Angular momenta commutation rules

The s, therefore, satisfy angular momentum commutation rules. Since each of these matrices has eigenvalues 1 and 0, they form a representation of the angular momentum operators for spin 1. 

Putting J = in the angular momentum commutation rules (3.75) we can verify that... 

A major difficulty for molecular as opposed to atomic systems arises from the fact that two different reference axis systems are important, the molecule-fixed and the space-fixed system. Many perturbation related quantities require calculation of matrix elements of molecule-fixed components of angular momentum operators. Particular care is required with molecule-fixed matrix elements of operators that include an angular momentum operator associated with rotation of the molecule-fixed axis system relative to the space-fixed system. The molecule-fixed components of such operators have a physical meaning that is not intuitively obvious, as reflected by anomalous angular momentum commutation rules. 

Use the angular momentum commutation rules L, L ] = ifiL, [L LJ = ihLx, [Lj, Lx] = ihLy, [L, l ] = 0, along with the analogous rules for the components of S and J, to prove the commutation relationships (2.45) that hold in the presence of spin-orbit coupling. Take the spin-orbit Hamiltonian to be Hgo = /( )L S, with f r) a function of the radial coordinates only. 

The Cartesian components of Ji and J2 obey the usual angular momentum commutation rules... 

The pictorial vector models of the uncoupled and coupled representations (Fig. E.l) embody the physical consequences of the angular momentum commutation rules. In the uncoupled representation, the vectors Jj and J2 can lie anywhere on their respective cones with their tips on the edges of the cones. Since these cones are invariant to rotations about the z axis, they represent... 

For a coupled spin system, the matrix of the Liouvillian must be calculated in the basis set for the spin system. Usually this is a simple product basis, often called product operators, since the vectors in Liouville space are spm operators. The matrix elements can be calculated in various ways. The Liouvillian is the conmuitator with the Hamiltonian, so matrix elements can be calculated from the commutation rules of spin operators. Alternatively, the angular momentum properties of Liouville space can be used. In either case, the chemical shift temis are easily calculated, but the coupling temis (since they are products of operators) are more complex. In section B2.4.2.7. the Liouville matrix for the single-quantum transitions for an AB spin system is presented. 

Any operator J, which satisfies the commutation rule Eq. (7-18), represents quantum mechanical angular momentum. Orbital angular momentum, L, with components explicitly given by Eq. (7-1), is a special example5 of J. 

The proof of the theorem affirming that J8 is a proper quantum mechanical angular momentum involves only an expansion of (Ji + J2) x (Ji + J2) with subsequent use of the commutation rules for Jj and J2, and the fact that Jj and J2 commute because they act in... 

A little reflection shows that the commutation relationships, recognized as one of the fundamental differences between classical and quantum systems, are common to all forms of angular momentum, including orbital, polarization and spin. It is of interest to note that the eigenvalues for all forms of angular momentum can be obtained directly from the commutation rules, without using special differential operators. To emphasize the commonality, angular momentum M of all forms will be represented here by three linear operators Mx, My and Mz, that obey the commutation rules ... 

Because the a matrices are proportional to angular momentum operators, they satisfy commutation rules of the type... 

The F,s) are characterized by their commutation rules with the z component of the total angular momentum... 

These operators obey commutation rules which are identical to the commutation rules for angular momentum or spin operators, hence the name quasi-spin... 

In quantum mechanics, spin is described by an operator which acts on a spin wavefunction of the electron. In the present case this operator describes an angular momentum with two possible eigenvalues along a reference axis. The first requirement fixes commutation rules for the spin components, and the second one leads to a representation of the spin operator by 2 x 2 matrices (Pauli matrices [Pau27]). One has... 

