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:
![{\displaystyle V(r)=-{\frac {e}{r^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1ad5208d8f9bb948a22eea2ff7628f5e61a001)
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
.
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:
![{\displaystyle -[{\frac {\hbar ^{2}}{2\mu }}\Delta +V(r)]\phi (r)=E\phi (r).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0013fa9db2dede0de9c7896e393397947aa54f)
Laplacian operator can be expressed as a function of
operator.
Theorem:
Laplacian operator
can be written as:
![{\displaystyle \Delta =-{\frac {1}{r^{2}}}L^{2}+{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6edac919d51cb916a730e7d187f4dcf25d2d6593)
Proof:
Here, tensorial notations are used (Einstein convention). By definition:
![{\displaystyle L_{i}=\epsilon _{ijk}x_{j}p_{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc589cee5cf94f73c3d8f7795d4dc3f90b64113d)
So:
The writing order of the operators is very important because operator do not
commute. They obey following commutation relations:
![{\displaystyle [x_{i},p_{j}]=i\hbar \delta _{ij}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/753921a44bc7d6b8a27297b359760a472e6b7596)
![{\displaystyle [x_{j},x_{k}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74d31a8d054100bfbbbb5322621a53ed53eca4ec)
![{\displaystyle [p_{j},p_{k}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd4617fa4bf3f818247af862f5b5f9b05aaed20)
From equation eqdefmomP, we have:
![{\displaystyle p_{k}=-i\hbar {\frac {\partial }{\partial x_{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd589814a27aad9a409fa1c27897bfef82fbfafb)
thus
![{\displaystyle x_{j}x_{j}p_{k}p_{k}=-x^{2}\hbar ^{2}\Delta .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9422ac357b08a595ce5b96db3fdb576ca3853708)
Now,
Introducing operator:
![{\displaystyle {\tilde {D}}=x_{k}{\frac {\partial }{\partial x_{k}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24e2b3df6cd16c199eda682d896e836f18d7b6b4)
we get the relation:
![{\displaystyle {IMP/label|eql2pri}L^{2}=-x^{2}\Delta +{\tilde {D}}^{2}+{\tilde {D}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bf6982bba29c097b34127363e6d8deff5b5676d)
Using spherical coordinates, we get:
![{\displaystyle {\tilde {D}}=r{\frac {\partial }{\partial r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d32438ad758a8d2ae6cd14fcd9998ed15c9483a3)
and
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13c14a81e92875fc971687290b7f3cab1adad92e)
So, equation eql2pri becomes:
![{\displaystyle \Delta =-{\frac {1}{r^{2}}}L^{2}+{\frac {\partial ^{2}}{\partial r^{2}}}+{\frac {2}{r}}{\frac {\partial }{\partial r}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6edac919d51cb916a730e7d187f4dcf25d2d6593)
Let us use the problem's symmetries:
Since:
commutes with operators acting on ![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
commutes with
operator
commutes with ![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
commutes with ![{\displaystyle H}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b)
we look for a function
that diagonalizes simultaneously
that is such that:
Spherical harmonics
can be introduced now:
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):
![{\displaystyle \phi (r)=R(r)Y_{l}^{m}(\theta ,\phi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f75e027e3a0f128fe020b6066342b3dbc47a0575)
problem becomes one dimensional:
![{\displaystyle -[{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ecc517022529bfb5edfcca7fbde45c134a8749b)
where
is indexed by
only.
Using the following change of variable:
, one gets the following spectral equation:
![{\displaystyle -[{\frac {\hbar ^{2}}{2\mu }}{\frac {d^{2}}{dr^{2}}}+V_{e}(r)]u_{kl}(r)=E_{kl}u_{kl}(r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/710d21aac88088d991d590cfe6021b7b1b42f7b5)
where
![{\displaystyle V_{e}(r)={\frac {l(l+1)}{2\mu r^{2}}}\hbar ^{2}+V(r)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3bb471923a6d28262969e2cef4c6f74eb0864d)
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.
