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

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

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}:

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