The Infinite Square Well with a Point Interaction: A Discussion on the Different Parameterizations

This chapter presents the two classical self-adjoint extension theories a la von Neumann and a la Kreı̆n, Višik, and Birman. We start with the discussion of the Friedrichs extension, which is in fact an independent quadratic form construction, and after that we present the theory of the Cayley transform for symmetric operators, on which von Neumann’s extension scheme was originally based and is indeed presented here. We also supplement this with the reinterpretation made by N. Dunford and J. T. Schwartz, where the emphasis is rather put on the abstract boundary value problem. Next, we discuss the Kreı̆n transform of positive operators and its role in Kreı̆n’s theory of self-adjoint extensions, that parallels the role of the Cayley transform in von Neumann’s extension theory. We then present in its entirety Kreı̆n’s extension theory of symmetric semi-bounded operators, following the original work by Kreı̆n, and also discussing the Ando-Nishio variant for the characterisation of the Kreı̆n-von Neumann extension. After this, the Višik-Birman parametrisation of self-adjoint extensions is analysed in its original form and in its more typical modern re-parametrisation, and all its fundamental ancillary results are discussed on invertibility of the extensions, their semi-boundedness, their spectrum, and Kreı̆n formula type resolvent identities. A detour on that sub-class of extensions that retain the same Friedrichs lower bound concludes such a review.

We describe the time evolution of a wave function in the infinite square well using a fractional revival formalism, and show that at all times the wave function can be described as a superposition of translated copies of the initial wave function. Using the model of a wave form propagating on a dispersionless string from classical mechanics to describe these translations, we connect the reflection symmetry of the square-well potential to a reflection symmetry in the locations of these translated copies, and show that they occur in a ``parity-conserving'' form. The relative phases of the translated copies are shown to depend quadratically on the translation distance along the classical path. We conclude that the time-evolved wave function in the infinite square well can be described in terms of translations of the initial wave-function shape, without approximation and without any reference to its energy eigenstate expansion. That is, the set of translated initial wave functions forms a Hilbert space basis for the time-evolved wave functions.

In the infinite square well, any wave function will return to its initial state at integer multiples of the revival time. Most quantum systems do not exhibit perfect revivals, but some exhibit partial revivals in which the wave function returns close to its initial state. Subsequent partial revivals usually deteriorate in quality. We discuss the reasons for the perfect revivals in the infinite square well and how a small change in the potential disrupts the revivals. As an example, we examine partial revivals of a Gaussian wave packet in an infinite square well with a step. First-order and second-order perturbation theory show that the rate at which revivals decay depends on the location of the step.

Electron capture and scaling anomaly in polar molecules

Transverse effects in optical systems have been intensively studied in last years/1,2/. But it is rather difficult to analyze resonator diffraction effects and especially to take into account the influence of space inhomogeneities on laser generation in the framework of standard approach. However one can solve this problems in a mathematically sound way using the self-adjoint extension theory. The extension theory technique provides almost analytically describing of different physical processes (see /3,4/ for details and examples). This technique was used in /5/ for chaotic behavior studies of quantum systems.

We construct families of squeezed quantum states on an interval (depending on boundary conditions, we interpret the interval as a circle or as the infinite square potential well) and obtain estimates of position and momentum dispersions for these states. A particular attention is paid to the possibility of proper localization of a particle in nanoscale space domains. One of the constructed family of squeezed states is based on the theta function. It is a generalization of the known coherent and squeezed states on the circle. Also we construct a family of squeezed states based on truncated Gaussian functions and a family of wave packets based on the discretization of an arbitrary continuous momentum probability distribution. The problem of finiteness of the energy dispersion for the squeezed states in the infinite well is discussed. Finally, we perform the limit of large interval length and the semiclassical limit. As a supplementary general result, we show that an arbitrary physical quantity has a finite dispersion if and only if the wave function of a quantum system belongs to the domain of the corresponding self-adjoint operator. This can be regarded as a physical meaning of the domain of a self-adjoint operator.

Quantum mechanics of a particle in an infinite square well under the influence of a time-dependent electric field is reconsidered. In some gauge, the Hamiltonian depends linearly on the momentum operator, which is symmetric but not self-adjoint when defined on a finite interval. In spite of this symmetric part, the Hamiltonian operator is shown to be self-adjoint. This follows from a theorem by Kato and Rellich which guarantees the stability of a self-adjoint operator under certain symmetric perturbations. The result, which has been assumed tacitly by other authors, is important in order to establish the equivalence of different Hamiltonian operators related to each other by quantum gauge transformations.

We study the spin-(1/2 Heisenberg antiferromagnet on an infinite square lattice. The calculational scheme known as ``paired-phonon analysis'' developed for strongly correlated quantum fluids is extended to a ``paired-magnon analysis'' to study quantum antiferromagnets. We define a complete and orthonormal set of multimagnon states and calculate the matrix elements of the Hamiltonian using a separability approximation. Our results obtained by diagonalizing the Hamiltonian matrix analytically are very similar to those obtained in spin-wave theory. We obtain -0.3290, for the ground-state energy per bond in units of the antiferromagnetic coupling and 0.303 for the ground-state staggered magnetization. These results compare well with the best-known estimates -0.334\ifmmode\pm\else\textpm\fi{}0.001 and 0.313, respectively. We derive the analytic form of the ground-state wave function in this approximation and find it to be of the same form as that assumed by Marshall in his variational studies.. The zero-point motion of long-wavelength excitations (spin waves) in the model, however, reflects a long-range tail in our wave function. We discuss the separability approximation by giving quantitative arguments which justify its validity.

