Operators, angular momenta commuting

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. 

In any quantum-mechanical problem, we have a set of operators F, <7,... which obey certain relations. (An example is the set of angular-momentum commutation relations.) What we want to show is that the matrices formed from these operators obey the same relations as the operators, so that we can, if we like, work with the matrices instead of with the operators. There are three ways of combining operators we can add two operators we can multiply an operator by a constant we can multiply two operators. We shall examine each process in turn. 

It was shown in Section 1.7 that when the operators Px, PY, Pz °t>ey general angular-momentum commutation relations, as in (5.41), then the eigenvalues of P2 and Pz are J(J+ )h2 and Mh, respectively, where M ranges from — J to J, and J is integral or half-integral. However, we exclude the half-integral values of the rotational quantum number, since these occur only when spin is involved. 

L, S, J AAA L, S, J L, S, J A J j =jl +j2 orbital, spin, and total angular momenta quantum mechanical operators corresponding to L, S, and J quantum numbers that quantize L2, S2, and J2 operator that obeys the angular momentum commutation relations total (j) and individual (ji, j2, ) angular momenta, when angular momenta are coupled... 

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 spin angular-momentum operators obey the general angular-momentum commutation relations of Section 5.4, and it is often helpful to use spin-angular-momentum ladder operators. 

The operator J for the total electronic angular momentum commutes with the atomic Hamiltonian, and we may characterize an atomic state by a quantum number /, which has the possible values [Eq. (11.39)]... 

For Hz, the operator L commutes with H. For a many-electron diatomic molecule, one finds that the operator for the axial component of the total electronic orbital angular momentum commutes with H. The component of electronic orbital angular momentum along the molecular axis has the possible values Mjh, where = 0, 1,... 

For polyatomic molecules the operator 5 for the square of the total electronic spin angular momentum commutes with the electronic Hamiltonian, and, as for diatomic molecules, the electronic terms of polyatomic molecules are classified as singlets, doublets, triplets, and so on, according to the value of 25 + 1. (The commutation of 5 and H holds provided spin-orbit interaction is omitted from the Hamiltonian for molecules containing heavy atoms, spin-orbit interaction is considerable, and 5 is not a good quantum number.)... 

As in ordinary spin algebra, the vector operator T obeys the angular momentum commutation relations. 

The Flamiltonian commutes widi the angular momentum operator as well as that for the square of the angular momentum I . The wavefiinctions above are also eigenfiinctions of these operators, with eigenvalues tndi li-zland It should be emphasized that the total angular momentum is L = //(/ + )/j,... 

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. 

It should be mentioned that if two operators do not commute, they may still have some eigenfunctions in common, but they will not have a complete set of simultaneous eigenfunctions. For example, the and Lx components of the angular momentum operator do not commute however, a wavefunction with L=0 (i.e., an S-state) is an eigenfunction of both operators. 

A. The Hamiltonian May Commute With Angular Momentum Operators... 

The electronic Hamiltonian commutes with both the square of the angular momentum operator r and its z-component and so the three operators have simultaneous eigenfunctions. Solution of the electronic Schrddinger problem gives the well-known hydrogenic atomic orbitals... 

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. 

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

The total momentum operator P, as well as the total angular momentum operator M, commute with H and hence are constants of the motion. However, they do not commute with another, their commutator being equal to... 

Hence U commutes with both position and momentum operators, and must, therefore, depend only on the spin operators. If s is a spin operator then since 8 is similar to an angular momentum operator... 

In non-relativistic Schrodinger theory every component of the orbital angular momentum L = r x p, as well as L2, commutes with the Hamiltonian H = p2/2m + V of a spinless particle in a central field. As a result, simultaneous eigenstates of the operators H, L2 and Lz exist in Schrodinger theory, with respective eigenvalues of E, l(l + l)h2 and mh. In Dirac s theory, however, neither the components of L, nor L2, commute with the Hamiltonian 10. 

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