Commutative operations

In a more favourable case, the wavefiinction ]i might indeed correspond to an eigenfiinction of one of the operators. If = //, then a measurement of A necessarily yields and this is an unambiguous result. Wliat can be said about the measurement of B in this case It has already been said that the eigenfiinctions of two commuting operators are identical, but here the pertinent issue concerns eigenfunctions of two operators that do not conmuite. Suppose / is an eigenfiinction of A. Then, it must be true that... 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

The fact that two operators commute is of great importance. It means that once a measurement of one of the properties is carried out, subsequent measurement of that property or of any of the other properties corresponding to mutually commuting operators can be made without altering the system s value of the properties measured earlier. Only subsequent measurement of another property whose operator does not commute with F,... 

On the other hand, as detailed further in Appendix C, if the two properties (F and G) do not commute, the second measurement destroys knowledge of the first property s value. After the first measurement, P is an eigenfunction of F after the second measurement, it becomes an eigenfunction of G. If the two non-commuting operators properties are measured in the opposite order, the wavefunction first is an eigenfunction of G, and subsequently becomes an eigenfunction of F. 

The inverter drive system that uses a current-controlled rectifier and parallel-capacitor commutation operates to both improve reliability and reduce cost. Such systems are built commercially for the ranges from 20 to 500 hp for the typical 20 1 constant-torque speed range. 

In this regard, we refer the readers to a few examples of Section 4 in which equality (58) holds true for commutative operators AaAp = Ap Aa-b) If the operators AaP) and A/j t) are non-commutative, then estimate... 

Before going further in more detail on this point, it is worth mentioning that the factorized operator (27) is self-adjoint and positive B = B > 0. To decide for yourself whether the obtained results are acceptable for commutative operators Ay and A2 and conditions (5)-(6), the first step is to discover from (30) the structure of the transition operator of scheme (26) such as... 

Remark 2 If A is a sum of p > 2 pairwise commutative operators such that... 

Relations (44,45) describe the general form of the N-order condition However, some terms must be eliminated from relation (45) because they do not occur when the anticommutation/commutation operations are carried out explicitly. We call these terms spin — forbidden because in all of them the spin correspondence which should exist between the creator and the annihilators forming the p-RO (which generates the p-RDM) is not maintained. These spin-forbidden terms are those having a transposition of at least two indices in their p-RDM. For instance ... 

Since commuting operators have simultaneous eigenfunctions, it follows that correct eigenfunctions of the BO Hamiltonian must also be eigenfunctions of and Sz with eigenvalues S (S + 1) andM5 = S,S 1,., —S. All 2S + 1 members of a... 

We now prove the converse, namely, that eigenfunctions of commuting operators can always be constructed to be simultaneous eigenfunctions. Suppose that Afi = atfi and that [A, 5] = 0. Since A and B commute, we have... 

We are now ready to obtain the set of simultaneous eigenfunctions for the commuting operators N and H. The ground-state eigenfunction 0) has already been determined and is given by equation (4.31). The series of eigenfunctions 11), 2),... are obtained from equations (4.34b) and (4.18b), which give... 

In this particular case, the algebra (2.5) is trivial, since the operator Jz obviously commutes with itself. Algebras formed by commuting operators are called Abelian. 

For a unihed representation, we introduce the space-dependent commutator operator... 

The operators x and commute with each other The condition for the two commuting operators is... 

As noted earlier, a cyclic group is Abelian, and each of its h elements is in a separate class. Therefore, it must have h one-dimensional irreducible representations. To obtain these there is a perfectly general scheme which is perhaps best explained by an example. It will be evident that the example may be generalized. Let us consider the group C5, consisting of the five commuting operations C5, C, C, C5, E we seek a set of five one-... 

Two quantities, represented by commuting operators, possess definite values at the same time. In atomic theory it is very important to find a full set of commuting operators and wave functions, because in this case we can unambiguously describe the system considered. Having defined a full set of wave functions xpt (i = 1,2we are in a position to expand the function of arbitrary state xp in terms of linear combination of these functions of the system considered, i.e. 

Thus, in the central field approximation the wave function of the stationary state of an electron in an atom will be the eigenfunction of the operators of total energy, angular and spin momenta squared and one of their projections. These operators will form the full set of commuting operators and the corresponding stationary state of an atomic electron will be characterized by total energy E, quantum numbers of orbital l and spin s momenta as well as by one of their projections. 

Expressions (15.64) and (15.65) imply that the set of operators W KkK at all possible values of ranks and their respective projections are complete in relation to the commutation operation. 

Bilinear combinations composed of creation and annihilation operators can be expressed in terms of triple tensors W KkK (see (15.59) and (15.61)). In this case, these tensors are generators of the Rsi+4 group, and relationship (15.64) determines the completeness condition for the generator set of this group in relation to the commutation operation. Out of this set we shall single out two subsets of operators - and Each of... 

Thus, with the mlNln2lNl configuration we can do without the wave functions derived using the vectorial coupling of momenta of individual shells and use the functions characterized by eigenvalues of the commuting operators N, T2, Tz, L2, Lz, S2, Sz... 

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