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 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:

Laplacian operator can be expressed as a function of $$L^2$$ operator.

Let us use the problem's symmetries: Since:

we look for a function $$\phi$$ that diagonalizes simultaneously $$H,L^2,L_z$$ that is such that: $$\begin{matrix} H\phi(r)&=&E\phi(r)\\ L^2\phi(r)&=&l(l+1)\hbar^2\phi(r)\\ L_z\phi(r)&=&m\hbar\phi(r) \end{matrix}$$ Spherical harmonics $$Y^m_l(\theta,\phi)$$ can be introduced now:
 * $$L_z$$ commutes with operators acting on $$r$$
 * $$L_z$$ commutes with $$L^2$$ operator $$L_z$$ commutes with $$H$$
 * $$L^2$$ commutes with $$H$$

Looking for a solution $$\phi(r)$$ 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 $$R(r)$$ 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:

where

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:

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

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:

where $$h_i=-\frac{\hbar^2}{2m}\Delta_i+W(r_i)$$.

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 ). 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.

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.

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. {{IMP/exmp| Consider again the helium atom ({{IMP/cite|ph:mecaq:Cohen73}}). From the symmetries of the problem, the basis chosen is:

{{IMP/eq|$$|1:n_1,l_1;2:n_2,l_2;L,m_L>\otimes|S,m_S>$$}}

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

{{IMP/eq|$$L\in\{l_1+l_2,l_1+l_2-1, \dots,|l_1-l_2|\}$$}}

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

{{IMP/eq|$$S\in\{0,1\}$$}}

Moreover, one has:

{{IMP/eq|$$m_L=m_{l_1}+m_{l_2}$$}}

and

{{IMP/eq|$$m_S=m_{s_1}+m_{s_2}$$}}

Table Tab. tabpauli represents in each box the value of $$m_Lm_S$$ for all possible values of $$m_L$$ and $$m_S$$. \begin{table}[hbt]{{IMP/label|tabpauli}} $$m_Lm_S$$ are presented for all possible values of $$m_L$$ and $$m_S$$. 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}{}& &$$1$$&$$1$$&$$0$$&$$0$$&$$-1$$&$$-1$$&$$m_{l_1}$$\\ \multicolumn{1}{c}{}& &$$+\frac{1}{2}$$&$$-\frac{1}{2}$$&$$+\frac{1}{2}$$&$$-\frac{1}{2}$$&$$+\frac{1}{2}$$&$$-\frac{1}{2}$$&$$m_{s_1}$$\\ \hline $$1$$&$$+\frac{1}{2}$$&&&&&&&\multicolumn{1}{c}{}\\ \cline{1-8} $$1$$&$$-\frac{1}{2}$$& 2 0&&&&&&\multicolumn{1}{c}{}\\ \cline{1-8} $$0$$&$$+\frac{1}{2}$$& 1 1& 1  0&&&&&\multicolumn{1}{c}{}\\ \cline{1-8} $$0$$&$$-\frac{1}{2}$$& 1 0& 1-1& 0  0&&&&\multicolumn{1}{c}{}\\ \cline{1-8} $$-1$$&$$+\frac{1}{2}$$& 0 1& 0  0 &-1  1&-1  0&&&\multicolumn{1}{c}{}\\ \cline{1-8} $$-1$$&$$-\frac{1}{2}$$& 0 0& 0-1&-1  0&-1-1&-2  0&&\multicolumn{1}{c}{}\\ \cline{1-8} $$m_{l_2}$$&$$m_{s_2}$$&\multicolumn{7}{c}{}\\ \cline{1-2} \end{tabular} \end{center} \end{table} One notes
 * center | frame |Pauli principle. Values of number

the spectral terms. In table Tab. tabpauli: $$^1S_0$$ term, terms $$^1D_2$$ on the   second diagonal and between those boxes (third and fifth diagonal) terms    $$^3P$$. $$L+S$$ odd. This result is the object of theorem theopair. Hund rule allows to order energy levels. }}
 * one recognizes in the low corner the
 * excluded terms ($$^3D$$) correspond to

{{IMP/pf| We will proof this result using symmetries. We have: $$\begin{matrix} & &=\sum_m \sum_{m'} \mathrel{<} l,l',m,m'|L,M_L\mathrel{>} \end{matrix}$$ Coefficients $$\mathrel{<} l,l',m,m'|L,M_L\mathrel{>} $$ are called Glebsh-Gordan\index{Glesh-Gordan coefficients} coefficients. If $$l=l'$$, it can be shown (see ({{IMP/cite|ph:mecaq:Cohen73}}) that: {{IMP/eq|$$\mathrel{<} l,l,m,m'|L,M_L\mathrel{>} =(-1)^L \mathrel{<} l,l,m',m|L,M_L\mathrel{>}.$$}} Action of $$P_{21}$$ on $$|1:n,l;2:n,l';L,M_L\mathrel{>}$$ can thus be written:
 * 1:n,l;2:n,l';L,M_L\mathrel{>} \\
 * 1:n,l,m;2:n',l',m'\mathrel{>}

{{IMP/eq|$$P_{21}|1:n,l;2:n,l';L,M_L\mathrel{>} =(-1)^L|1:n,l;2:n,l';L,M_L\mathrel{>} $$}} Physical ket obtained is: $$\begin{matrix} & &= \left\{ \begin{array}{ll} 0&\mbox{ if }L+S\mbox{ is odd }\\ \end{array} \right. \end{matrix}$$ }}
 * n,l,n,l;L,M_L;S,M_S\mathrel{>}\\
 * 1:n,l;2:n,l';L,M_L\mathrel{>} \otimes |S,M_S\mathrel{>} &\mbox{ if }L+S\mbox{ is even }

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  that operator $$T_2$$ can be written $$T_2=\xi(r_i)\vec l_i\vec s_i$$. It can also be shown that operator $$\vec J=\vec L+\vec S$$ 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 $$\vec L,\vec S,\vec J$$.