Contracted Schrödinger equation in quantum phase‐space

The phase space formulation of quantum mechanics is equivalent to standard quantum mechanics where averages are calculated by way of phase space integration as in the case of classical statistical mechanics. We derive the quantum hierarchy equations, often called the contracted Schrödinger equation, in the phase space representation of quantum mechanics which involves quasi‐distributions of position and momentum. We use the Wigner distribution for the phase space function and the Moyal phase space eigenvalue formulation to derive the hierarchy. We show that the hierarchy equations in the position, momentum, and position‐momentum representations are very similar in structure. © 2017 Wiley Periodicals, Inc.     

Controlling Photons in a Box and Exploring the Quantum to Classical Boundary (Nobel Lecture)

Microwave photons trapped in a superconducting cavity constitute an ideal system to realize some of the thought experiments imagined by the founding fathers of quantum physics. The interaction of these trapped photons with Rydberg atoms crossing the cavity illustrates fundamental aspects of measurement theory. The experiments performed with this “photon box” at Ecole Normale Supérieure (ENS) belong to the domain of quantum optics called “Cavity Quantum Electrodynamics”. We have realized the non‐destructive counting of photons, the recording of field quantum jumps, the preparation and reconstruction of “Schrödinger cat” states of radiation and the study of their decoherence, which provides a striking illustration of the transition from the quantum to the classical world. These experiments have also led to the demonstration of basic steps in quantum information processing, including the deterministic entanglement of atoms and the realization of quantum gates using atoms and photons as quantum bits. This lecture starts by an introduction stressing the connection between the ENS photon box and the ion trap experiments of David Wineland, whose accompanying lecture recalls his own contribution to the field of single particle control. I give then a personal account of the early days of Cavity Quantum Electrodynamics before describing the main experiments performed at ENS during the last twenty years and concluding by a discussion comparing our work to other researches dealing with the control of single quantum particles.     

On the Inversion of Quantum Mechanical Systems: Determining the Amount and Type of Data for a Unique Solution

The inverse problem of extracting a quantum mechanical potential from laboratory data is studied from the perspective of determining the amount and type of data capable of giving a unique answer. Bound state spectral information and expectation values of time-independent operators are used as data. The Schrödinger equation is treated as finite dimensional and for these types of data there are algebraic equations relating the unknowns in the system to the experimental data (e.g., the spectrum of a matrix is related algebraically to the elements of the matrix). As these equations are polynomials in the unknown parameters of the system, it is possible to determine the multiplicity of the solution set. With a fixed number of unknowns the effect of increasing the number of equations on the multiplicity of solutions is assessed. In general, if the number of the equations matches the number of the unknowns, the solution set is denumerable. A result on the solvability of polynomial equations is extended to the case where the number of equations exceeds the number of unknowns. We show that if one has more equations than the number of unknowns, generically a unique solution exists. Several examples illustrating these results are provided.     

Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Within the framework of supersymmetric quantum mechanics method, we study by an algebraic method the arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential by a proper approximation to nonlinear centrifugal term. The explicitly analytical formula of energy levels is derived, and the corresponding bound state wave functions are presented. The function analysis method is used to rederive the same energy levels of the quantum system under consideration to check the validity of this algebraic method. In addition, it is shown from numerical results of energy levels that above certain α parameter depending on the choices of potential parameters V₁ and V₂ the hyperbolical molecular potential cannot trap a particle as it becomes weaker and the energy level starts to turn positive when the potential parameter α becomes larger. © 2014 Wiley Periodicals, Inc.     

Scattering state of the multiparameter potential with an improved approximation for the centrifugal term in D‐dimensions

Recently, multi‐parameter potential has been introduced and had been discussed as special cases of other potential model, that is why we are interested to the study of such a potential. In order to study this potential, the D‐dimensional Schrödinger has been presented in detail and the scattering state with any arbitrary J‐state due to this potential has been investigated approximately. After this step, we have discussed analytically the scattering and bound state for some special cases in D‐dimensional situations which play important roles in physics. © 2015 Wiley Periodicals, Inc.