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Exponential Operators and Parameter Differentiation in Quantum Physics

### Abstract

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 *e*^{A}e^{B} 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.

© 1967 The American Institute of Physics

Received 23 August 1966
Published online 21 December 2004

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2004-12-21

2016-09-30

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