Introduction to Mathematical Physics/Quantum mechanics/Some observables

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Hamiltonian operators[edit]

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[edit]

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[edit]

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[edit]

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