Standard representations of finite rotation matrices (FRM) are defined by expressions that are wedded to two particular coordinate frames such as, e.g., are involved in the definition of Euler angles. We present here representations for the FRM in invariant tensor forms comprised of vectors defined in a space-fixed coordinate frame $K.\mathrm{}$ Three explicit expressions for a FRM are presented: First, in terms of tensor products of the spherical or Cartesian basis vectors of frame $K,$ second in a differential form containing the tensor products of gradient operators, and, third, as the superposition of so-called “minimal” bipolar harmonics depending on any pair of unit vectors a, b connected with a frame $K.\mathrm{}$ Based on these results for the FRM, the transformation rule for an irreducible tensor set under space rotations may be written in terms of bipolar harmonics. Our results are especially useful for analyzing angular distributions in atomic processes involving a precise accounting of all effects of photon and target polarizations. Four examples are considered as illustrations of the techniques presented. First, an invariant representation of the photon polarization tensor is found in terms of linear and circular polarization degrees of the photon beam. Second, an invariant decomposition of tripolar harmonics of second rank in terms of very simple, rank 2 tensors is presented. Third, a convenient parametrization is proposed for the polarization state multipoles of a polarized atomic target. Fourth, a simple invariant formula is derived for the angular distribution of polarized photons resulting from electric dipole photon emission by an arbitrary polarized atom.

Exact boundary conditions at finite distance for the solutions of the time-dependent Schrödinger equation are derived. A numerical scheme based on the Crank-Nicholson method is proposed to illustrate its applicability in several examples.

We study the application of the semiclassical coherent-state path-integral technique to the problem of scattering. A numerical method based on initial-value trajectories and their stability information is proposed to solve the root search for the complex trajectories that determine the semiclassical propagator. Very good agreement between the numerically exact quantum and semiclassical results for the correlation function can be achieved for scattering from one-dimensional barrier as well as well-type potentials by using just one single complex trajectory at each time.

Partial revivals of wave packets are investigated by means of the action-angle Wigner function, which is formulated in terms of action quantum numbers and angle variables. In the eigenbasis of the Hamiltonian, the phase shifts of the expansion coefficients of the wave packet are approximated by a quadratic function of action quantum numbers. Under this approximation, the evolution of the action-angle Wigner function of the wave packet, similarly to the classical Liouville density, can be viewed as a result of each phase-space point moving along a classical orbit. The partial and full revivals of wave packets are direct consequences of two facts: (i) The action-angle Wigner function is distributed only on discrete tori of the phase space and (ii) phase-space points of nearby tori move against each other with a constant speed.

An eight-component formalism is proposed for the relativistic two-fermion problem. In QED it extends the applicability of the Dirac equation with hyperfine interaction to the positronium case. The use of exact relativistic two-body kinematics entails a $\mathrm{CP}$-invariant spectrum that is symmetric in the total center-of-mass system energy. It allows the extension of recent ${\alpha }^6$ recoil corrections to the positronium case and implies recoil corrections to the fine and hyperfine structures and to the Bethe logarithm.

A model for a quantum register $\mathcal{R}$ made of $N$ replicas of a $d$-dimensional quantum system (cell) coupled with the environment is studied by means of a Born-Markov master equation (ME). Dissipation and decoherence are discussed in various cases in which a subdecoherent encoding can be rigorously found. For the quantum-bit case $\mathrm{}(d=2\mathrm{})\mathrm{}$ we have solved, for small $N,$ the ME by numerical direct integration and studied, as a function of the coherence length ${\xi }_c$ of the bath, fidelity and decoherence rates of states of the register. For large enough ${\xi }_c$ the singlet states of the global su(2) pseudo spin algebra of the register (noiseless at ${\xi }_c=\infty )$ are shown to have a much smaller decoherence rates than the rest of the Hilbert space.

A null test of quantum mechanics is proposed. It is based on an arrangement in which a source sends two signals to two detectors. At each detector one of two different quantities can be measured. If the source satisfies three conditions, all of which can be verified experimentally, then, according to quantum mechanics, a particular outcome is forbidden. This result follows only from the way in which quantum mechanics constructs probabilities, and does not involve any dynamics. Looking for this outcome is a test of quantum mechanics. It is shown how this test can be realized using two-photon down-conversion.

In the probability representation of quantum mechanics, quantum states are represented by a classical probability distribution, the marginal distribution function (MDF), whose time dependence is governed by a classical evolution equation. We find and explicitly solve, for a wide class of Hamiltonians, equations for the Green function of such an equation, the so-called classical propagator. We elucidate the connection of the classical propagator to the quantum propagator for the density matrix and to the Green function of the Schrödinger equation. Within the new description of quantum mechanics we give a definition of coherence solely in terms of properties of the MDF and we test the definition recovering well known results. As an application, the forced parametric oscillator is considered. Its classical and quantum propagator are found, together with the MDF for coherent and Fock states.