A prolate {\gamma}-rigid regime of the Bohr-Mottelson Hamiltonian within the minimal length formalism, involving an infinite square well like potential in {\beta} collective shape variable, is developed and used to describe the spectra of a variety of vibrational-like nuclei. The effect of the minimal length on the energy spectrum and the wave function is duly investigated. Numerical calculations are performed for some nuclei revealing a qualitative agreement with the available experimental data.

If V(r) is homogeneous in the position coordinates, then the Schr\"odinger equation with H(t)=T+\ensuremath{\alpha}(t)V(r) is separable in time if \ensuremath{\alpha}(t) is properly chosen. This fact makes the time evolution of such systems much easier to compute than it would otherwise be. Examples of such systems are the hydrogen atom with time-varying nuclear charge, the harmonic oscillator with a changing spring constant, and the infinite square well with moving walls. If \ensuremath{\alpha}(t) is made periodic by concatenating ``properly chosen'' segments, then a unitary operator U can be defined which evolves the system through one period. If the wave function is expressed as an N-term truncated expansion of eigenfunctions of H(t), then U is an N\ifmmode\times\else\texttimes\fi{}N matrix which can be computed straightforwardly. The properties of U are studied by examining the statistical properties of its eigenvectors and eigenvalues, whose phases are the quasienergies. Numerical examples are given for the Fermi-Ulam cosmic accelerator (FUCA), which is a particle in a one-dimensional box with a moving wall, with various values for relevant parameters, which can lead to Poisson, Gaussian orthogonal ensemble, Gaussian unitary ensemble, or other quasienergy statistics, corresponding to inte- grable, chaotic, and non-time-reversible chaotic situations. Poincar\'e sections for the classical FUCA are also displayed. They show regimes of regular and of chaotic motion, whose correspondence with the statistics of U is sought with limited success.

We consider a point interaction model for two particles in an external field, which is similar to the Skornyakov-Ter-Martirosyan model and is treated using the theory of self-adjoint extensions of symmetric operators. The corresponding energy operator has an infinite set of bound states with the energy values receding to-∞.

We present a general transfer matrix approach for the propagation of guided waves in presence of inhomogeneities or «scatterers». We particularly address the problem of the coupling with radiation modes leading to a leakage of the guided wave to the surrounding bulk medium at each scattering. Examples are surface acoustic waves, electromagnetic or acoustic excitations or evanescent electromagnetic waves near a boundary, guided waves in optical fibers… From symmetry and conservation laws, we obtain, in the case of symmetric scatterers, the general form of the transfer matrix in terms of four independent real parameters. For ID-periodic lattices of identical scatterers, we show that leakage vanishes at the band edge: this coherent effect stems from the complete destructive interference between the converted radiations at each scatterer. This result demonstrates that coherent leakage is deeply different from a usual «dissipation» effect. Finally, we discuss the competition between Anderson localization and coherent leakage in the presence of disorder. Near the band edges and in the presence of disorder, the «attenuation length» due to the leakage is much larger than the localization length. We suggest an experimental situation where these effects could be observed Etude generale de la propagation d'ondes guidees en presence d'inhomogeneites et de diffuseurs. Exemples: ondes acoustiques de surface, excitations electromagnetiques et acoustiques, onde evanescentes proches d'une frontiere, ondes guidees dans les fibres optiques

The spectral line shapes for the HD rotational spectra perturbed by He and Ar gases, which show interference between allowed and collisionally induced dipole transitions, are calculated including the effects of inelastic rotational collisions. The general parametrization of the line shape, including inelastic collisions, requires six independent real parameters as opposed to four in the theory of Herman, Tipping, and Poll [Phys. Rev. A 20, 2006 (1979)]. Semiclassical calculations based on classical trajectories indicate the importance of the inelastic effects and show qualitative agreement with the experimental parameters. However, it is clear that a full explanation of experimental data will eventually require a full quantum treatment.

Tests are described for determining whether certain functions are the boundary values of analytic functions taking the upper half plane into itself. In one theorem the functions are real-valued and defined on an open interval and a single point. In other theorems, the functions are defined on a real interval taking values in an arc γ in the upper half plane. In every case these results are closely related to Loewner's theorem in which the putative boundary values are required to be real-valued. As in Loewner's theory, the quadratic inequalities are finite sums involving at most the value of the function and its first derivative at points. These results are closely related to the Pick-Nevanlinna inequalities for functions defined on a sequence of points in the upper half plane; the Pick-Nevanlinna inequalities characterize those functions that are restrictions of analytic functions taking the upper half plane into itself.

A well-known tool in conventional (von Neumann) quantum mechanics is the self-adjoint extension technique for symmetric operators. It is used, e.g., for the construction of Dirac–Hermitian Hamiltonians with point-interaction potentials. Here we reshape this technique to allow for the construction of pseudo-Hermitian (J-self-adjoint) Hamiltonians with complex point interactions. We demonstrate that the resulting Hamiltonians are bijectively related to the so-called hypermaximal neutral subspaces of the defect Krein space of the symmetric operator. This symmetric operator is allowed to have arbitrary but equal deficiency indices ⟨n, n⟩. General properties of the operators for these Hamiltonians are derived. A detailed study of -operator parametrizations and Krein type resolvent formulae is provided for J-self-adjoint extensions of symmetric operators with deficiency indices ⟨2, 2⟩. The technique is exemplified on 1D pseudo-Hermitian Schrödinger and Dirac Hamiltonians with complex point-interaction potentials.