# Introduction to Mathematical Physics/Quantum mechanics/Some observables

## Hamiltonian operators

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

${\displaystyle E_{c}={\frac {p^{2}}{2m}}.}$

Its quantum equivalent, the hamiltonian ${\displaystyle H}$ is:

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

Remark: Passage relations Quantification rules ([#References

## Position operator

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

${\displaystyle R_{x}\phi (x,y,z)=x\phi (x,y,z)}$

${\displaystyle R_{y}\phi (x,y,z)=y\phi (x,y,z)}$

${\displaystyle R_{z}\phi (x,y,z)=z\phi (x,y,z)}$

## Momentum operator

In the same way, to "classical" momentum of a particle is associated a set of three observables ${\displaystyle P=(P_{x},P_{y},P_{z})}$. Action of operator ${\displaystyle P_{x}}$ is defined by \index{momentum operator}:

eqdefmomP

${\displaystyle P_{x}\phi ={\frac {\hbar }{i}}{\frac {\partial }{\partial x}}\phi }$

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

${\displaystyle [R_{i},R_{j}]=0}$

${\displaystyle [P_{i},P_{j}]=0}$

${\displaystyle [R_{i},P_{j}]=i\hbar \delta _{ij}}$

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

## Kinetic momentum operator

Definition:

A kinetic momentum \index{kinetic moment operator} ${\displaystyle J}$, is a set of three operators ${\displaystyle J_{x},J_{y},J_{z}}$ that verify following commutation relations \index{commutation relations}:

${\displaystyle [J_{i},J_{l}]=i\hbar \epsilon _{kil}J_{k}}$

that is:

${\displaystyle [J_{x},J_{y}]=i\hbar J_{z}}$

${\displaystyle [J_{y},J_{z}]=i\hbar J_{x}}$

${\displaystyle [J_{z},J_{x}]=i\hbar J_{y}}$

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

Example:

Orbital kinetic momentum

Theorem:

Operator defined by ${\displaystyle L_{i}=\epsilon _{ijk}R_{j}P_{k}}$ 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 ${\displaystyle M}$:

${\displaystyle M={\frac {\mu _{B}}{\hbar }}L}$

Example:

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

Postulate:

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

${\displaystyle M_{S}=2{\frac {\mu _{b}}{\hbar }}S}$