The Atomic Hamiltonian

The Hamiltonian of an atom can be divided into three parts  

We can remove the degeneracy of each level by applying an external magnetic field (the Zeeman effect). If B is the applied field, we have the additional term in the Hamiltonian [Eq. (6.131)] 

Evaluation of S (Bethe and Jackiw, p. 169) gives as the final weak-field result  

Thus the external field splits each level into 2/ + 1 states, each state having a different value of Mj. 

We have based the discussion on a scheme in which we first added the individual electronic orbital angular momenta to form a total-orbital-angular-momentum vector and did the same for the spins L = S,- L, and S = 2i S,. We then combined L and S to get J. This scheme is called Russell-Saunders couplit (or L-S coupling) and is appropriate where the spin-orbit interaction energy is small compared with the interelec-tronic repulsion energy. The operators L and S commute with + W,ep, but when is included in the Hamiltonian, L and no longer commute with H. (J does commute with + //rep + Q ) If the spin-orbit interaction is small, then L and S almost commute with (t, and L-S coupling is valid. 

FIGURE 11.6 Effect of inclusion of successive terms in the atomic Hamiltonian for the ls2p helium configuration, fig is not part of the atomic Hamiltonian but is due to an applied magnetic field. 

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The sum in m goes from 1 to 5, because there are five different atomic d orbitals 4>, . In the usual notation, these orbitals are labeled dxy, dxz, dzy, dx2 yi, and d-izi ri. The normalized atomic orbitals are eigenfunctions of the atomic Hamiltonian T + V, with energy e°. As a first approximation, the overlap integrals of the atomic orbitals across neighboring sites can be neglected. 

If (3.16) is to be a good approximation, the effect of the radiation s electric field must be a small addition to the atomic Hamiltonian. For a radiation intensity of 1 watt/cm2, we have for the radiation s electric field, using (3.29),... 

While using (4.14) and exact wave functions. This supports the conclusion of Drake [83] that for electric dipole transitions, by considering the commutator of with the atomic Hamiltonian in the Pauli approximation, we obtain Qwith relativistic corrections of order v2/c2 (see (4.18)-(4.20)). However, for many-electron atoms and ions, one has to use approximate (e.g., Hartree-Fock) wave functions, and then this term gives non-zero contribution, conditioned by the inaccuracy of the model adopted. 

Of course, the last term in (53) can have any value, 0 < G(x, x fy< oo, and for this reason cannot have any physical significance. Taking the atomic Hamiltonian as... 

Let us outline briefly a possible way to calculate the normal modes of a molecule, and the relation between the positions of individual atoms and collective variables. We assume, that the atomic configuration of a system is determined mainly by the elastic forces, which are insensitive to the transport electrons. The dynamics of this system is determined by the atomic Hamiltonian... 

One can see that the full Hamiltonian consists of three terms, two which describe separately the parts for the atom and the field, and one which represents a coupling between the field (vector potential A) and terms from the atom (operator V,-). Obviously, it is this mixed term which is responsible for the photon-atom interaction. Provided perturbation theory can be applied, this term then acts as a transition operator between undisturbed initial and final states of the atom. Following this approach, one has to verify whether the disturbance caused by the electromagnetic field in the atom is small enough such that perturbation theory is applicable. Hence, one has to compare the terms which contain the vector potential A with an energy ch that is characteristic for the atomic Hamiltonian ... 

The parameters discussed above are often called free ion parameters for the atomic Hamiltonian, because they take the same form in the calculations of the free ion as they do in a crystal environment. However, it should be recognized that these parameters are modified by the crystal environment. 

The configuration-interaction approximation to i) results from diagonal-ising the atomic Hamiltonian H in the M-dimensional basis Ip ). 

The atomic Hamiltonian, Hq, couples radial values j to j and 7 1 and is diagonal in I while the interaction term Hi couples angular momentum / to / 1 and is diagonal in j. 

The atomic Hamiltonian is modified by the operator of the spin-orbit coupling... 

