Angular momentum vectors

A particle moving with momentum p at a position r relative to some coordinate origin has so-called orbital angular momentum equal to L = r x p. The three components of this angular momentum vector in a cartesian coordinate system located at the origin mentioned above are given in terms of the cartesian coordinates of r and p as follows ... 

Figure 1.5 Direction of the angular momentum vector p for an electron in an orbit...

An effect of space quantization of orbital angular momentum may be observed if a magnetic field is introduced along what we now identify as the z axis. The orbital angular momentum vector P, of magnitude Pi, may take up only certain orientations such that the component (Pi) along the z axis is given by... 

Figure 5.5 The rotational angular momentum vector P for (a) a linear molecule and (b) the prolate symmetric rotor CH3I where is the component along the a axis...

Each electron in an atom has two possible kinds of angular momenta, one due to its orbital motion and the other to its spin motion. The magnitude of the orbital angular momentum vector for a single electron is given, as in Equation (1.44), by... 

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. 

It depends on both the angle a of the angular momentum vector rotation and other Euler angles F and q, which determine the molecule s axis shift. Besides, the angle F is also the azimuth of the change in angular momentum A/ = J(t + 0) — J(t — 0), which is the result of collision. 

In this case, it is expressed via polar coordinates of the angular momentum vector in a plane, perpendicular to the diatomic molecular axis. The function, which is present in (6.6), can be determined by (6.7) as follows ... 

In addition to CO(v = 0—2,7) populations, Houston and Kable recorded CO Doppler profiles to measure the translational energy release, and the vector correlation between the recoil velocity vector and the angular momentum vector of CO. Together, these data paint a compelling picture that two pathways to CH4 + CO are operative. The rotationally hot CO population (85% of total CO)... 

Figure 5.5 The motion in a magnetic field B of the orbital angular momentum vector L.

Hyperfine structure arises through the interaction of the electron spin with a nuclear spin. Consider first the interaction of the electron spin with a single magnetic nucleus of spin , In an applied magnetic field the nuclear spin angular momentum vector, of magnitude (/ / -f l)]l/2, precesses around the direction of the field in an exactly analogous way to that of the electron spin. The orientations that the nuclear spin can take up are those for which the spin in the z-direction, M, has components of ... 

In a manner similar to that by which the atomic states were designated as s, p, d, or /, the letters S, P, D, and F correspond to the values of 0, 1, 2, and 3, respectively, for the angular momentum vector, L. After the values of the vectors L, S, and / have been determined, the overall angular momentum is described by a symbol known as a term symbol or spectroscopic state. This symbol is constructed as Ps+1)Lj where the appropriate letter is used for the L value as listed earlier, and the quantity (2S + 1) is known as the multiplicity. For one unpaired electron, (2S + 1) = 2, and a multiplicity of 2 gives rise to a doublet. For two unpaired electrons, the multiplicity is 3, and the state is called a triplet state. 

Spin I> 0 nuclei possess a magnetic dipole or dipole moment, n, which arises from a spinning, charged particle. Nuclei that have a nonzero spin will also have a magnetic moment, and the direction of that magnetic moment is collinear with the angular momentum vector associated with the nucleus. This can be expressed as... 

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. 

Here, the relevant angular momentum vectors and quantum numbers are L (7), the total orbital angular momentum of the atom, obtained as the vector coupling of those corresponding to the core and to the outer electron(s), S (S), the total spin, and J (7), the total angular momentum for a given atomic level [8] ... 

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