Commutator identities

Exercise. For the present example the integral in (4.7b) turns out to be very simple. First prove the commutator identity [IB, W ] = a IB and then derive from it B (t) = eaf B. The result is... 

The /-selection rules are more difficult to obtain. The trick is to take matrix elements of the double commutator identity Eq. (38e). Apparently this trick is due to Dirac, Pauli, and Guttinger (see, e.g., Slater, 1960, Appendix 31). Using Eq. (31), the left side of Eq. (38e) can be expressed as... 

The following general commutator identities are quite useful for simplifying the commutation relations, which occur in the determination of realizations of a Lie algebra. The well-known relations... 

Recall that the commutator of A and B is defined d [A,B] = AB - BA [Eq. (3.7)]. The following commutator identities are helpful in evaluating commutators these identities are easily proved by writing out the commutators in detail (Problem 5.1) ... 

Now we evaluate the commutators of with each of its components, using commutator identities of Section 5.1. 

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... 

Consequently, Eqs. (43) and (59) are identical, for applications in a 3D parameter space, except that the vector product in the former is expressed as a commutator in the latter. Both are computed as diagonal elements of combinations of strictly off-diagonal operators and both give gauge independent results. Equally, however, both are subject to the limitations with respect to the choice of surface for the final integration that are discussed in the sentence following Eq. (43). 

The total Hamiltonian operator H must commute with any pemiutations Px among identical particles (X) due to then indistinguishability. For example, for a system including three types of distinct identical particles (including electrons) like Li2 Li2 with a conformation, one must satisfy the following commutative laws ... 

Let us examine a special but more practical case where the total molecular Hamiltonian, H, can be separated to an electronic part, W,.(r,s Ro), as is the case in the usual BO approximation. Consequendy, the total molecular wave function fl(R, i,r,s) is given by the product of a nuclear wave function X uc(R, i) and an electronic wave function v / (r, s Ro). We may then talk separately about the permutational properties of the subsystem consisting of electrons, and the subsystemfs) formed of identical nuclei. Thus, the following commutative laws Pe,Hg =0 and =0 must be satisfied X =... 

The symbol [R, S] is used to denote what is called the commutator of the operators R and S. There are several useful identities that can be proven for commutators among operators A, B, C, and D, scalar numbers k, and integers n. Several of these are given as follows ... 

R = (i/ r) require translations t in addition to rotations j/. The irreducible representations for all Abelian groups have a phase factor c, consistent with the requirement that all h symmetry elements of the symmetry group commute. These symmetry elements of the Abelian group are obtained by multiplication of the symmetry element./ = (i/ lr) by itself an appropriate number of times, since R = E, where E is the identity element, and h is the number of elements in the Abelian group. We note that N, the number of hexagons in the ID unit cell of the nanotube, is not always equal h, particularly when d 1 and dfi d. 

The 1 operator is the identity, while Py generates all possible permutations of two electron coordinates, Pyi all possible permutations of three electron coordinates etc. It may be shown that the antisymmetrizing operator A commutes with H, and that A acting twice gives the same as A acting once, multiplied by the square root of N factorial. 

Theorem B.—Any four-by-four matrix that commutes with a set of y is a multiple of the identity. 

The proof of this theorem follows from theorem A A four-by-four matrix that commutes with the y commuted with their products and hence with an arbitrary matrix. However, the only matrices that commute with every matrix are constant multiples of the identity. Theorem B is valid only in four dimensions, i.e., when N = 4. In other words the irreducible representations of (9-254) are fourdimensional. 

The requirement that det 8 = 1, implies that Tr T = 0. Note that T is then uniquely determined by this requirement and Eq. (9-383). For assume that there were two such T s that satisfied Eq. (9-383). This difference would then commute with y , and, hence, by theorem A, their difference would be a constant multiple of the identity. But both of these T s can have trace zero only if this constant is equal to zero. This unique T is given by... 

The courve of pH optima determination indicated a presence of acidic exopolygalacturonase form as it was in Fraction A but with slight shift to pH 3.6 (Fig. 2). It was impossible to commute this enzyme form with the acidic exopolygalacturonase from Fraction A because of its molecular mass about 30000 and action pattern identical with form with pH optimum 5.4. Further characterization of this form was not made because of its low content in lyophilizate. 

The odd expansion coefficients are block-adiagonal and hence c j I c I [g.3+ k3] = 0. This means that the coefficient of x on the right hand side is identically zero. (Later it will be shown that 0 and that could be nonzero.) Since the parity matrix commutes with the block-diagonal even coefficients, the reduction condition gives... 

The group contains the identity, E, multiplication by which commutes with all other members of the group (EA = AE) (identity). 

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