In this paper, a method of generalizing the Bell inequality is presented that makes it possible to include detector inefficiency directly in the original Bell inequality. To enable this, the concept of “change of ensemble” will be presented, providing both qualitative and quantitative information on the nature of the “loophole” in the proof of the original Bell inequality. In a local hidden-variable model lacking change of ensemble, the generalized inequality reduces to an inequality that quantum mechanics violates as strongly as the original Bell inequality, irrespective of the level of efficiency of the detectors. A model that contains change of ensemble lowers the violation, and a bound for the level of change is obtained. The derivation of the bound in this paper is not dependent upon any symmetry assumptions such as constant efficiency, or even the assumption of independent errors.

A connection is made between a recently introduced Lagrange-distributed approximating-functional and the Paley-Wiener sampling theorem. The Lagrange-distributed approximating-functional sampling is found to provide much superior results to that of Paley-Wiener sampling. The relations between discrete variable representation and Lagrange-distributed approximating functionals are discussed. The latter is used to provide an even spaced, interpolative grid representation of the Hamiltonian, in which the kinetic energy matrix has a banded, Toeplitz structure. In this paper we demonstrate that the Lagrange-distributed approximating-functional representation is an accurate and reliable representation for use in fast-Fourier-transform wave-packet propagation methods and apply it to the time-independent wave-packet reactant-product decoupling method, calculating state-to-state reaction probabilities for the two-dimensional (collinear) and three-dimensional $\mathrm{}(J=0\mathrm{})\mathrm{}$ ${\mathrm{H}+\mathrm{H}}_2$ reactions. The results are in very close agreement with those of previous calculations. We also discuss the connection between the distributed approximating-functional method and the existing mathematical formalism of moving least-squares theory.

A one-parameter solvable model for three bosons subject to $\delta$-function attractive interactions in one dimension with periodic boundary conditions is studied. The energy levels and wave functions are classified and given explicitly in terms of three momenta. In particular, eigenstates and eigenvalues are described as functions of the model parameter $c$. Some of the states are given in terms of complex momenta and represent dimer or trimer configurations for large negative $c$. The asymptotic behavior for small and large values of the parameter, and at thresholds between real and complex momenta, is provided. The properties of the potential energy are also discussed.

In analogy with its classical counterpart, a noisy quantum channel is characterized by a loss, a quantity that depends on the channel input and the quantum operation performed by the channel. The loss reflects the transmission quality: if the loss is zero, quantum information can be perfectly transmitted at a rate measured by the quantum source entropy. By using block coding based on sequences of $n$ entangled symbols, the average loss (defined as the overall loss of the joint $n$-symbol channel divided by $n$, when $\overrightarrow{n}\infty )$ can be made lower than the loss for a single use of the channel. In this context, we examine several upper bounds on the rate at which quantum information can be transmitted reliably via a noisy channel, that is, with an asymptotically vanishing average loss while the one-symbol loss of the channel is nonzero. These bounds on the channel capacity rely on the entropic Singleton bound on quantum error-correcting codes [Phys. Rev. A 56, 1721 (1997)]. Finally, we analyze the Singleton bounds when the noisy quantum channel is supplemented with a classical auxiliary channel.

In bulk quantum computation one can manipulate a large number of indistinguishable quantum computers by parallel unitary operations and measure expectation values of certain observables with limited sensitivity. The initial state of each computer in the ensemble is known but not pure. Methods for obtaining effective pure input states by a series of manipulations have been described by Gershenfeld and Chuang (logical labeling) [Science 275, 350 (1997)] and Cory et al. (spatial averaging) [Proc. Natl. Acad. Sci. USA 94, 1634 (1997)] for the case of quantum computation with nuclear magnetic resonance. We give a different technique called temporal averaging. This method is based on classical randomization, requires no ancilla quantum bits, and can be implemented in nuclear magnetic resonance without using gradient fields. We introduce several temporal averaging algorithms suitable for both high-temperature and low-temperature bulk quantum computing and analyze the signal-to-noise behavior of each. Most of these algorithms require only a constant multiple of the number of experiments needed by the other methods for creating effective pure states.

We consider the problem of compression of the quantum information carried by ensemble of mixed states. We prove that for arbitrary coding schemes the least number of quantum bits (qubits) per message needed to convey the signal states asymptotically faithfully is bounded from below by the Holevo function $S\left(\varrho \right)-{\sum }_i{p}_{i}S\left({\varrho }_i\right)$. We also show that a compression protocol can be composed with another one, provided that the latter offers perfect transmission. Such a compound protocol is applied to the case of binary source. It is conjectured to reach the obtained bound. Finally, we point out that in the case of mixed signal states there could be a difference between the maximal compression rates at the coding schemes that are “blind” to the signal and the ones that assume the knowledge about the identities of the signal states.

