One-dimensional Wave Functions Research Articles

Efficient solution of the non-unitary time-dependent Schrodinger equation on a quantum computer with complex absorbing potential

We explore the possibility of adding complex absorbing potential at the boundaries when solving the one-dimensional real-time Schrödinger evolution on a grid using a quantum computer with a fully quantum algorithm described on a n qubit register. Due to the complex potential, the evolution mixes real- and imaginary-time propagation and the wave function can potentially be continuously absorbed during the time propagation. We use the dilation quantum algorithm to treat the imaginary-time evolution in parallel to the real-time propagation. This method has the advantage of using only one reservoir qubit at a time, that is measured with a certain success probability to implement the desired imaginary-time evolution. We propose a specific prescription for the dilation method where the success probability is directly linked to the physical norm of the continuously absorbed state evolving on the mesh. We expect that the proposed prescription will have the advantage of keeping a high probability of success in most physical situations. Applications of the method are made on one-dimensional wave functions evolving on a mesh. Results obtained on a quantum computer identify with those obtained on a classical computer. We finally give a detailed discussion on the complexity of implementing the dilation matrix. Due to the local nature of the potential, for n qubits, the dilation matrix only requires 2n CNOT and 2n unitary rotation for each time step, whereas it would require of the order of 4n+1 C-NOT gates to implement it using the best-known algorithm for general unitary matrices.

Quantum chaos in supersymmetric quantum mechanics: An exact diagonalization study

We use exact diagonalization to study energy level statistics and out-of-time-order correlators (OTOCs) for the simplest supersymmetric extension $\hat{H}_S = \hat{H}_B \otimes I + \hat{x}_1 \otimes \sigma_1 + \hat{x}_2 \otimes \sigma_3$ of the bosonic Hamiltonian $\hat{H}_B = \hat{p}_1^2 + \hat{p}_2^2 + \hat{x}_1^2 \, \hat{x}_2^2$. For a long time, this bosonic Hamiltonian was considered one of the simplest systems which exhibit dynamical chaos both classically and quantum-mechanically. Its structure closely resembles that of spatially compactified pure Yang-Mills theory. Correspondingly, the structure of our supersymmetric Hamiltonian is similar to that of spatially compactified supersymmetric Yang-Mills theory, also known as the BFSS model. We present numerical evidence that a continuous energy spectrum of the supersymmetric model leads to monotonous growth of OTOCs down to the lowest temperatures, which is also expected for the BFSS model from holographic duality. This growth is saturated by low-energy eigenstates with effectively one-dimensional wave functions and a completely non-chaotic energy level distribution. We observe a sharp boundary separating these low-energy states from the bulk of chaotic high-energy states. Our data suggests, although with a limited confidence, that at low temperatures the OTOC growth might be exponential over a finite range of time, with the corresponding Lyapunov exponent scaling linearly with temperature. In contrast, the gapped low-energy spectrum of the bosonic Hamiltonian leads to oscillating OTOCs at low temperatures without any signatures of exponential growth. We also find that the OTOCs for the bosonic Hamiltonian are never sufficiently close to the classical Lyapunov distance. On the other hand, the OTOCs for the supersymmetric system agree with the classical limit reasonably well over a finite range of temperatures and evolution times.

Bending wavefunctions for linear molecules

The bending motion of a linear triatomic molecule has two unique characteristics: the bending mode is doubly degenerate and only positive values of the bending angle, expressed by the bond angle supplement ρ¯, can be observed. The double degeneracy requires the wavefunction to be described as a two-dimensional oscillator. In the present work, we first review the conventional expressions based on two, symmetrically equivalent normal coordinates. Then we discuss an alternative expression for the bending wavefunction in terms of two geometrical coordinates, the bond angle supplement ρ¯ (=π-τ⩾0, where τ is the bond angle) and the rotation angle χ (0⩽χ<2π) describing rotation of the molecule around the molecular axis. In this formalism, defined for the (ρ¯,χ) polar-coordinate space with volume element ρ¯dρ¯dχ, the one-dimensional wavefunction resulted through re-normalization for χ has zero amplitude at ρ¯=0, and the ro-vibrational average of the bending angle, i.e., the expectation value 〈ρ¯〉, attains a non-zero, positive value for any ro-vibrational state including the vibrational ground state. This conclusion appears to cause some controversy since much conventional spectroscopic wisdom insists on 〈ρ¯〉 having the value zero.

