New Convenient Way for Deriving Exponential Operators' Disentangling Formulas and Their Applications (II)

Abstract

In the preceding paper [Commun. Theor. Phys. 51 (2009) 321] we have recommended a convenient method for disentangling exponential operators in the form of exp{B + C}, trying to find an operator A that satisfies [A, B] = C, and [A,[A,B]] = 0, then from the Baker–Hausdorff formula we have exp{B + C} = exp{B+[A, B]} = eA eB e−A. After arranging eA eB = eB eA eW, the disentangling exp{B + C} = eB eW is obtained. In this work we use this method to two-mode case, especially, derive the normal ordering form of exp[h(a†a + b†b) + ga†b† + kab] without appealing to Lie algebra method.

New Convenient Way for Disentangling Exponential Operator Composed of Angular Momentum Operators

“…( 5), this is equivalent to the disentangling It must be noted that the purpose of this disentangling is not for obtaining a nomal ordering, so this task is different from what we have done in Ref. [2]. We now do it without appealing to the SU(2) Lie algebra theory obeyed by ((a…”

“…Instead, in our preceding paper [1] we have recommended an alternate method for disentangling some exponential operators, it can be summarized as follows. For decomposing exp{B + C}, we try to find an operator A, which satisfies [A, B] = C, and [A, [A, B]] = 0, then from the Baker-Hausdorff formula [2] e A B e…”