Norm-preserving mappings, effective operators

Effective Operators and Classification of Mapping Operators Effective Operators Generated by Norm-Preserving Mappings Effective Operators Generated by Non-Norm-Preserving Mappings which Produce a Non-Hermitian Effective Hamiltonian... 

C. Effective Operators Generated by Norm-Preserving Mappings... 

We now turn to the effective operator definitions generated by the non-norm-preserving mappings K, L). As mentioned in Section II.B, expression (2.16), in which is replaced by ), applies if the a )o are used, while expression (2.14), with (/) ) replaced by ), applies if the are incorporated into new model eigenfunctions. We first discuss the effective operator expressions that may be obtained from (2.16). 

Section II and Table I show that state-independent effective operators can be obtained with norm-preserving mappings K, or with any of the three kinds of non-norm-preserving mappings K, L), K, L), and K, L). This section first proves that the commutation relations between two arbitrary operators cannot generally be conserved upon transformation to any of these state-independent effective operators. A determination is then made of operators whose commutation relations are preserved by at least some state-independent effective operator definitions, and a few applications are then presented. Particular interest is focused on operators which commute with H, including constants of the motion. 

The key step in deriving (4.7) is the commutation of P with H. Clearly, a similar reasoning applies when replacing H with any operator that commutes with H because such an operator also commutes with P. Therefore, this leads to Theorem V as follows state-independent effective operators produced by norm-preserving mappings conserve the commutation relations between H and an arbitrary operator B and between B and any operator that commutes with H. Given particular choices of P,... 

Hence, the commutation relation between A and B is conserved iff the right hand sides of Eqs. (4.8) and (4.9) are equal to each other, thereby leading to Theorem VII as follows the commutation relation between two operators A and B is preserved upon transformation to state-independent effective operators obtained with norm-preserving mappings iff A and B satisfy... 

Another important application of Theorem V is that (Corollary V.2) the dipole length and dipole velocity transition moments are equivalent when computed with state-independent effective operators obtained with norm-preserving mappings. According to definition A (see Table I), these computations evaluate o( p /8)o, and (a r )3)oWith... 

The derivations of Theorems V and VI for the preservation of [ A, B] consist of (1) the replacement by P of the product KK that is present between A and B in [A, B], (2) the commutation of A with P, and (3) the incorporation of P into K and K. Step (1) uses Eq. (4.6), which is valid only for norm-preserving mappings. With other definitions, however, the products analogous to KK may be replaced by P if (criterion 1) they satisfy the fundamental relation (2.13). Step (2) clearly applies to any effective operator definitions. So does step (3) since P can be combined with any mapping operators using Eqs. (2.3) and (2.6). Hence, all effective operator definitions that fulfill criterion 1 conserve [A, B] if [A, B] =0. When these definitions, like A, produce the associated effec-... 

This is also true for the non-norm-preserving mappings (k, 1) and k, 1) of Table 11 which yield effective operators that are not presented here [71]. 

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