The quantum numbers J and Mj

Before moving to further examples, we must address the interaction between the total angular orbital momentum, L, and the total spin angular momentum, 5. To do so, we define the total angular momentum quantum number, J. Equation 21.11 gives the relationship for the total angular momentum for a multi-electron species. 

The quantum number J takes values L + 5),(L + 5 1)... L - 5, and these values fall into the series 0,1, 2. .. or 5,, . .. (like j for a single electron, J for the multi-electron system must be positive or zero). It follows that there are (25 + 1) possible values of / for 5 L, and 2L + 1) possible values for L S. The value of Mj denotes the component of the total angular momentum along the axis. Just as there are relationships between 5 and Ms, and between L and Ml, there is one between J and Mj  

We are now in a position to write full term symbols which include information about 5, L and J. The notation for a full term symbol is  

The value of Mj denotes the component of the total angular momentum along the z axis. Just as there are relationships between 5 and Ms, and between L and Ml, there is one between J and Mj  

A term symbol ( triplet P zero ) signifies a term with L=l, (25+1) = 3 (i.e. 5= 1), and /=0. Different values 

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We now consider many-electron atoms. We will assume Russell-Saunders coupling, so that an atomic state can be characterized by total electronic orbital and spin angular-momentum quantum numbers L and S, and total electronic angular-momentum quantum numbers J and Mj. (See Section 1.17.) The electric-dipole selection rules for L, J, and Mj can be shown to be (Bethe and Jackiw, p. 224)... 

In comparatively small electric fields molecules selected with respect to their rotational quantum numbers j and mj rotate in space but have preferential orientations of their molecular axis. The preferential orientations depend on the chosen set of quantum numbers j and mj and are defined with respect to the direction of the external field. The method of state selection is well known to molecular beam spectroscopists and is not restricted to polar... 

We use the quantum number / to label the rotational energy levels of the molecule, and (as with the atomic quantum number /) it can have integer values from 0 to 1 and on up. The rotational wavefunctions are again the spherical harmonics, but now we index them by the rotational quantum numbers J and Mj where Mj gives the projection of the rotational angular... 

The functions f and must belong to reps of the sphere group each characterized by a quantum number J and (2J -f- 1) quantum numbers Mj. The product of two basis functions will belong to a direct product representation which is reducible in the usual fashion, into other basis functions that are linear combinations of the products each one being characterized by a J and set of Mj quantum numbers. 

The light pulses are assumed to satisfy the impact excitation condition. Then the hyperfine interaction can be neglected during the time of excitation. Therefore within this time the atoms can reasonably be described by a system of basic states with quantum numbers J, I, mj, and mj. The corresponding level scheme for the 63 81 2 groxond state and the excited 6p Pi state for cesium with nuclear spin 1 2 is shown in Fig. 1. The sublevels of a common fine structure state are degenerate in energy. 

According to the SP formalism, nearly each electron is treated by its own wavefunction with a quantum number j and magnetic number mj. (The collinear approximation is also implemented in the method). This permits treatment of open shell system. 

The general formula and the individual cases as presented in Eq. (97) indicate that indeed the number of conical intersections in a given snb-space and the number of possible sign flips within this sub-sub-Hilbert space are interrelated, similar to a spin J with respect to its magnetic components Mj. In other words, each decoupled sub-space is now characterized by a spin quantum number J that connects between the number of conical intersections in this system and the topological effects which characterize it. 

In this volume, principal consideration is given to the lighter elements, so that the Russell-Saunders (549) vector model of the atom is used. In this model a multielectron atom is assumed to have the quantum numbers n, L = lif Ml, 8 = siy (or n, L, J = L + S, Mj). This implies stronger and Si-Sj coupling than U-Si coupling. It follows from Pauli s principle that for a closed shell =... 

We now apply these general results to the specific problem of HF. Apart from the Stark effect, we shall otherwise ignore the very small matrix elements which are off-diagonal in J, and evaluate the terms for J = 1, Mj = 0, 1, /H = h = 1 /2. As Weiss pointed out, the total magnetic component Mz = Mj + MF + Mh is a good quantum number and may be used to set up a decoupled representation in which states of different Mz value are diagonalised separately. For J = 1 there are twelve primitive basis states, which we write below in the form MF, Mh, Mj). 

Note that in the expressions which follow, A, E, and 2 are signed quantities. All of the matrix elements are strictly diagonal in F and MF (or in J and Mj for the nuclear spin-free problem), and we ignore any matrix elements off-diagonal in S and I. The symbol q denotes vibronic state quantum numbers not otherwise specified in the basis states. 

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