Introduction to Mathematical Physics/N body problem in quantum mechanics/Atoms

One nucleus, one electron edit


This case corresponds to the study of hydrogen atom.\index{atom} It is a particular case of particle in a central potential problem, so that we apply methods presented at section #secpotcent to treat this problem. Potential is here:



It can be shown that eigenvalues of hamiltonian   with central potential depend in general on two quantum numbers   and  , but that for particular potential given by equation eqpotcenhy, eigenvalues depend only on sum  .

Rotation invariance edit


We treat in this section the particle in a central potential problem ([#References|references]). The spectral problem to be solved is given by the following equation:


Laplacian operator can be expressed as a function of   operator.

Theorem: Laplacian operator   can be written as:


Proof: Here, tensorial notations are used (Einstein convention). By definition:




The writing order of the operators is very important because operator do not commute. They obey following commutation relations:


From equation eqdefmomP, we have:






Introducing operator:


we get the relation:


Using spherical coordinates, we get:




So, equation eql2pri becomes:


Let us use the problem's symmetries:


  •   commutes with operators acting on  
  •   commutes with   operator   commutes with  
  •   commutes with  

we look for a function   that diagonalizes simultaneously   that is such that:


Spherical harmonics   can be introduced now:


Spherical harmonics   are eigenfunctions common to operators   and  . It can be shown that:


Looking for a solution   that can\footnote{Group theory argument should be used to prove that solution actually are of this form.} be written (variable separation):


problem becomes one dimensional:



where   is indexed by   only. Using the following change of variable:  , one gets the following spectral equation:




The problem is then reduced to the study of the movement of a particle in an effective potential  . To go forward in the solving of this problem, the expression of potential   is needed. Particular case of hydrogen introduced at section sechydrog corresponds to a potential   proportional to   and leads to an accidental degeneracy.

One nucleus, N electrons edit

This case corresponds to the study of atoms different from hydrogenoids atoms. The Hamiltonian describing the problem is:


where   represents a spin-orbit interaction term that will be treated later. Here are some possible approximations:

N independent electrons edit

This approximation consists in considering each electron as moving in a mean central potential and in neglecting spin--orbit interaction. It is a ``mean field approximation. The electrostatic interaction term


is modelized by the sum  , where   is the mean potential acting on particle  . The hamiltonian can thus be written:


where  .


More precisely,   is the linear operator acting in the tensorial product space   and defined by its action on function that are tensorial products:


It is then sufficient to solve the spectral problem in a space   for operator  . Physical kets are then constructed by anti symmetrisation (see example exmppauli of chapter chapmq) in order to satisfy Pauli principle.\index{Pauli} The problem is a central potential problem (see section #secpotcent). However, potential   is not like   as in the hydrogen atom case and thus the accidental degeneracy is not observed here. The energy depends on two quantum numbers   (relative to kinetic moment) and   (rising from the radial equation eqaonedimrr). Eigenstates in this approximation are called electronic configurations.


For the helium atom, the fundamental level corresponds to an electronic configuration noted  . A physical ket is obtained by anti symetrisation of vector:


Spectral terms edit

Let us write exact hamiltonian   as:


where   represents a correction to   due to the interactions between electrons. Solving of spectral problem associated to   using perturbative method is now presented.


It is here assumed that  . This assumption is called  --  coupling approximation.

To diagonalize   in the space spanned by the eigenvectors of  , it is worth to consider problem's symmetries in order to simplify the spectral problem. It can be shown that operators  ,  ,   and   form a complete set of observables that commute.


Consider again the helium atom ([ph:mecaq:Cohen73]). From the symmetries of the problem, the basis chosen is:


where   is the quantum number associated to the total kinetic moment\index{kinetic moment}:


and   is the quantum number associated to total spin of the system\index{spin}:


Moreover, one has:




Table Tab. tabpauli represents in each box the value of   for all possible values of   and  . \begin{table}[hbt]




For an atom with two electrons, states such that   is odd are excluded.


We will proof this result using symmetries. We have:


Coefficients   are called Glebsh-Gordan\index{Glesh-Gordan coefficients} coefficients. If  , it can be shown (see ([ph:mecaq:Cohen73]) that:


Action of   on   can thus be written:


Physical ket obtained is:


Fine structure levels edit

Finally spectral problem associated to


can be solved considering   as a perturbation of  . It can be shown ([ph:atomi:Cagnac71]) that operator   can be written  . It can also be shown that operator   commutes with  . Operator   will have thus to be diagonilized using eigenvectors   common to operators   and  . each state is labelled by:


where   are azimuthal quantum numbers associated with operators  .