# 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).

Example:

Classical energy of a free particle is

$E_c=\frac{p^2}{2m}.$

Its quantum equivalent, the hamiltonian $H$ is:

$H=\frac{P^2}{2m}.$

Remark: Passage relations Quantification rules ([#References

## 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:

$R_x\phi(x,y,z)=x\phi(x,y,z)$

$R_y\phi(x,y,z)=y\phi(x,y,z)$

$R_z\phi(x,y,z)=z\phi(x,y,z)$

## 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}:

eqdefmomP

$P_x\phi=\frac{\hbar}{i}\frac{\partial}{\partial x} \phi$

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

$[R_i,R_j]=0$

$[P_i,P_j]=0$

$[R_i,P_j]=i\hbar \delta_{ij}$

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 $B$.

## Kinetic momentum operator

Definition:

A kinetic momentum \index{kinetic moment operator} $J$, is a set of three operators $J_x,J_y,J_z$ that verify following commutation relations \index{commutation relations}:

$[J_i,J_l]=i\hbar\epsilon_{kil}J_k$

that is:

$[J_x,J_y]=i\hbar J_z$

$[J_y,J_z]=i\hbar J_x$

$[J_z,J_x]=i\hbar J_y$

where $\epsilon_{ijk}$ is the permutation signature tensor (see appendix secformultens). Operator $J$ is called a vector operator (see appendix chapgroupes.

Example:

Orbital kinetic momentum

Theorem:

Operator defined by $L_i=\epsilon_{ijk}R_jP_k$ 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 $M$:

$M=\frac{\mu_B}{\hbar}L$

Example:

Postulates for the electron. We have seen at section secespetat that state space for an electron (a fermion of spin $s=1/2$) is the tensorial product orbital state space and spin state space. One defines an operator $S$ 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 $S$ is a kinetic moment.

Postulate:

Electron is a particle of spin $s=1/2$ and it has an intrinsic magnetic moment \index{magnetic moment}:

$M_S=2\frac{\mu_b}{\hbar}S$