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 closedform 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 Poincare 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 Poincare 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 know very little about species distributions in prokaryotic marine plankton. Such information is very interesting in its own right, and ignorance of it is also beginning to hamper process studies, such as those on viral infection. New DNA- and RNA-based approaches avoid many prior limitations. Here we discuss four such applications: (1) cloning and sequencing of 16S rRNA genes to produce lists of what types of organisms are present; (2) quantification of these individual types in marine samples by nucleic acid hybridization, including single cell fluorescence; (3) quantitative comparison by DNA-DNA hybridization of entire microbial communities in terms of shared common types, without knowledge of community components; and (4) finding cultures that are representative of native communities. Several previously uncharacterized types of bacteria and archaea (probably including novel phyla) are present in marine plankton. Evidence from both the Atlantic and Pacific suggests that as-of-yet uncultivated archaea may dominate the deep sea, and thus may be the most abundant group of organisms on Earth. Such archaea are in surface waters as well, and can be visualized with fluorescent probes and enriched at room temperature with addition of organic nutrients. Community hybridization shows that variability of microbial community compositions in time and space is high. Although most native bacteria do not grow in culture, some proteobacterial cultures appear by genomic hybridization to be representative of certain communities. These and other results indicate the utility of DNA- and RNA-based methods.

Quantum-statistical proofs are given that the isolated (or Kubo) susceptibility tensor is positive indefinite and is bounded from above by the adiabatic susceptibility tensor, while the isothermal susceptibility tensor is positive definite and is bounded from below by the adiabatic susceptibility tensor. The results apply to either the static dielectric or magnetic cases. Biasing fields and permanent dipole moments may be present if desired. Criteria for equality of the various susceptibilities are established. Contact is made with work of Falk, Caspers, Mountain, Klein, Rosenfeld, and Saito.

The m, n matril( element of an arbitrary potential function V(q) in the one-dimensional harmonic-oscillator representation is shown to be given by (ay) is the Fourier transform of V(q). This formula is specialized to the cases where V(q) is given by qi, q i e-i'YQ2, e ia • q , and q-l sin(aXq), wherej is a nonnegative integer and 'Y, z, and A are real parameters. Results are compared, where possible, with previous work.

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