Coherent States of Harmonic and Reversed Harmonic Oscillator

A one-dimensional wave function is assumed whose logarithm is a quadratic form in the configuration variable with time-dependent coefficients. This trial function allows for general time-dependent solutions both of the harmonic oscillator (HO) and the reversed harmonic oscillator (RO). For the HO, apart from the standard coherent states, a further class of solutions is derived with a time-dependent width parameter. The width of the corresponding probability density fluctuates, or "breathes" periodically with the oscillator frequency. In the case of the RO, one also obtains normalized wave packets which, however, show diffusion through exponential broadening with time. At the initial time, the integration constants give rise to complete sets of coherent states in the three cases considered. The results are applicable to the quantum mechanics of the Kepler-Coulomb problem when transformed to the model of a four-dimensional harmonic oscillator with a constraint. In the classical limit, as was shown recently, the wave packets of the RO basis generate the hyperbolic Kepler orbits, and, by means of analytic continuation, the elliptic orbits are also obtained quantum mechanically.

Tunneling through a barrier with the phase–amplitude method

A previous study (Rawitscher, 2015) of the solution of Milne’s non linear equation for the phase and amplitude of a one-dimensional wave function is extended to the case where the incident energy is less than the potential (Barrier region). The numerical method again consists in implementing a spectral expansion of the amplitude in terms of a number of Chebyshev polynomials. The method is applied to a Morse-type potential, for energies in a resonance region, and for one energy below the resonance region, and a strong repulsive Coulomb potential. The results are compared with highly accurate direct solutions of the Schrödinger equation, and were found to be accurate to 1:10−6.

Accurate wave functions in the presence of long-ranged potentials

For attractive potentials of the form 1/r 3 or 1/r obtaining accurate wave functions at long distances can be numerically challenging because of the many oscillations involved. In this case high accuracy can be achieved using the phase-amplitude representation of the one-dimensional wave function.

How quantum are non-negative wavefunctions?

We consider wavefunctions which are non-negative in some tensor product basis. We study what possible teleportation can occur in such wavefunctions, giving a complete answer in some cases (when one system is a qubit) and partial answers elsewhere. We use this to show that a one-dimensional wavefunction which is non-negative and has zero correlation length can be written in a “coherent Gibbs state” form, as explained later. We conjecture that such holds in higher dimensions. Additionally, some results are provided on possible teleportation in general wavefunctions, explaining how Schmidt coefficients before measurement limit the possible Schmidt coefficients after measurement, and on the absence of a “generalized area law” [D. Aharonov et al., in Proceedings of Foundations of Computer Science (FOCS) (IEEE, 2014), p. 246; e-print arXiv.org:1410.0951] even for Hamiltonians with no sign problem. One of the motivations for this work is an attempt to prove a conjecture about ground state wavefunctions which have an “intrinsic” sign problem that cannot be removed by any quantum circuit. We show a weaker version of this, showing that the sign problem is intrinsic for commuting Hamiltonians in the same phase as the double semion model under the technical assumption that TQO-2 holds [S. Bravyi et al., J. Math. Phys. 51, 093512 (2010)].

CORRECTION OF THE GROUND STATE ENERGY OF ONE-DIMENSIONAL GROSS-PITAEVSKII EQUATION WITH GAIN-LOSS TERM

We consider the correction of ground state energy of one-dimensional Gross-Pitaevskii equation by adding a gain-loss term as a time-dependent external potential. The interesting purpose of this term is that it can be used to explain the experimental results especially in the nonlinear fiber optics regarding the pulse propagation and collapse-revival of the condensate in the Bose-Einstein condensation. In the Bose-Einstein condensation itself, the function can represent that condensate can interact with the normal atomic cloud. Some analytical solutions have been obtained by choosing an ansatz solution of the wave function and its solution can be dark or bright soliton. Since the Gross-Pitaevskii equation can be treated as a macroscopic quantum oscillator, we can use time-dependent perturbation theory as in ordinary quantum mechanics to find the ground state energy correction if we assume other terms to be very small. In addition, time-dependent potential allows a transition from one energy level to others. In this case, we expand the solution of nonstationary one-dimensional wave function as a linear superposition of harmonic oscillator normalized eigen functions. To get the recursive formulas, we suggest an option to formulate the coefficients after inserting the initial condition which must be satisfied such as in quantum mechanics.

