Introduction to Mathematical Physics/Quantum mechanics/Some observables

Hamiltonian operators

Hamiltonian operator \index{hamiltonian operator} has been introduced as the infinitesimal generator times $i\hslash$ 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). Template:IMP/exmp Template:IMP/rem

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 $\varphi$ of the orbital Hilbert space: Template:IMP/eq

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 $B$.