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

One nucleus, one electron

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 $H$ with central potential depend in general on two quantum numbers $k$ and $l$, but that for particular potential given by equation eqpotcenhy, eigenvalues depend only on sum $n=k+l$.

Rotation invariance

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: Template:IMP/eq Laplacian operator can be expressed as a function of $L^2$ operator. Template:IMP/thm Template:IMP/pf Let us use the problem's symmetries:

Since:

• $L_z$ commutes with operators acting on $r$
• $L_z$ commutes with $L^2$ operator $L_z$ commutes with $H$
• $L^2$ commutes with $H$

we look for a function $\varphi$ that diagonalizes simultaneously $H,L^2,L_z$ that is such that:

$\begin{array}{ccc}H\varphi (r)& =& E\varphi (r)\\ L^2\varphi (r)& =& l(l+1){\hslash }^2\varphi (r)\\ L_z\varphi (r)& =& m\hslash \varphi (r)\end{array}$

Spherical harmonics $Y_l^m(\theta ,\varphi )$ can be introduced now: Template:IMP/defn Looking for a solution $\varphi (r)$ that can\footnote{Group theory argument should be used to prove that solution actually are of this form.} be written (variable separation): Template:IMP/eq problem becomes one dimensional: Template:IMP/label Template:IMP/eq where $R(r)$ is indexed by $l$ only. Using the following change of variable: $R_l(r)=\frac{1}{r}{u}_l(r)$, one gets the following spectral equation: Template:IMP/eq where Template:IMP/eq The problem is then reduced to the study of the movement of a particle in an effective potential $V_e(r)$. To go forward in the solving of this problem, the expression of potential $V(r)$ is needed. Particular case of hydrogen introduced at section sechydrog corresponds to a potential $V(r)$ proportional to $1/r$ 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: Template:IMP/eq where $T_2$ 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 Template:IMP/eq is modelized by the sum $\sum W(r_i)$, where $W(r_i)$ is the mean potential acting on particle $i$. The hamiltonian can thus be written: Template:IMP/eq where $h_i=-\frac{{\hslash }^2}{2m}{\mathrm{\Delta }}_i+W(r_i)$. Template:IMP/rem It is then sufficient to solve the spectral problem in a space $E_i$ for operator $h_i$. 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 $W(r_i)$ is not like $1/r$ as in the hydrogen atom case and thus the accidental degeneracy is not observed here. The energy depends on two quantum numbers $l$ (relative to kinetic moment) and $n$ (rising from the radial equation eqaonedimrr). Eigenstates in this approximation are called electronic configurations. Template:IMP/exmp

Spectral terms

Let us write exact hamiltonian $H$ as:

where $T_1$ represents a correction to $H_0$ due to the interactions between electrons. Solving of spectral problem associated to $H_1=H_0+T_1$ using perturbative method is now presented. Template:IMP/rem To diagonalize $T_1$ in the space spanned by the eigenvectors of $H_0$, it is worth to consider problem's symmetries in order to simplify the spectral problem. It can be shown that operators $L^2$, $L_z$, $S^2$ and $S_z$ form a complete set of observables that commute. Template:IMP/exmp

Fine structure levels

Finally spectral problem associated to

can be solved considering $T_2$ as a perturbation of $H_1=H_0+T_1$. It can be shown (Template:IMP/cite) that operator $T_2$ can be written $T_2=\xi (r_i){\overrightarrow{l}}_i{\overrightarrow{s}}_i$. It can also be shown that operator $\overrightarrow{J}=\overrightarrow{L}+\overrightarrow{S}$ commutes with $T_2$. Operator $T_2$ will have thus to be diagonilized using eigenvectors $|J,m_J>$ common to operators $J_z$ and $J^2$. each state is labelled by:

where $L,S,J$ are azimuthal quantum numbers associated with operators $\overrightarrow{L},\overrightarrow{S},\overrightarrow{J}$.