The interaction of the quadrupole electric moment of the muonic molecules with the Coulomb field of the embedding ${\mathrm{D}}_2$ molecule gives rise to corrections to the energy levels that depend on the spin and significantly modify the muonic molecule fine structure. We present the first-order calculation of the fine and hyperfine structure of the energy levels of the weakly bound states $(dt\mu {)}_{11}$ and $(dd\mu {)}_{11}$ that accounts for the quadrupole finite-size corrections as well. The reported results are of interest for the upcoming high-precision measurement of the energy levels and the formation rate of the muonic molecules $\mathrm{dt}\mu$ and $\mathrm{dd}\mu$.

The doubly excited ${5s5p(}^3{P}_1)np$ autoionizing Rydberg states of In have been studied for principal quantum number $n$ ranging from 16 to 30, lying $\sim 5.3$ eV above the first ionization limit. The experiment was achieved with a resonant three-photon excitation in combination with the isolated core excitation method. The resulting spectra exhibit an occurrence of five series. Precise measurements of the resonance frequencies yield the values of quantum defects and series limits, allowing us to assign them as the autoionizing Rydberg series converging to one of the $F^{\prime }=7/2,$ 9/2, and 11/2 hyperfine structures of the ${\mathrm{In}}^{+}$ $5s5p{}^3{P}_1$ state. The observation of the hyperfine effects of the autoionizing resonances is made possible through their narrow linewidth.

Using angular-momentum-projected mean-field theory, we develop $1/N$ expansion solutions for the vibron model of diatomic molecules. Analytic expressions of spectroscopic accuracy are derived for rotational-vibrational energy levels and for the intensities of transitions among them. The results are used in a systematic study of diatomic molecules in the vibron model with a view to finding appropriate Hamiltonians for a realistic description of rotation-vibration spectra.

We investigate the hydrogen molecule in a strong parallel magnetic field using a fully numerical Hartree-Fock approach. We find that for magnetic fields below $4.2\times {10}^4$ T the ground state of ${\mathrm{H}}_2$ is the strongly bound singlet state ${}^1{\Sigma }_g,$ for magnetic fields stronger than $3\times {10}^6$ T the ground state of the molecule is the strongly bound triplet ${}^3{\Pi }_u,$ and for magnetic fields between $4.2\times {10}^4$ T and $3\times {10}^6$ T the symmetry of the ground state is the triplet state ${}^3{\Sigma }_u,$ which is characterized by repulsion at intermediate internuclear distances and by a weak quadrupole-quadrupole interaction between atoms at large internuclear separation. In this region of magnetic field strength the hydrogen molecule is bound weakly, if at all; the hydrogen atoms behave like a weakly nonideal gas of Bose particles and can form a superfluid phase predicted in earlier works [Korolev and Liberman, Phys. Rev. Lett. 72, 270 (1994)]. For magnetic fields between $\approx 3\times {10}^5$ T and $3\times {10}^6$ T the triplet state ${}^3{\Pi }_u$ is found to be metastable. This state may be responsible for an unknown excitonic line observed experimentally [Timofeev and Chernenko, JETP Lett. 61, 617 (1995)].

A precise measurement of the $5d{}^2{D}_j$ $(j=\frac{3}{2},\frac{5}{2})$ hyperfine structure in atomic Cs is reported. A pump and delayed-probe method based on quantum-beat spectroscopy has been used, whereby the linear polarization degree of the probe signal is determined as a function of probe delay. From the measured polarization beat frequencies for delays up to about 180 ns, the magnetic-dipole and electric-quadrupole coupling constants $A$ and $B$ for the two levels are obtained: $A=-21.24\left(5\right)\mathrm{}\mathrm{MHz}$ and $B=0.2\mathrm{}\left(5\right)\mathrm{}\mathrm{MHz}$ for the $5d{}^2{D}_{5/2}$ level and $A=48.78\mathrm{}\left(7\right)\mathrm{}\mathrm{MHz}$ and $B=0.1\mathrm{}\left(7\right)\mathrm{}\mathrm{MHz}$ for the $5d{}^2{D}_{3/2}$ level.

We propose an alternative method of calculating so-called optimized effective potentials (OEP’s) by directly exploiting the property of the total energy $E$ of an interacting $N$-electron system to attain a minimum for the true potential of the associated Kohn-Sham equations if $E$ is expressed as a functional of the occupied Kohn-Sham orbitals that solve these equations. The method is based on forming the difference between the sought-for true potential and some local spin-density reference potential corrected to yield the known large $r$ behavior of the OEP. This difference is expanded in terms of attenuated sinusoidal functions that decay exponentially beyond the range of orbital localization. By using this expansion $E$ becomes a function of the expansion coefficients whose values are determined by searching for the minimum of $E.\mathrm{}$ This is achieved by employing a variant of a steepest descent method. Due to the flexibility of the method, the exchange-only virial relation can easily be incorporated by performing the minimization in a suitably modified way. The total energy results for a set of atoms (Be-Xe) differ only by about ${10}^{-3}\mathrm{Ry}$ from those obtained by other authors using different techniques. We have also successfully extended our method to the relativistic case. For the treatment of extended systems we propose a combination of our scheme with an existing approximate OEP method.