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Journal of Mathematical Physics

The Baker-Hausdorff Formula and a Problem in Crystal Physics

George H. Weiss, A. A. Maradudin

Published Jul 1, 1962• 160 citations
Publisher

Abstract

A derivation is given of the Baker-Hausdorff formula for z = ln exey where x and y are noncommuting operators. This result is then used to obtain an expression for z in the case that X = P2 + Q2 and y = αP + βQ, where P and Q are two operators whose commutator [P,Q] = c is a c number. With the aid of this result a very simple derivation of the expression for the intensity of x rays scattered by the thermal vibrations of a crystal is presented.

DOI
10.1063/1.1724280

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A derivation is given of the Baker-Hausdorff formula for z = ln exey where x and y are noncommuting operators. This result is then used to obtain an expression for z in the case that X = P2 + Q2 and y = αP + βQ, where P and Q are two operators whose commutator [P,Q] = c is a c number. With the aid of this result a very simple derivation of the expression for the intensity of x rays scattered by the thermal vibrations of a crystal is presented.
Two operator identities involving a q-commutator, [A,B]?AB+qBA, where A and B are two arbitrary (generally noncommuting) linear operators acting on the same linear space and q is a variable that commutes with these two operators, are formulated, proved and discussed. The first identity is a direct generalization of the Baker?Hausdorff lemma, whereas the second involves the time derivative of the exponential function of a time-dependant operator. It is indicated how these two identities can be used to good advantage to treat the Foldy?Wouthuysen transformation of the Dirac Hamiltonian for a particle in an external electromagnetic field.
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.
Abstract The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y, according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X+Y is not equivalent to e X e Y , but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators {X, Y, Z}, such that [X, Y] = κ Z for scalar κ and cyclic permutations, here it is proven that e a X+b Y is equivalent to e p Z e q X e−p Z for scalar p and q. Extensions for e a X+b Y+c Z are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.