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

One nucleus, one electronEdit

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

V(r)=-\frac{e}{r^2}

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 invarianceEdit

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:

-[\frac{\hbar^2}{2\mu}\Delta+V(r)]\phi(r)=E\phi(r).

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

Theorem: Laplacian operator \Delta can be written as:

\Delta=-\frac{1}{r^2}L^2+\frac{\partial^2}{\partial r^2}+\frac{2}{r}\frac{\partial}{\partial r}

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

L_i=\epsilon_{ijk}x_jp_k

So:

\begin{matrix}
L_iL_i&=&\epsilon_{ijk}\epsilon_{ilm}x_{j}p_k x_l p_m\\
&=&(\delta_{jl}\delta_{km}-\delta_{jm}\delta{kl})x_{j}p_kx_l p_m\\
&=&x_jx_jp_kp_k-x_jp_kx_kp_j
\end{matrix}

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

[x_i,p_j]=i \hbar \delta_{ij}

[x_j,x_k]=0

[p_j,p_k]=0

From equation eqdefmomP, we have:

p_k=-i\hbar \frac{\partial}{\partial x_k}

thus

x_jx_jp_kp_k=-x^2\hbar^2\Delta.

Now,

\begin{matrix}
x_jp_kx_kp_j&=&[i\hbar\delta_{ik}+p_kx_j]x_kp_j\\
&=&i\hbar x_kp_k+p_kx_jx_kp_j\\
&=&i\hbar x_kp_k+p_kx_kx_jp_j
\end{matrix}

Introducing operator:


\tilde D=x_k\frac{\partial}{\partial x_k}

we get the relation:

{{IMP/label|eql2pri}}
L^2=-x^2\Delta+\tilde D^2+\tilde D

Using spherical coordinates, we get:


\tilde D=r\frac{\partial}{\partial r}

and


\tilde D^2=(r\frac{\partial}{\partial r})(r\frac{\partial}{\partial
r})=r^2\frac{\partial^2}{\partial r^2}+\frac{\partial}{\partial r}

So, equation eql2pri becomes:


\Delta=-\frac{1}{r^2}L^2+\frac{\partial^2}{\partial
r^2}+\frac{2}{r}\frac{\partial}{\partial r}

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 \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:

Definition:

Spherical harmonics Y^m_l(\theta,\phi) are eigenfunctions common to operators L^2 and L_z. It can be shown that:

\begin{matrix}
L^2Y^m_l&=&l(l+1)Y^m_l\\
L_zY^m_l&=&mY^m_l
\end{matrix}

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):

\phi(r)=R(r)Y^m_l(\theta,\phi)

problem becomes one dimensional:

eqaonedimrr

-[\frac{\hbar^2}{2\mu}(\frac{d^2}{dr^2}+\frac{2}{r}\frac{\partial}{\partial
r})+\frac{l(l+1)}{2\mu r^2}\hbar^2+V(r)]R_{l}(r)=E_{kl}R_{l}(r)

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:

-[\frac{\hbar^2}{2\mu}\frac{d^2}{dr^2}+V_e(r)]u_{kl}(r)=E_{kl}u_{kl}(r)

where

V_e(r)=\frac{l(l+1)}{2\mu r^2}\hbar^2+V(r)

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 electronsEdit

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


H=\sum_i-\frac{\hbar^2}{2m}\Delta_i-
\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r_i}+ \sum_{j\mathrel{>}
i}\frac{1}{4\pi\epsilon_0}\frac{e^2}{r_{ij}}+T_2

where T_2 represents a spin-orbit interaction term that will be treated later. Here are some possible approximations:

N independent electronsEdit

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


-\frac{1}{4\pi\epsilon_0}\frac{Ze^2}{r_i}+\sum_{j\mathrel{>}
i}\frac{1}{4\pi\epsilon_0}\frac{e^2}{r_{ij}}

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:


H_0=\sum_ih_i

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

Remark:

More precisely, h_i is the linear operator acting in the tensorial product space \otimes_{i=1}^N E_i and defined by its action on function that are tensorial products:


[1_1\otimes\dots\otimes 1_{i-1}\otimes h_i \otimes 1_{i+1}\dots\otimes
1_{N}] 
(\phi_1\otimes\dots\phi_N)
=h_i(\phi_1)\otimes\dots\phi_N

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.

Example:

For the helium atom, the fundamental level corresponds to an electronic configuration noted 1s^2. A physical ket is obtained by anti symetrisation of vector:


|1:n,l,m_l,m_s>\otimes|2:n,l,m_l,m_s>

Spectral termsEdit

Let us write exact hamiltonian H as:

H=H_0+T_1+T_2

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.

Remark:

It is here assumed that T_2<<T_1. This assumption is called L--S coupling approximation.

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.

Example:

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

|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}:

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}:

S\in\{0,1\}

Moreover, one has:

m_L=m_{l_1}+m_{l_2}

and

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]

tabpauli

Theorem:

theopair

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

Proof:

We will proof this result using symmetries. We have:

\begin{matrix}
|1:n,l;2:n,l';L,M_L\mathrel{>}  \\
& &=\sum_m \sum_{m'}  \mathrel{<} l,l',m,m'|L,M_L\mathrel{>}
|1:n,l,m;2:n',l',m'\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 ([ph:mecaq:Cohen73]) that:

\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:

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}
|n,l,n,l;L,M_L;S,M_S\mathrel{>}\\
& &= \left\{
\begin{array}{ll}
0&\mbox{ if  }L+S\mbox{ is odd }\\
|1:n,l;2:n,l';L,M_L\mathrel{>} \otimes |S,M_S\mathrel{>} &\mbox{ if  }L+S\mbox{ is even }
\end{array} 
\right.
\end{matrix}

Fine structure levelsEdit

Finally spectral problem associated to

H=H_0+T_1+T_2

can be solved considering T_2 as a perturbation of H_1=H_0+T_1. It can be shown ([ph:atomi:Cagnac71]) 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 </math>T_2</math> will have thus to be diagonilized using eigenvectors |J,m_J> common to operators J_z and J^2. each state is labelled by:

^{2S+1}L_{J}

where L,S,J are azimuthal quantum numbers associated with operators \vec L,\vec S,\vec J.

Last modified on 30 May 2009, at 17:40