Ginzburg–Landau equations induced from multi-dimensional bichromatic waves

In this paper we consider an envelope function for the multi-dimensional bichromatic wave u b ( x , t ) defined by a Fourier transformation and show that it satisfies a kind of Ginzburg–Landau equation under some conditions for the spectrum function S ( k ) and the angular function ω ( k ) . The result is an extension of Nohara’s work [B.T. Nohara, Derivation and consideration of governing equations of envelope surface created by directional, nearly monochromatic waves, International Journal of Nonlinear Dynamics and Chaos in Engineering Systems 31 (4) (2003) 375–392; B.T. Nohara, Governing equations of envelope surface created by directional, nearly monochromatic waves, Journal of Japan Society of Industrial and Applied Mathematics 13 (2) (2003) 213–224 (in Japanese)] which treated one-dimensional wave functions.

Vector solitons in nearly one-dimensional Bose-Einstein condensates

We derive a system of nonpolynomial Schr\"odinger equations for one-dimensional wave functions of two components in a binary self-attractive Bose-Einstein condensate loaded in a cigar-shaped trap. The system is obtained by means of the variational approximation, starting from the coupled three-dimensional (3D) Gross-Pitaevskii equations and assuming, as usual, the factorization of 3D wave functions. The system can be obtained in a tractable form under a natural condition of symmetry between the two species. A family of vector (two-component) soliton solutions is constructed. Collisions between orthogonal solitons (ones belonging to the different components) are investigated by means of simulations. The collisions are essentially inelastic. They result in strong excitation of intrinsic vibrations in the solitons, and create a small orthogonal component (``shadow") in each colliding soliton. The collision may initiate collapse, which depends on the mass and velocities of the solitons.

Investigation of the dynamics of two coupled oscillators with mixed quantum-classical methods

The dynamics of two coupled oscillators can become quite complex if anharmonic potential energy functions are employed. This type of system therefore represents a good model for an investigation of the performance of mixed quantum-classical methods. In this work, the motion of two coupled particles with a mass ratio of one to ten is studied with three different mixed quantum-classical methods in the presence of anharmonic potential terms for a comparison with exact quantum mechanical calculations. The mixed quantum-classical approaches include the multitrajectory Ehrenfest, the mixed quantum-classical Bohmian (MQCB), and the so-called coupled Schrödinger equations (CSE) formalisms. The analysis shows that while the description of a weakly anharmonic system by the Ehrenfest and MQCB schemes is accurate if proper sampling techniques are applied, both approximations break down rapidly if the anharmonic terms are increased. The performance of the simple CSE prescription, which corresponds to a reduction of the full two-dimensional wave function to two one-dimensional wave functions representing two quantum oscillators coupled via the potential energy in a classical fashion, decreases if the width of the initial wave packet is enlarged. The dependence of the CSE method on the diffuseness of the initial wave packet is therefore opposite to that of the MQCB method, which is more accurate for wide wave packets. Overall, the multitrajectory Ehrenfest ansatz is found to be most successful in reproducing the exact quantum results.

Can resonances occur in the photodissociation continuum of a diatomic molecule? The role of potential discontinuities

Continuum resonances are standard fare in the instructional literature for quantum mechanics, where they arise from the continuity conditions imposed on one-dimensional wavefunctions for piecewise-constant potential energy functions. Such resonance structure weakens progressively as the discontinuity in the potential is smoothed, showing that the structure is specifically attributable to the discontinuity. Since diatomic molecular potential energy curves seldom vary rapidly on the distance scale of the period of the wavefunction, such continuum resonances are not expected in absorption continua. A historically interesting prediction of such structure in the Schumann–Runge continuum (B ← X) of O2 is attributed to the inadvertent incorporation of discontinuity in the B-state potential curve employed in the computations.Key words: quantum mechanics, continuum resonance, diatomic absorption, photodissociation continuum, numerical methods.

An exactly solvable model where properties of a fermion assembly are dominated by the highest occupied level: case of harmonic confinement inddimensions

It is shown explicitly that the fermion particle density (r) in d dimensions for isotropic harmonic confinement and odd d is determined by the one-dimensional wavefunctions of the highest occupied state. Knowledge of the Dirac density matrices for d = 1 and 2 suffices to completely determine the general d-dimensional matrices.

One-Dimensional Relativistic Problems on the Bound States and Degradation for a Superposition of Two δ Potentials

For a superposition of two δ potentials, exact one-dimensional wave functions of the bound states with quantization conditions and exact wave functions of the degradation state with transmission and reflection coefficients have been found. The phenomenon of complete reflection taking place for potentials of this type is investigated.

