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Communications in Theoretical Physics

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

Fan Hong-Yi, Liyun Hu

Published Feb 1, 2009• 13 citations
Publisher

Abstract

We recommend a new convenient method for disentangling some exponential operators and derive a set of new operator identities. Especially, we derive the normal odering form of exp [fa † a + ga †2 + ka 2 ] without appealing to Lie algebra method. Application of these formulas in solving some dynamic Hamiltonian is presented.
DOI
10.1088/0253-6102/51/2/26

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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]} = e A e B e −A . After arranging e A e B = e B e A e W , the disentangling exp{B + C} = e B e W 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.
In the preceding paper ( Commun. Theor. Phys. 51 (2009) 321 ) we have recommended a convenient method for disentangling exponential operators. In this work we use this method for disentangling exponential operators composed of angular momentum operators. We mainly disentangle the form of exp[2hJ z + gJ + + kJ − ] as the ordering exp(...J + )exp(...J z )exp(...J − ), we employ the Schwinger Bose realization J − = b † a, J + = a † b, J z = (a † a − b † b)/2 to fulfil this task, without appealing to Lie algebra method. Note that this operator's disentangling is different from its decomposition in normal ordering.
在前面的纸(Commun。Theor。Phys。51 (2009 ) 321 ) 我们为解开指数的操作员推荐了一个方便方法。在这个工作，我们使用这个方法解开由尖动量操作员组成的指数的操作员。我们主要解开 exp 的形式[ 2hJz +gJ++ kJ ]作为订的 exp (...J+) exp (... Jz ) exp (... J )，我们采用 Schwinger Bose 实现 J = ba ，J+= ab ， Jz =( aa bb )完成这项任务的 /2 ，没有呼吁谎言代数学方法。注意这个操作员的解开与它在正常的订的分解不同。
For two operators [Formula: see text] and [Formula: see text] which obey [Formula: see text]we shall prove [Formula: see text]which means we want to merge two exponential operators’ product into one exponential operator. This kind of operator identity is useful for calculating the quantum entropy [Formula: see text] since [Formula: see text] when [Formula: see text] but [Formula: see text] thus the merging operator formula is demanded. In this way, several exponential operators’ product can also be merged. We shall employ the method of integration within normally ordered product (IWOP) to derive the merging operator identity.