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

__toc__ One nucleus, one electron sechydrog 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: eqpotcenhy 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 secpotcent 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. Laplacian operator can be written as: Here, tensorial notations are used (Einstein convention). By definition: So: The writing order of the operators is very important because operator do not commute. They obey following commutation relations: From equation eqdefmomP, we have: thus Now, Introducing operator: we get the relation: Using spherical coordinates, we get: and So, equation eql2pri becomes: Let us use the problem's symmetries: Since: 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: eqaonedimrr where is indexed by only. Using the following change of variable: , one gets the following spectral equation: where 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 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 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 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: and Table Tab. tabpauli represents in each box the value of for all possible values of and . \begin{table}[hbt]tabpauli center frame Pauli principle. Values of number are presented for all possible values of and . Pauli principle implies that some boxes are empty: they correspond to states for which two particles have the same quantum numbers.} \begin{center} \begin{tabular}{ r r r r r r r r l } \cline{3-9} \multicolumn{1}{c}{}& &&&&&&&\\ \multicolumn{1}{c}{}& &&&&&&&\\ \hline &&&&&&&&\multicolumn{1}{c}{}\\ \cline{1-8} && 2 0&&&&&&\multicolumn{1}{c}{}\\ \cline{1-8} && 1 1& 1 0&&&&&\multicolumn{1}{c}{}\\ \cline{1-8} && 1 0& 1-1& 0 0&&&&\multicolumn{1}{c}{}\\ \cline{1-8} && 0 1& 0 0 &-1 1&-1 0&&&\multicolumn{1}{c}{}\\ \cline{1-8} && 0 0& 0-1&-1 0&-1-1&-2 0&&\multicolumn{1}{c}{}\\ \cline{1-8} &&\multicolumn{7}{c}{}\\ \cline{1-2} \end{tabular} \end{center} \end{table} One notes the spectral terms. In table Tab. tabpauli: one recognizes in the low corner the term, terms on the second diagonal and between those boxes (third and fifth diagonal) terms . excluded terms () correspond to odd. This result is the object of theorem theopair. Hund rule allows to order energy levels. Hund's rule: the level of minimal energy in a given configuration has the largest possible value of and for this value of , the largest value of . theopair 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 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 .