Commutators angular momentum

The quantum generalization of the APR Hamiltonian results after supplementing this classical Hamiltonian with a non-commuting angular momentum part [Lj, p] = -ihSji which introduces quantum dispersion and thus qualitatively new effects due to additional fluctuations and tunnehng. 

The reason there are so many Hund s cases is that each Hund s case corresponds to an arrangement of terms in H in order of relative importance. For each arrangement there is a different Hund s case. Since each Hund s case is associated with a complete set of commuting angular momentum operators, explicitly defined transformations between any two Hund s case basis sets may be specified independent of the details of a particular molecular example. 

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. 

Moreover, because there are only two eigenstates, it follows from the completeness property, the vanishing of (n VQ// n) and the angular momentum commutation relations that... 

The presence of two angular momenta has as a consequence that only their sum, representing the total angular momentum in the case considered, necessary commutes with the Hamiltonian of the system. Thus only the quantum number K, associated with the sum, N, of and Lj,... 

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. 

As illustrated above, any p2 configuration gives rise to iD , and levels which contain nine, five, and one state respectively. The use of L and S angular momentum algebra tools allows one to identify the wavefunctions corresponding to these states. As shown in detail in Appendix G, in the event that spin-orbit coupling causes the Hamiltonian, H, not to commute with L or with S but only with their vector sum J= L +... 

Because the total angular momentum P still commutes with Hj-ot, each such eigenstate will contain only one J-value, and hence Tn can also be labeled by a J quantum number ... 

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. 

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... 

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 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. 

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... 

The addition of similar relations for the x- and -components of these angular momentum veetors leads to the result that [J, L S] = 0, so that J and L S commute. Furthermore, we may easily show that commutes with L S because eaeh term m = I + S +2 u S commutes with L S. Thus, J and commute with H in equation (7.33) and J and are eonstants of motion. 

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