Angular momenta orbital

According to the correspondence principle, the angular momentum operator I in quantum mechanics is formally defined in close analogy to classical mechanics. 

The projection of the momentum operator onto the position operator is given by 

The explicit evaluation of the scalar product then yields 

We rewrite part of the second term on the right-hand side of this equation as 

With the definition of a radial momentum in Eq. (4.108), we can now write the kinetic energy operator in spherical coordinates as 

1 Ligand Field Quenching of Orbital Angular Momentum 

The canonical real d-orbital wave functions are linear combinations of the complex eigen functions (solutions) of the hydrogenic Schrodinger equation and the orbital angular momentum operator. Thus, instead of a set of five degenerate orbitals that may be indexed by values of orbital 

The most convenient basis set in a Hgand field consists of the real d-orbitals, though as will be seen, the choice is not unique. The real d-orbitals are linear combinations of complex wave functions. The bra-ket notation below is used  

Notice that an unpaired electron in dxy or d yi in this geometry cannot contribute orbital angular momentum because each of the orbitals consists of equal parts of mi = 2. It might seem that this is true for the pair dxz nd dy as well, but because they are still degenerate in this geometry, it is possible to take linear combinations of these two wavefunctions and still have solutions to the Schrodinger equation. 

in a fairly straightforward way, it is possible to predict the magnetic properties of many metal complexes based on symmetry and electron count. And, often, the orbital contribution is quenched by the ligand field. However, a very interesting example of a complex that behaves in the completely opposite way, that is in which there is almost no quenching of the orbital angular momentum is [Fe C(SiMe3)3 2]. The oxidation state of the iron is formally 2+, In the free ion, the electron 

We first consider a particle of mass m moving according to the laws of classical mechanics. The angular momentum L of the particle with respect to the origin of the coordinate system is defined by the relation 

The square of the magnitude of the vector L is given in terms of these components by 

If a force F acts on the particle, then the torque T on the particle is defined as 

If there is no force acting on the particle, the torque is zero. Consequently, the rate of change of the angular momentum is zero and the angular momentum is conserved. 

The quantum-mechanical operators for the components of the orbital angular momentum are obtained by replacing px, Py, Pz in the classical expressions (5.2) by their corresponding quantum operators. 

Imagine a particle with position vector ra in a three-dimensional (3D) orthonormal coordinate system with axes x, y, z and a wave function P(ia) = I ixa, ya, za)- If the particle is rotated about one of the axes x, y, or z through an infinitesimal angle 8cp, the new position coordinates r a are given in first order by 

Similar relations are obtained for y a and z a. Thus, if the particle with position vector ra is rotated about a general axis through Sep, its new position coordinates are given by 

To first order, any function T(ra) is transformed under this coordinate rotation into 

On the other hand, the entity conserved in a closed system due to the isotropy of space is the orbital angular momentum of the system. Apart from a constant factor, the operator fa x Va must therefore correspond to the orbital angular momentum. Further, the angular momentum is an observable (i.e., real valued). Thus the corresponding operator ought to be Hermitian. An operator is said to be Hermitian if it obeys the turn-over rule, that is, 

The corresponding many-particle operators are obtained by summing over the indices of the individual particles 

The factor h is introduced to make a dimensionless operator with cartesian components 

This corresponds to the classical result that the angular 

Space quantization of orbital angular momenturn showing the 21+1 projections of the angular momentum vector on the z-axis. The figure is drawn for the case 1=2, 

Solutions of equation (3.38) which remain finite as p- - and also as p-3-0, and are therefore physically acceptable, have the form 

From equations (3.35), (3.37), and(3.40) we see that the negative energy eigenvalues are given by 

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There are significant differences between tliese two types of reactions as far as how they are treated experimentally and theoretically. Photodissociation typically involves excitation to an excited electronic state, whereas bimolecular reactions often occur on the ground-state potential energy surface for a reaction. In addition, the initial conditions are very different. In bimolecular collisions one has no control over the reactant orbital angular momentum (impact parameter), whereas m photodissociation one can start with cold molecules with total angular momentum 0. Nonetheless, many theoretical constructs and experimental methods can be applied to both types of reactions, and from the point of view of this chapter their similarities are more important than their differences. 

The simplest case arises when the electronic motion can be considered in temis of just one electron for example, in hydrogen or alkali metal atoms. That electron will have various values of orbital angular momentum described by a quantum number /. It also has a spin angular momentum described by a spin quantum number s of d, and a total angular momentum which is the vector sum of orbital and spin parts with... 

For the El state, the projeetion A = 1 of the eleetron orbital angular momentum along the intemuelear axis ean eouple with the projeetion S = to yield two spin-orbit levels, witii D = jand i The NO(X n)... 

P, Jy, and J , are the components of the total orbital angular momentum J of the nuclei in the IX frame. The Euler angles a%, b, cx appear only in the P, P and P angular momentum operators. Since the results of their operation on Wigner rotation functions are known, we do not need then explicit expressions in temis of the partial derivatives of those Euler angles. 

When the molecule is not in a S state there is an interaction between the rotation of the molecule and S and/or L, and the details of coupling the angular momenta are involved. Most nonsinglet molecules with electronic orbital angular momentum A = 0 obey Hund s case (b) coupling. In Case (b), the electronic orbital angular momentum combines with the nuclear orbital angular... 

For all produet states of this Ms value, one identifies the highest Ml value (this gives a value of the total orbital angular momentum, L, that ean arise for this S). For the above example, the highest Ml within the Ms =1 states is Ml = 1 (not Ml = 2), henee L=1. 

Their symmetry labels can be obtained by vector coupling (see Appendix G) the spin and orbital angular momenta of the two subsystems. The orbital angular momentum coupling... 

The above m funetions are appropriate whenever one wishes to deseribe orbitals that are eigenfunetions of L, the eomponent of orbital angular momentum along the z-axis. 

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 ... 

The functions are known as the angular wave functions or, because they describe the distribution of p over the surface of a sphere of radius r, spherical harmonics. The quantum number n = l,2,3,...,oo and is the same as in the Bohr theory, is the azimuthal quantum number associated with the discrete orbital angular momentum values, and is... 

Unlike the total energy, the quantum mechanical value Pi of the orbital angular momentum is significantly different from that in the Bohr theory given in Equation (1.8). It is now given by... 

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 1.9 Space quantization of orbital angular momentum for T = 3...

We saw in Section 1.3.2 and in Figure 1.5 how the orbital angular momentum of an electron can be represented by a vector, the direction of which is determined by the right-hand screw mle. 

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... 

Flowever, the values of the total orbital angular momentum quantum number, L, are limited or, in other words, the relative orientations of f j and 2 are limited. The orientations which they can take up are governed by the values that the quantum number L can take. L is associated with the total orbital angular momentum for the two electrons and is restricted to the values... 

It can be shown quite easily that, for a filled sub-shell such as 2p or L = 0. Space quantization of the total orbital angular momentum produces 2L - - 1 components with M] = L, L —, —L, analogous to space quantization of f. In a filled sub-shell... 

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