Commutation. Law

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 =... 

Law One a -j- b = b + a (commutative law of addition). In the synthesis of polymer blends, the polymers (elements) may be added in any order to get the same chemical mixture. Thus, the elements are interchangeable with respect to position, and may be written in any order. 

When dealing with basic arithmetic and combined operations, it is helpful to understand three basic number laws The commutative law, the associative law, and the distributive law. Sometimes these three laws are referred to as properties (such as the Commutative Property). 

Canonically-conjugate observables do not commute. Corresponding to a generalised position coordinate q there is a generalised momentum p. The commutation law is... 

The order of the arrangement of the matrices in products, such as those occurring in Eqs. (62) and (63), must be maintained since the commutative law of multiplication does not hold for matrices in general, i.e., PG GP. 

One exception is the commutative law. In general, matrix multiplication is Non-commutative ... 

These commutation laws (Born and Jordan, 1925) take here the place of the quantum conditions in Bohr s theory. The considerations by which their adoption is justified, as also the further development of matrix mechanics as a formal calculus, are for brevity omitted here. In the next section, however ( 4, p. 121), it will be found that the analogous commutation laws in wave mechanics are mere matters of course. In Appendix XV (p. 291), taking the harmonic oscillator as an example, we show how and why they lead to the right result. 

Operators, or entities which operate on any function, that is, which when applied to this function, generate another function, can be represented in the most diverse ways. Heisenberg s matrices are simply one definite kind of representation of such operators another kind is the set of difhirential coefficients corresponding to the momentum components and the energy. In the latter kind of representation the Born-Jordan commutation laws admit of a simple interpretation here, by what we have just fvien, pq — qp simply means the application of the differential operator... 

The difference between classical mechanics and quantum mechanics, as was explained in the text, lies in the fact that the quantities p and q are no longer regarded as ordinary functions of the time, but stand for matrices, the element q,nm of which denotes the quantum amplitude associated with the transition from one energy-level to another, E. Its square, just like q, the square of the amplitude in classical mechanics, is a measure of the intensity of the line of the spectrum emitted in this transition. When we introduce the matrices into the classical equations of motion, we must also bring in the commutation law... 

The individual terms of the principal diagonal represent the energies of the individual stiites, and in fact the element W n gives the energy of the n-th state. The energy matrix involves in the first place the elements of the not yet completely determined co-ordinate matrix. These may, liowever, be obtained by means of the commutation law fq — qjp -= hj lTri, and the energy levels may then be found. 

If, following the above rules, w(i substitute the co-ordinate matrix and the momentum matrix in the commutation law, wc readily obtain the matrix... 

The so-called fundamental laws of algebra are /. The law of association The number of things in any group is independent of the order. II. The commutative law ... 

These are necessary and sufficient conditions for a set of elements to form a group. It is evident that operations /, A, B, C, D, and E form a group in this sense. It should be noted that the commutative law of multiplication does not necessarily hold. For example. Table 1-2 shows that CD DC. 

Another definition of interest here involves the subalgebra G of a given Lie algebra G. One has to consider a subset G CG that is closed with respect to the same commutation laws of G,... 

The algebraic structure of the product (3.1) can be understood in terms of the separate commutation laws for the two (independent) U,(2) (i = 1, 2) Lie algebras. These are obtained directly from the commutation laws (2.54), which are here extended to include two families of boson... 

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