Mutually commuting operators

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,... 

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... 

Now we may think about adding px, Py, Pz, to the above set of operators. The operators H, px, Pv, Pzjfi and Jz do not represent a set of mutual commuting operators. The reason for this is that p,xyJv for which is a consequence of the fact that, in general, rotation and translation operators do not commute as shown in Fig. F.l. 

As indicated earlier, the commutator of with any of the angular momentum component operators happens to be zero, which means that also commutes with L, L, and L. The component operators, though, do not commute with each other. Therefore, the largest set of mutually commuting operators for the rigid rotator problem consists of three, the Hamiltonian, the operator, and any one of the component operators. 

A corollary to Theorem 8.3 is that a set of functions can be found that are simultaneously eigenfunctions of a set of mutually commuting operators. Symmetry operators in Abelian... 

Let us consider first (cf. p. 92) a set of product functions whose factors refer to two systems with mutually commuting operators Sf, Su, Si, Szj. The first and second factors are assumed to be normalized eigenfunctions of the operators with subscripts 1 and 2 respectively, with quantum numbers Si, Mi and S2, M2, and for given values of 5i and S2 there will be (2Si + 1)(2. + 1) possible products. The problem is how to combine these products in order to obtain eigenfunctions of the total spin operators and defined in (4.1.1). For present purposes, noting that S (or = jt, y, z) is a sum of two parts Si , S2 , it is useful to write... 

Note that 7s, fi, 77., and ffj operate on the corresponding degrees of freedom, and hence mutually commute. Note especially that L and N always assume integer values. 

The fact that, L, and H all commute with one another (i.e., are mutually commutative) makes the series of measurements described in the above examples more straightforward than if these operators did not commute. 

In the first experiment, the fact that they are mutually commutative allowed us to expand the 64 % probable eigenstate with L=1 in terms of functions that were eigenfunctions of the operator for which measurement was about to be made without destroying our knowledge of the value of L. That is, because and can have simultaneous eigenfunctions, the L = 1 function can be expanded in terms of functions that are eigenfunctions of both and L. This in turn, allowed us to find experimentally the... 

Because the total Hamiltonian of a many-electron atom or molecule forms a mutually commutative set of operators with S2, Sz, and A = (V l/N )Ep sp P, the exact eigenfunctions of H must be eigenfunctions of these operators. Being an eigenfunction of A forces the eigenstates to be odd under all Pp. Any acceptable model or trial wavefunction should be constrained to also be an eigenfunction of these symmetry operators. 

In this chapter, we applied the product operator formalism of Sprensen et al. (1983) to calculate polarization-transfer functions between two energy-matched spins. However, if three or more spins are coupled, the evolution of the density operator cannot be derived with the simple rules of the standard product operator formalism, since coupling terms such as 7t/]2(2/] /2 ) and 77/23(2/2 /3 ), which appear in the Hamiltonian, do not mutually commute. 

Since all these four operators mutually commute, the total wave function is simultaneously an eigenfunction of H and px, Py, Pz - i.e-. the energy and the momentum of the center of mass can both be measured (without making any error) in a space-fixed coordinate system (see Appendix I available at hooksite.elsevier.com/978-0-444-59436-5). From its definition, the momentum of the center of mass is identical with the total momentum. ... 

Now, we need to determine the set of the operators that all mutually commute. Only then can all the physical quantities to which the operators correspond have definite values when measured. Also, the wave function can be an eigenfunction of all of these operators and it can be labeled by the quantum numbers, each corresponding to an eigenvalue of the operators in question. We cannot choose as these operators the whole set of H, Jx, J, Jz, because as it was shown earlier, Jx, Jy, Jz do not commute among themselves (although they do with H and J ). 

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