This case corresponds to the study of atoms different from hydrogenoids
atoms. The Hamiltonian describing the problem is:
![{\displaystyle 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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9f5ce073115a2ca6df905371d51fdb8a51dd519)
where
represents a spin-orbit interaction term that will be treated
later. Here are some possible approximations:
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
![{\displaystyle -{\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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d6b7af8fc1fe865cfa01c79cf70f89ebf5edf7)
is modelized by the sum
, where
is the mean potential
acting on particle
. The hamiltonian can thus be written:
![{\displaystyle H_{0}=\sum _{i}h_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/400a3c37e49a19ccf521a4b77a30452dc7e19759)
where
.
Remark:
More precisely,
is the linear operator acting in the tensorial
product space
and defined by its action on function that are
tensorial products:
![{\displaystyle [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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5dcb3e32a72b96cba903b6eb61bf01db478c4a0c)
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.
Example:
For the helium atom, the fundamental level corresponds to an electronic
configuration noted
. A physical ket is obtained by anti symetrisation of
vector:
![{\displaystyle |1:n,l,m_{l},m_{s}>\otimes |2:n,l,m_{l},m_{s}>}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b788b1f47b85cb3c60a45f3138c2be9a8844c9)
Let us write exact hamiltonian
as:
![{\displaystyle H=H_{0}+T_{1}+T_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d9c6b97f3c4b0042e9505734f6746da380b2b2)
where
represents a correction to
due to the interactions between electrons. Solving of spectral problem associated to
using perturbative method is now presented.
Remark:
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.
Example:
Consider again the helium atom ([ph:mecaq:Cohen73]).
From the symmetries of the problem, the basis chosen is:
![{\displaystyle |1:n_{1},l_{1};2:n_{2},l_{2};L,m_{L}>\otimes |S,m_{S}>}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01e5778700a97f175273e1dd2e49461172247d7d)
where
is the quantum number associated to the total kinetic
moment\index{kinetic moment}:
![{\displaystyle L\in \{l_{1}+l_{2},l_{1}+l_{2}-1,\dots ,|l_{1}-l_{2}|\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea6e95d587ee37eb03c43914f6ad33647c208005)
and
is the quantum number associated to total spin of the
system\index{spin}:
![{\displaystyle S\in \{0,1\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12ca2bb5c01c9067835542c3efb90ebeacb43f95)
Moreover, one has:
![{\displaystyle m_{L}=m_{l_{1}}+m_{l_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c91c6a4ec4da97f876efafc274e37bf8a2619b9)
and
![{\displaystyle m_{S}=m_{s_{1}}+m_{s_{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f3a18543af80037ea7c31a2a5c23a08f9be69cd)
Table Tab. tabpauli represents in each box the value of
for all possible values of
and
.
\begin{table}[hbt]
Theorem:
For an atom with two electrons, states such that
is odd are excluded.
Proof:
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:
![{\displaystyle {\mathrel {<}}l,l,m,m'|L,M_{L}{\mathrel {>}}=(-1)^{L}{\mathrel {<}}l,l,m',m|L,M_{L}{\mathrel {>}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62dc3fe578f4a2cdc7866d3c8542cf8f6edb9ee1)
Action of
on
can thus be written:
![{\displaystyle P_{21}|1:n,l;2:n,l';L,M_{L}{\mathrel {>}}=(-1)^{L}|1:n,l;2:n,l';L,M_{L}{\mathrel {>}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a247ec39e7eb7ee4fa1c8a9c8255c558bf4bc4)
Physical ket obtained is:
Finally spectral problem associated to
![{\displaystyle H=H_{0}+T_{1}+T_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d9c6b97f3c4b0042e9505734f6746da380b2b2)
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:
![{\displaystyle ^{2S+1}L_{J}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/deff807413c1f99b3142d6344e0312f3714bd55c)
where
are azimuthal quantum numbers associated with operators
.