Total angular momentum vector

There is appreciable coupling between the resultant orbital and resultant spin momenta. This is referred to as LS coupling and is due to spin-orbit interaction. This interaction is caused by the positive charge Ze on the nucleus and is proportional to Z". The coupling between L and S gives the total angular momentum vector J. 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

Fig. 2-12.—The interaction of the orbital angular momentum vectors for two p electrons (h = 1, l — 1) to form the resultant total angular momentum vector, with values corresponding to 0, 1, and 2 for the total angular momentum quantum number L.

In general the value of the 0-factor is neither 1 nor 2, but has some other value. In case that the electronic state of the atom is such as to approximate closely to Russell-Saunders coupling the value of the 0-factor can be, calculated in a simple way. The total angular momentum vector of the atom is the resultant of the vector corresponding to... 

Fig. 4. Possible orientations of die total angular momentum vector i relative to the direction of an externally applied magnetic field B and the magnitudes of the associated magnetic quantum state vectors my...

Each electronic state is defined by three angular momenta total orbital angular momentum vector L, total spin angular momentum vector S, and total angular momentum vector J. These vectors have the following definitions ... 

Similarly, the total angular momentum vector J also has its z component Jz, with its corresponding magnitude being Mj a.u., where... 

In Figure 2.7 the arrows indicate possible orientations of the total angular momentum vector such that the component in the line-of-force direction is always a rational fraction of the total measure. The possible vectors are identified by their projection on the radius of the unit circle as fractions k/n. The quantum number k = 0 is considered meaningless. In Sommerfeld s words [8] ... 

For many-electron light atoms, the Russell-Saunders coupling rules prevail One combines the orbital angular momenta lt of each electron, treated as a vector, to form the total orbital angular momentum quantum number (and vector) L = h one similarly couples the spin angular momentum quantum numbers s, into a total spin angular momentum quantum number S = s > then one adds L and S to get the total angular momentum vector... 

Figure 7.2 The total angular momentum vectors J obtained from the sum of L and S for. v — j and. v = —i...

For an electron having both orbital and spin angular momentum, the total angular momentum vector is given by ... 

The z component of the total angular momentum vector is now j(h/2w) and there are (2j - - 1) possible orientations in space. 

Its total angular momentum vector remains invariant (because of the isotropy of space). 

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