We therefore, once and for all, take the decision to use model potentials which are fitted to the results of calculations of the electronic structure of atomic cores, which are as accurate as it is possible to make them both numerically and in terms of correctness of the atomic Hamiltonian. 

Although the individual orbitd-angular-momentum operators L, do not commute with the atomic Hamiltonian (11.1), one can show (Bethe and Jackiw, pp. 102-103) that L does commute with the atomic Hamiltonian [provided spin-orbit interaction (Section 11.6) is neglected]. We can therefore characterize an atomic state by a quantum number L, where L(L -I- 1) is the square of the magnitude of the toted electronic orbital angular momentum. The electronic wave function il/ of an atom satisfies L tfr = L(L -I- The total-electronic-orbital-angular-momentum quantum number L of an atom is specified by a code letter, as follows ... 

The atomic Hamiltonian (t of (11.1) (which omits spin-orbit interaction) does not involve spin and therefore commutes with the total-spin operators 5 and S. The fact that commutes with ft is not enough to show that the atomic wave functions are eigenfunctions of 5 The Pauli antisymmetry principle requires that each tp must be an eigenfunction of the exchange operator with eigenvalue —1 (Section 10.3). Hence must also commute with if we are to have simultaneous eigenfunctions of H, S, and ic. Problem 11.16 shows that [. , 4fc] = 0, so the atomic wave functions are eigenfunctions of We have = S S + l)feV each atomic state can be characterized by a total-electronic-spin quantum number S. 

Parity of Atomic States. Consider the atomic Hamiltonian (11.1). We showed in Section 7.5 that the parity operator O commutes with the kinetic-energy operator. The quantity 1/r/ in (11.1) is = (jc + Replacement of each coordinate by its... 

The operator J for the total electronic angular momentum commutes with the atomic Hamiltonian, and we may characterize an atomic state by a quantum number /, which has the possible values [Eq. (11.39)]... 

For the atomic Hamiltonian (11.1), all states that belong to the same term have the same energy. However, when spin-orbit interaction (Section 11.6) is included in H, one finds that the value of the quantum number J affects the energy slightly. Hence states that belong to the same term but that have different values of/will have slightly... 

The atomic Hamiltonian (11.1) does not involve electron spin. In reality, the existence of spin adds an additional term, usually small, to the Hamiltonian. This term, called the spinr-orbit interaction, splits an atomic term into levels. Spin-orbit interaction is a relativistic effect and is properly derived using Dirac s relativistic treatment of the electron. This section gives a qualitative discussion of the origin of spin-orbit interaction. 

When a proper relativistic derivation of the spin-orbit-interaction term Hs.o. the atomic Hamiltonian is carried out, one finds that for a one-electron atom (see Bethe and Jackiw, Chapters 8 and 23)... 

For lathanide ions, the spin-orbit interaction is the next most important component in the atomic Hamiltonian. Within the 4f" configuration, it is the sum over the 4f shell of (r)/ s where I and s are the individual orbital and spin angular momenta of each electron. The spin-orbit parameter is given by... 

In the following I want to attempt a sort of unification of different sources of optical rotation and dichroism and show that far from being a narrow specialist s area of laser spectroscopy it is an enormously rich and varied field of study. I will therefore take as my starting point the famous and well-known dispersion relations and develop from these the form of the Faraday, Stark and PNC optical rotation. I shall also consider very briefly the extension of these ideas to the case of Doppler-free polarimetry and later I shall discuss how the use of lasers themselves brings in a variety of problems, in particular that of saturation. Finally, I will say something about the form of the weak interaction in so far as it enters the atomic Hamiltonian as a weak (no pun really intended ) perturbation. 

The presence of a neutral weak current interaction between the electrons and nucleons in an atom gives rise to a parity non-conserving part of the atomic Hamiltonian which can be written as... 

The atomic multiplet DMFT method is a generalization of the Hubbard-I approximation (Flubbard, 1963,1964a, 1964b) to a full f-manifold of 14 orbitals. The model adopted for the isolated atom considers only these 14 degrees of freedom, and their interaction through the Coulomb force in the atomic Hamiltonian... 

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