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The Philosophical Magazine A Journal of Theoretical Experimental and Applied Physics

Taylor's theorem for shift operators

R. A. Sack

Published Jun 1, 1958• 30 citations
Publisher

Abstract

Abstract An expansion equivalent to Taylor's series is derived for functions of two-operators of the form F(A + B) where B isa shift operator with regard to A; the result is obtained as a special case of a theorem due to Kermack and McCrea. The development in powers of Binvolves finite differences of the function F(A) instead of derivatives as in the case of commuting variables. Exponential functions can be factorized into two terms, each depending on one operator only. The result allows a Gaussian potential energy to be expressed as a product of exponentials involving only creation and destruction operators. Explicit results are derived for the matrix elements of such a Gaussian function in the representations which diagonalize the Hamiltoniana of the harmonic oscillator in one and two dimensions; they can be expressed in terms of Jacobi polynomials (finite hypergeometric series).

DOI
10.1080/14786435808236839

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Abstract An expansion equivalent to Taylor's series is derived for functions of two-operators of the form F(A + B) where B isa shift operator with regard to A; the result is obtained as a special case of a theorem due to Kermack and McCrea. The development in powers of Binvolves finite differences of the function F(A) instead of derivatives as in the case of commuting variables. Exponential functions can be factorized into two terms, each depending on one operator only. The result allows a Gaussian potential energy to be expressed as a product of exponentials involving only creation and destruction operators. Explicit results are derived for the matrix elements of such a Gaussian function in the representations which diagonalize the Hamiltoniana of the harmonic oscillator in one and two dimensions; they can be expressed in terms of Jacobi polynomials (finite hypergeometric series).
An expansion equivalent to Taylor's series is derived for functions of two-operators of the form F(A + B) where B isa shift operator with regard to A; the result is obtained as a special case of a theorem due to Kermack and McCrea. The development in powers of Binvolves finite differences of the function F(A) instead of derivatives as in the case of commuting variables. Exponential functions can be factorized into two terms, each depending on one operator only. The result allows a Gaussian potential energy to be expressed as a product of exponentials involving only creation and destruction operators. Explicit results are derived for the matrix elements of such a Gaussian function in the representations which diagonalize the Hamiltoniana of the harmonic oscillator in one and two dimensions; they can be expressed in terms of Jacobi polynomials (finite hypergeometric series).
As is well known, the binomial theorem is a classical mathematical relation that can be straightforwardly proved by induction or through a Taylor expansion, albeit it remains valid as long as [A,B]=0. In order to generalize such an important equation to cases where [A,B]≠0, an algebraic approach based on Cauchy’s integral theorem in conjunction with the Baker–Campbell–Hausdorff series is presented that allows a partial extension of the binomial theorem when the commutator [A,B]=c, where c is a constant. Some useful applications of the new proposed generalized binomial formula, such as energy eigenvalues and matrix elements of power, exponential, Gaussian, and arbitrary f(x̂) functions in the one-dimensional harmonic oscillator representation are given. The results here obtained prove to be consistent in comparison to other analytical methods.
In this work, the commutator of any two reasonable functions of several pairs of canonical conjugate operators is obtained as a sum of terms of partial derivatives of those functions (equations 9, 10 or 11). When applied to quantum mechanics, first term in the sum is formally equivalent to Poisson bracket in classical mechanics, which is a well-known result. The novelty respect other papers is the type of functions of operators considered: equations mentioned are proved valid when each function is an infinite series of positive and negative powers of these operators, as long as every series converges. Therefore, this formalism can be applied to functions such as the Coulomb potential, where the inverse of radial distance cannot be expressed as a Taylor series.
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.
We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.