Fourier grid Hamiltonian multiconfigurational self-consistent-field: A method to calculate multidimensional hydrogen vibrational wavefunctions

The Fourier Grid Hamiltonian Multiconfigurational Self-Consistent-Field (FGH-MCSCF) method for calculating vibrational wavefunctions is presented. This method is designed to calculate multidimensional hydrogen nuclear wavefunctions for use in mixed quantum/classical molecular dynamics simulations of hydrogen transfer reactions. The FGH-MCSCF approach combines a MCSCF variational method, which describes the vibrational wavefunctions as linear combinations of configurations that are products of one-dimensional wavefunctions, with a Fourier grid method that represents the one-dimensional wavefunctions directly on a grid. In this method a full configuration interaction calculation is carried out in a truncated one-dimensional wavefunction space [analogous to complete active space self-consistent-field (CASSCF) in electronic structure theory]. A state-averaged approach is implemented to obtain a set of orthogonal multidimensional vibrational wavefunctions. The advantages of the FGH-MCSCF method are that it eliminates the costly calculation of multidimensional integrals, treats the entire range of the hydrogen coordinates without bias, avoids the expensive diagonalization of large matrices, and accurately describes ground and excited state hydrogen vibrational wavefunctions. This paper presents the derivation of the FGH-MCSCF method, as well as a series of test calculations on systems comparing its performance with exact diagonalization schemes.

Schröedinger.m: A Mathematica Package for Solving the Time-Independent Schrödinger Equation

Schroedinger.m enables the user of Mathematica, a software system available from Wolfram Research, Inc. (), to define a potential and determine bound-state energy eigenvalues for a one-dimensional Hamiltonian. Students new to both quantum mechanics and Mathematica find Schroedinger.m easy to use. It can be used in a junior-level physical chemistry course to provide examples or to provide the means for students in such a course to carry out numerical experiments.The commands defined in the package require only elementary knowledge of Mathematica. However, the more Mathematica one knows, the further one can go with the applications. Knowing even a very few basic Mathematica functions enables a student to use solutions from this package to do some fairly sophisticated computations with one-dimensional wave functions.

Exponentiated random walks, supersymmetry and localization

We examine the nature and properties of the “exponentiated random walk” one-dimensional wavefunction Ψ0=exp[−x(x)], previously introduced in the context of the supersymmetric mappings of a classical Langevin random field problem. Three main results are presented. The first is that the state Ψ0 is extended, although it is the exact groundstate of a disordered one-dimensional quantum problem. The second is that in that problem supersymmetry is neither truly unbroken, or truly broken, we call this a situation of marginal unbroken supersymmetry and identify a class of other problems with the same property. The third result is obtained by studying the local behaviour of the wave function Ψ0 by means of generalized Lyapunov exponents. Locally, Ψ0 exhibits exponential localization, with a localization length identical to that of weak localization in the 1-dimensional Anderson problem.

A finite element approach to molecular vibration

In this thesis, piecewise continuous Hermite cubic polynomials are proposed as basis functions for the solution of one-dimensional, molecular vibrational problems. The technique, referred to as the finite element method as discussed by Strang and Fix (1973), introduces greater flexibility into the form of the wavefunction solution than is allowed by global basis functions, and enables a clear demonstration of convergence. The general theory of molecular vibrations is outlined in Chapter Two of this thesis while the finite element method and its application to one-dimensional problems is discussed in Chapter Three. The one-dimensional wavefunction solutions form a basis for multidimensional molecular vibrational problems following the formulation of Carney and Porter (1976), described in Chapter Four. In particular, the triatomic molecule, H„ , is used as a model for the prediction and comparison of vibrational assignments. Two sets of potential surface data have been employed in this thesis to aid in the spectral assignments of H„ and D„ . For comparative reasons, the data of Carney and Porter (1974) was adopted, while for predictive purposes, the data of Burton, von Nagy-Felsobuki, Doherty and Hamilton (1983b) was used. Methods of fitting the potential surface data also affect the results. A sixth order power series fit of the form proposed by Simons, Parr and Finlan (1973) was adapted to the potential surface data. However for reasons discussed further in Chapter Five, this fit was modified to provide an improved physical form. Statistical and physical criteria for evaluating the various fits are presented in Chapter Five along with contour plots and one-dimensional cuts through the surface. Finally, in Chapter Six, the theoretical vibrational + + assignment predictions of H„ and D, are presented. For the H3 results, comparisons are made between the proposed experimental assignments of Steinmetzger (1982) and the various theoretical predictions using different potential surface data and fits, and different basis functions and methods. The concluding Chapter Seven summarises the problems encountered and solutions proposed and the advantages found in adapting the finite element method to multidimensional molecular vibrational problems.