Ursula aus Yale sieht beim s Orbital keine Entartung - schade !

06/09/2011 - 18:28 von Fast Fury | Report spam
Spin-Orbit Interaction
Although neglected up to this lecture, the interaction between the
electron-spin and the orbital angular momentum must also be included
in the atomic Hamiltonian. Such interaction is described according to
the spin-orbit Hamiltonian defined as follows,


(50)




where is the Coulombic potential of the electron in the field of the
atom. Note that the spin-orbit interaction is proportional to . A
proper derivation of Eq. (50) requires a relativistic treatment of the
electron which is beyond the scope of these lectures.
Note: A classical description of such interaction also gives a
perturbation proportional to . This is because from the reference
frame of the electron, the nucleus is a moving charge that generates a
magnetic field , proportional to . Such magnetic field interacts with
the spin magnetic moment . Therefore, the interaction between and
is proportional to . Unfortunately, however, the proportionality
constant predicted by such classical model is incorrect, and a proper
derivation requires a relativistic treatment of the electron as
mentioned earlier in this section.

In order to compute the spin-orbit Hamiltonian of a many-electron
atom, it is necessary to compute first an approximate effective
potential of each electron in the total electric field of electrons
and nuclear charges. Then, we can compute the sum over all electrons
as follows,


(51)




The correction of eigenfunctions and eigenvalues, due to the spin-
orbit coupling, is usually computed according to perturbation theory
after solving the atomic eigenvalue problem in the absence of the spin-
orbit interaction. For example, the spin-orbit correction to the
eigenvalue of state for a one-electron atom is,


(52)




Note that the product can be written in terms of and as follows,
because, and, since the unperturbed wave function is an eigenfunction
of , and ,



Therefore,



It is important to note that, due to the spin-orbit coupling, the
total energy of a state depends on the value of the total angular
momentum quantum number . Furthermore, each of the energy levels is (2J
+1) times degenerate, as determined by the possible values of . For
example, when L=1, and S=1/2, then the possible values of J are 1/2
and 3/2, since (J=L+S, L+S-1, ..., L-S).

The spin orbit interaction is, therefore, responsible for splitting of
spectroscopic lines in atomic spectra.

It is possible to remove the degeneracy of energy levels by applying
an external magnetic field that perturbs the system as follows,
where , with , and The external perturbation is, therefore, described
by the following Hamiltonian,




The energy correction according to first-order perturbation theory
is:



where and is a proportionality constant. Therefore, the perturbation
of an external magnetic field splits energy level characterized by
quantum number J into 2J+1 energy sub-levels. These sub-levels
correspond to different possible values of , as described by the
following diagram:





Exercise 45: (A). Calculate the energy of the spectroscopic lines
associated with transitions 3S 3P for Na in the absence of an
external magnetic field. (B). Calculate the spectroscopic lines
associated with transitions 3S 3P for Na atoms perturbed by an
external magnetic field as follows:




and with




 

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#1 Fast Fury
06/09/2011 - 23:15 | Warnen spam
On 6 Sep., 18:28, Fast Fury wrote:
Spin-Orbit Interaction
Although neglected up to this lecture, the interaction between the
electron-spin and the orbital angular momentum must also be included
in the atomic Hamiltonian. Such interaction is described according to
the spin-orbit Hamiltonian defined as follows,

 (50)

where  is the Coulombic potential of the electron in the field of the
atom. Note that the spin-orbit interaction is proportional to . A
proper derivation of Eq. (50) requires a relativistic treatment of the
electron which is beyond the scope of these lectures.
Note: A classical description of such interaction also gives a
perturbation proportional to . This is because from the reference
frame of the electron, the nucleus is a moving charge that generates a
magnetic field , proportional to . Such magnetic field interacts with
the spin magnetic moment  . Therefore, the interaction between  and
is proportional to . Unfortunately, however, the proportionality
constant predicted by such classical model is incorrect, and a proper
derivation requires a relativistic treatment of the electron as
mentioned earlier in this section.

In order to compute the spin-orbit Hamiltonian of a many-electron
atom, it is necessary to compute first an approximate effective
potential  of each electron  in the total electric field of electrons
and nuclear charges. Then, we can compute the sum over all electrons
as follows,

 (51)

The correction of eigenfunctions and eigenvalues, due to the spin-
orbit coupling, is usually computed according to perturbation theory
after solving the atomic eigenvalue problem in the absence of the spin-
orbit interaction. For example, the spin-orbit correction to the
eigenvalue of state  for a one-electron atom is,

 (52)

Note that the  product can be written in terms of  and  as follows,
because,  and, since the unperturbed wave function is an eigenfunction
of ,  and ,

Therefore,

It is important to note that, due to the spin-orbit coupling, the
total energy of a state depends on the value of the total angular
momentum quantum number . Furthermore, each of the energy levels is (2J
+1) times degenerate, as determined by the possible values of . For
example, when L=1, and S=1/2, then the possible values of J are 1/2
and 3/2, since (J=L+S, L+S-1, ..., L-S).

The spin orbit interaction is, therefore, responsible for splitting of
spectroscopic lines in atomic spectra.

It is possible to remove the degeneracy of energy levels by applying
an external magnetic field that perturbs the system as follows,
where , with , and  The external perturbation is, therefore, described
by the following Hamiltonian,

The energy correction according to first-order perturbation theory
is:

where  and  is a proportionality constant. Therefore, the perturbation
of an external magnetic field splits energy level characterized by
quantum number J into 2J+1 energy sub-levels. These sub-levels
correspond to different possible values of , as described by the
following diagram:

Exercise 45: (A). Calculate the energy of the spectroscopic lines
associated with transitions 3S  3P for Na in the absence of an
external magnetic field. (B). Calculate the spectroscopic lines
associated with transitions 3S  3P for Na atoms perturbed by an
external magnetic field  as follows:

and  with

­--



Die FU Penner aus Berlin ( ich berichtete darüber ) meinen :

Ich habe ein Termschema (Wasserstoff-Atom) gegeben und zwar ohne und
mit Feinstruktur-Aufspaltung.

(1) Nun soll ich erklàren durch welchen Effekt die Energie des
Grundzustands abgesenkt wird.

(2) Als nàchstes soll ich erklàren, in wieviele Feinstrukturlinien die
-Linie aufspaltet? Das sind doch 5, oder nicht?

(3) Nun soll ich angeben welche Übergànge erlaubt sind

Ich habe folgende Übergànge notiert:

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