Defining commutation relations

These relations are called the defining commutation relations for the Lie algebra and the coefficients ciJk are called the structure constants. A different choice of basis leads to a different but equivalent set of structure constants. We also say that the are generators of the Lie algebra. The rather difficult problem of classifying different Lie algebras will not be considered here (see, e.g., Wybourne, 1974 Gilmore, 1974), since those we shall use arise quite naturally from physical considerations. 

Once a Lie algebra has been defined in the abstract sense via the defining commutation relations Eq. (3), it is of practical interest to find physical realizations of the generators in terms of position and momentum operators, which also satisfy these defining commutation relations. We shall call such a set of concrete operators a realization of the Lie algebra. In practice, as we shall see, we often work backwards by starting with a set of concrete operators... 

As an example consider the important three-dimensional Lie algebra often denoted as so(3) or su(2), whose defining commutation relations can be cast in the form... 

Well-known realizations of the generators of this Lie algebra are given by the three components of the orbital angular momentum vector L = r x p, the three components of the spin S = a realized in terms of the Pauli spin matrices (Schiff, 1968), or the total one-electron angular momentum J = L + S. The components of each of these vector operators satisfy the defining commutation relations Eq. (4) if we use atomic units. We should also note that the vector cross-product example mentioned earlier also satisfies Eq. (4) if we define E = iey, j = 1, 2, 3. 

Given a Lie algebra with defining commutation relations Eq. (3), we can consider the generators , as operators acting on some n-dimensional vector space W. If j> i = 1,..., n is a basis set for W then... 

Thus, they are closed under commutation and form a three-dimensional Lie algebra. By taking the appropriate linear combinations of these operators we can obtain operators satisfying the so(2, 1) defining commutation relations,... 

Except for the factor —2H, Eqs. (162a)-(162c) are the defining commutation relations for the Lie algebra so(4) given in Section IV. ... 

Since there is a large number of defining commutation relations for so(4, 2) it is desirable to have a compact notation for them. The commutation relations can be expressed in a convenient form if we define an antisymmetric set of operators... 

Since the electronic eigenvalues (the adiabatic PESs) are uniquely defined at each point in configuration space we have m(0) = m(P), and therefore Eq. (32) implies the following commutation relation ... 

The commutation relations involving operators are expressed by the so-called commutator, a quantity which is defined by... 

A tensor operator under the algebra G 3 G, T, is defined as that operator satisfying the commutation relations... 

It follows from the fermion commutation relations that the entries of a fe-matrix are related by a system of linear equalities. For example, consider the pair transport operator = 2 b a a b + h ala b ), which moves a spin-up, spin-down pair of electrons between sites p,v of A. If we define w o = ... 

It can be shown from the commutation relations that a function cannot be an eigenfunction of more than one component of M, but can be an eigenfunction of M2 and one component of M simultaneously. To find expectation values from these wave functions, we need to define an operation of integration. We shall let the symbol (9A m > Mx9Am) represent such an operation. For electron-orbital motion where s a function of xyz, this operation becomes... 

To define a Lie algebra homomorphism, it suffices to define it on basis elements of 01 and check that the commutation relations are satisfied. Because the homomorphism is linear, it is defined uifiquely by its value at basis elements. Because the bracket is linear, if the brackets of basis elements satisfy the equality in Definition 8.7, then any linear combination of basis elements will satisfy equality in Definition 8.7. 

The operator a(i) in the Heisenberg algebra, of course, corresponds to the operator Pa[i constructed in Chapter 8. But our commutator relation (8.14) differs from the standard one, we need to modify operators. In fact, it is more natural to change also the sign of the bilinear form. Hence we define... 

A vectorial product will be defined below by (5.14), and V as a tensor of first rank is defined by (2.12). Operator L may be defined also in a more general way by the commutation relations of its components. Such a definition is applicable to electron spin s, as well. Therefore, we can write the following commutation relations between components of arbitrary angular momentum j ... 

Further we shall show that such linear combinations have a certain rank in the quasispin space of n lN2n2lN2 configuration. The commutation relations between tensors (17.8) or (17.9) are completely defined by the anticommutation relations for creation and annihilation operators. These relations can be written as... 

A new momentum operator Pj a must therefore be introduced, defined in such a way to be canonically conjugated to A.y-/V through the commutation relations... 

Finally, we make a few additional remarks. First, note that a pure number state is a3j= state whose phase 0k is evenly distributed between 0 and 2n. This is a consequence of the commutation relation [3] between Nk and e,0 <. Nevertheless, dipole mafKi w elements calculated between number states are (as all quantum mechanical amplitudes) well-defined complex numbers, and as such they have well-defined phajje j S Thus, the phases of the dipole matrix elements in conjunction with the mode ph f i f/)k [Eq. (12.15)] yield well-defined matter + radiation phases that determine the outcome of the photodissociation process. As in the weak-field domain, if only gJ one incident radiation mode exists then the phase cancels out in the rate expres4<3 [Eq. (12.35)], provided that the RWA [Eqs. (12.44) and (12.45)] is adoptedf However, in complete analogy with the treatment of weak-field control, if we irradh ate the material system with two or more radiation modes then the relative pb between them may have a pronounced effect on the fully interacting state, phase control is possible. 

The operation must be understood as hermitian conjugation on B and P.) In addition, the new operators P, P must satisfy the boson commutation relation satisfied by B and B The relations (1.13) may also be expressed in a compact form through the use of a matrix q>, defined as... 

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