We shall defer a discussion of symmetry until Section 6.11. The other operator is Fz, which was defined in Eq. 6.2. Because a and /3 are eigenfunctions of Iz, the product functions are eigenfunctions of F,. By using the well-established commutation rules for angular momentum, it can be shown that Fz and X commute, so Eq. 6.20 is applicable. For the two-spin case, Eq. 6.1 shows that the four basis functions are classified according to Fz = 1, 0, or —1. Only (f)2 and 3, which have the same value of Fz, can mix. Thus only X23 and X-i2 might be nonzero all 10 other off-diagonal elements of the secular equation are clearly zero and need not be computed. 

A more-general type of angular momentum operator J obeys the same commutation rules (3.60) as L. These rules follow from rotational invariance. They are... 

Here, we call the attention of the reader that our Eq. (24) in the previous interlude correspond to Eqs. (32) in Ref. [3] for the cartesian components of the angular momentum in the body frame and the inertial frame, respectively, in terms of the Euler angles. Notice that the angles if and

commutation rules from Eq. (22) in Ref. [3], for the analysis of the rotations of asymmetric molecules are as follows ... 

The matrices are so chosen that they are subject to the same commutation rules as the ordinary components of angular momentum ( 5, p. 127) further, in this case the 2 -direction is the specially distinguished direction, in which the spin, represented by the fourth coordinate, has set itself. For is a diagonal matrix with the proper values (diagonal elements) +1, —1 the 2 -component of the angular momentum has therefore one of the fixed values +1 or —-1. On the other hand, and Sy are not diagonal matrices their values are therefore not measurable simultaneously with but are only statistically determinate. 

Orbital hybridization, like the Bohr model of the hydrogen atom in its ground state, is an effort to dress up a defective classical model by the assumption ad hoc of quantum features. The effort fails in both cases because the quantum-mechanics of angular momentum is applied incorrectly. The Bohr model assumes a unit of quantized angular momentum for the electron which is presumed to orbit the nucleus in a classical sense. Quantum-mechanically however, it has no orbital angular momentum. The hybridization model, in turn, spurns the commutation rules of quantized... 

Angular Momentum Components Defined by Normal and Anomalous Commutation Rules. 73... 

The quantum numbers that are listed in any basis set must be eigenvalues of operators that form a set of mutually commuting operators. Watson (1999) analyzes the commutation rules among the magnitude, A2, and molecule frame component, Aa, angular momentum operators, where A = N, N+, 1, and explains why... 

If V were isotropic (i.e., if atom C would have C AB interaction energy independent of f), then of course we might say that there is no coupling between the rotation of C and the rotation of AB. We would have then a separate conservation law for the first and the second angular momentum and the corresponding commutation rules (cf. Chapter 2 and Appendix F available at booksite.elsevier.com/978-0-444-59436-5) ... 

Hund s third rule is a relativistic correction to the first two rules, introducing a splitting of the terms given by the previous rules. The energy operator (Hamiltonian) commutes with the square of the total angular momentum J = L - - S, and therefore, the energy levels depend rather on the total momentum Jp = J J + This means that they depend on the mumal orientation of L and S (this is a relativistic effect due to the spin-orbit coupling in the Hamiltonian). The vectors L and S add in quantum mechanics in a specific way (see... 

The components of the angular momentum operators satisfy the following commutation rules ... 

Simple as they may appear, the classical Hamiltonians developed for rigid rotors in the preceding section are conceptually new. In our discussion of diatomic rotations in Chapter 3, the rotational states JM> were obtained as eigenfunctions of the space-fixed angular momentum operators and P =P 4-Jl + P. The space-fixed angular momentum components J, Jy, p obey the familiar commutation rules... 

To obtain the selection rules on AJ and AM, we exploit the properties of vector operators. All quantities that transform like vectors under three-dimensional rotations have operators exhibiting commutation rules that are identical to those shown by the space-fixed angular momentum operators f,c->... 

In view of this fact we make the assumption that the electron has spin angular momentum, represented by a set of operators S , Sy, S , and which are analogous to the operators Ma , My, M , and for orbital angular momentum and obey the same commutation rules. According to the discussion in the previous paragraph, we assume that there is... 

---