Introduction to Mathematical Physics/Quantum mechanics/Some observables

Hamiltonian operators
Hamiltonian operator \index{hamiltonian operator} has been introduced as the infinitesimal generator times $$i\hbar$$ 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).

Position operator
Classical notion of position $$r$$ of a particle leads to associate to a particle a set of three operators (or observables) $$R_x,R_y,R_z$$ called position operators\index{position operator} and defined by their action on a function $$\phi$$ of the orbital Hilbert space:

Momentum operator
In the same way, to "classical" momentum of a particle is associated a set of three observables $$P=(P_x,P_y,P_z)$$. Action of operator $$P_x$$ is defined by \index{momentum operator}:

Operators $$R$$ and $$P$$ verify commutation relations called canonical commutation relations \index{commutation relations} :

where $$\delta_{ij}$$ is Kronecker symbol (see appendix secformultens) and where for any operator $$A$$ and $$B$$, $$[A,B]=AB-BA$$. Operator $$[A,B]$$ is called the commutator of $$A$$ and $$B$$.