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Journal of Physics A Mathematical and General

Ordered operator expansions by comparison

Wolfgang Witschel

Published Feb 1, 1975• 53 citations
Publisher

Abstract

Ordered operator expansions for operators forming physically important low-dimensional Lie algebras are derived in a simple unified way. Starting with the Zassenhaus formula for the disentangling of exponential operators, series expansions of both undisentangled and disentangled exponentials and comparison of the operator coefficients of equal powers of an ordering parameter alpha leads to ordered operator expansions. This 'comparison method' gives an alternative simple derivation of some already known formulae and a number of new formulae in the physical and chemical applications of the harmonic oscillator and for master equation problems with nearest-neighbour transition probabilities. The 'comparison method' cannot be applied to the angular momentum algebra directly. By a slight modification it can be used to derive from one matrix element or trace of J x k J y l J z n all possible combinations k, l, n by simply comparing powers of ordering parameters.

DOI
10.1088/0305-4470/8/2/003

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Ordered operator expansions for operators forming physically important low-dimensional Lie algebras are derived in a simple unified way. Starting with the Zassenhaus formula for the disentangling of exponential operators, series expansions of both undisentangled and disentangled exponentials and comparison of the operator coefficients of equal powers of an ordering parameter alpha leads to ordered operator expansions. This 'comparison method' gives an alternative simple derivation of some already known formulae and a number of new formulae in the physical and chemical applications of the harmonic oscillator and for master equation problems with nearest-neighbour transition probabilities. The 'comparison method' cannot be applied to the angular momentum algebra directly. By a slight modification it can be used to derive from one matrix element or trace of J x k J y l J z n all possible combinations k, l, n by simply comparing powers of ordering parameters.
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.
Elementary parameter-differentiation techniques are developed to systematically derive a wide variety of operator identities, expansions, and solutions to differential equations of interest to quantum physics. The treatment is largely centered around a general closed formula for the derivative of an exponential operator with respect to a parameter. Derivations are given of the Baker-Campbell-Hausdorff formula and its dual, the Zassenhaus formula. The continuous analogs of these formulas which solve the differential equation dY(t)/dt = A(t) Y(t), the solutions of Magnus and Fer, respectively, are similarly derived in a recursive manner which manifestly displays the general repeated-commutator nature of these expansions and which is quite suitable for computer programming. An expansion recently obtained by Kumar and another new expansion are shown to be derivable from the Fer and Magnus solutions, respectively, in the same way. Useful similarity transformations involving linear combinations of elements of a Lie algebra are obtained. Some cases where the product eAeB can be written as a closed-form single exponential are considered which generalize results of Sack and of Weiss and Maradudin. Closed-form single-exponential solutions to the differential equation dY(t)/dt = A(t) Y(t) are obtained for two cases and compared with the corresponding multiple-exponential solutions of Wei and Norman. Normal ordering of operators is also treated and derivations, corollaries, or generalization of a number of known results are efficiently obtained. Higher derivatives of exponential and general operators are discussed by means of a formula due to Poincaré which is the operator analog of the Cauchy integral formula of complex variable theory. It is shown how results obtained by Aizu for matrix elements and traces of derivatives may be readily derived from the Poincaré formula. Some applications of the results of this paper to quantum statistics and to the Weyl prescription for converting a classical function to a quantum operator are given. A corollary to a theorem of Bloch is obtained which permits one to obtain harmonic-oscillator canonical-ensemble averages of general operators defined by the Weyl prescription. Solutions of the density-matrix equation are also discussed. It is shown that an initially canonical ensemble behaves as though its temperature remains constant with a ``canonical distribution'' determined by a certain fictitious Hamiltonian.
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.