Operator is a kinetic moment. Electron is a particle of spin and it has an intrinsic magnetic moment \index{magnetic moment}: