Introduction to Mathematical Physics/Quantum mechanics/Some observables

Hamiltonian operatorsEdit

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).

Example:

Classical energy of a free particle is

 

Its quantum equivalent, the hamiltonian   is:

 

Remark: Passage relations Quantification rules ([#References

Position operatorEdit

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 operatorEdit

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 operatorEdit

Definition:

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.

Example:

Orbital kinetic momentum

Theorem:

Operator defined by   is a kinetic momentum. It is called orbital kinetic momentum.

Proof:

Let us evaluate (see ([#References

Postulate:

To orbital kinetic momentum is associated a magnetic moment  :

 

Example:

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.

Postulate:

Operator   is a kinetic moment.

Postulate:

Electron is a particle of spin   and it has an intrinsic magnetic moment \index{magnetic moment}: