Commutation rules angular momentum operators

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. 

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. 

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

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... 

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. 

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... 

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->... 

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. 

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 ... 

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. 

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... 

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... 

The orientational dependence of the crystal-field Hamiltonian V is obtained by considering the properties under rotation of the tensor cf and therefore also the tensor bf Since the terms are the components of an irreducible tensor operator of rank k, they satisfy the same commutation rule with respect to the angular momentum J as the... 

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