Quantum Mechanical Three-body Problem Research Articles

Emergent Inflation of the Efimov Spectrum under Three-Body Spin-Exchange Interactions.

We resolve the unexpected and long-standing disagreement between experiment and theory in the Efimovian three-body spectrum of ^{7}Li, commonly referred to as the lithium few-body puzzle. Our results show that the discrepancy arises out of the presence of strong nonuniversal three-body spin-exchange interactions, which enact an effective inflation of the universal Efimov spectrum. This conclusion is obtained from a thorough numerical solution of the quantum mechanical three-body problem, including precise interatomic interactions and all spin degrees of freedom for three alkali-metal atoms. Our results show excellent agreement with the experimental data regarding both the Efimov spectrum and the absolute rate constants of three-body recombination, and in addition reveal a general product propensity for such triatomic reactions in the Paschen-Back regime, stemming from Wigner's spin conservation rule.

Tensor product scheme for computing bound states of the quantum mechanical three-body problem

We develop a computationally and numerically efficient method to calculate binding energies and corresponding wave functions of quantum mechanical three-body problems in low dimensions. Our approach exploits the tensor structure of the multidimensional stationary Schrödinger equation, being expressed as a discretized linear eigenvalue problem. In one spatial dimension, we solve the three-body problem with the help of iterative methods. Here the application of the Hamiltonian operator is represented by dense matrix–matrix products. In combination with a newly-designed preconditioner for the Jacobi–Davidson QR, our highly accurate tensor method offers a significantly faster computation of three-body energies and bound states than other existing approaches. For the two-dimensional case, we additionally make use of a hybrid distributed/shared memory parallel implementation to calculate the corresponding three-body energies. Our novel method is of high relevance for the analysis of few-body systems and their universal behavior, which is only governed by the particle masses, overall symmetries, and the spatial dimensionality. Our results have straightforward applications for ultracold atomic gases that are widespread and nowadays utilized in quantum sensors.

Variational Methods in the Quantum Mechanical Three-Body Problem with a Coulomb Interaction

Variational Methods in the Quantum Mechanical Three-Body Problem with a Coulomb Interaction

Theory of exciton-electron scattering in atomically thin semiconductors

The realization of mixtures of excitons and charge carriers in van-der-Waals materials presents a new frontier for the study of the many-body physics of strongly interacting Bose-Fermi mixtures. In order to derive an effective low-energy model for such systems, we develop an exact diagonalization approach based on a discrete variable representation that predicts the scattering and bound state properties of three charges in two-dimensional transition metal dichalcogenides. From the solution of the quantum mechanical three-body problem we thus obtain the bound state energies of excitons and trions within an effective mass model which are in excellent agreement with Quantum Monte Carlo predictions. The diagonalization approach also gives access to excited states of the three-body system. This allows us to predict the scattering phase shifts of electrons and excitons that serve as input for a low-energy theory of interacting mixtures of excitons and charge carriers at finite density. To this end we derive an effective exciton-electron scattering potential that is directly applicable for Quantum Monte-Carlo or diagrammatic many-body techniques. As an example, we demonstrate the approach by studying the many-body physics of exciton Fermi polarons in transition-metal dichalcogenides, and we show that finite-range corrections have a substantial impact on the optical absorption spectrum. Our approach can be applied to a plethora of many-body phenomena realizable in atomically thin semiconductors ranging from exciton localization to induced superconductivity.

Squeezing the Efimov effect

The quantum mechanical three-body problem is a source of continuing interest due to its complexity and not least due to the presence of fascinating solvable cases. The prime example is the Efimov effect where infinitely many bound states of identical bosons can arise at the threshold where the two-body problem has zero binding energy. An important aspect of the Efimov effect is the effect of spatial dimensionality; it has been observed in three dimensional systems, yet it is believed to be impossible in two dimensions. Using modern experimental techniques, it is possible to engineer trap geometry and thus address the intricate nature of quantum few-body physics as function of dimensionality. Here we present a framework for studying the three-body problem as one (continuously) changes the dimensionality of the system all the way from three, through two, and down to a single dimension. This is done by considering the Efimov favorable case of a mass-imbalanced system and with an external confinement provided by a typical experimental case with a (deformed) harmonic trap.

Permutation-symmetric three-particle hyper-spherical harmonics based on the S3 ⊗ SO(3)rot ⊂ O(2)⊗SO(3)rot ⊂ U(3)⋊S2 ⊂ O(6) subgroup chain

We construct the three-body permutation symmetric hyperspherical harmonics to be used in the non-relativistic three-body Schrödinger equation in three spatial dimensions (3D). We label the state vectors according to the S3⊗SO(3)rot⊂O(2)⊗SO(3)rot⊂U(3)⋊S2⊂O(6) subgroup chain, where S3 is the three-body permutation group and S2 is its two element subgroup containing transposition of first two particles, O(2) is the “democracy transformation”, or “kinematic rotation” group for three particles; SO(3)rot is the 3D rotation group, and U(3),O(6) are the usual Lie groups. We discuss the good quantum numbers implied by the above chain of algebras, as well as their relation to the S3 permutation properties of the harmonics, particularly in view of the SO(3)rot⊂SU(3) degeneracy. We provide a definite, practically implementable algorithm for the calculation of harmonics with arbitrary finite integer values of the hyper angular momentum K, and show an explicit example of this construction in a specific case with degeneracy, as well as tables of K≤6 harmonics. All harmonics are expressed as homogeneous polynomials in the Jacobi vectors (λ,ρ) with coefficients given as algebraic numbers unless the “operator method” is chosen for the lifting of the SO(3)rot⊂SU(3) multiplicity and the dimension of the degenerate subspace is greater than four – in which case one must resort to numerical diagonalization; the latter condition is not met by any K≤15 harmonic, or by any L≤7 harmonic with arbitrary K. We also calculate a certain type of matrix elements (the Gaunt integrals of products of three harmonics) in two ways: 1) by explicit evaluation of integrals and 2) by reduction to known SU(3) Clebsch–Gordan coefficients. In this way we complete the calculation of the ingredients sufficient for the solution to the quantum-mechanical three-body bound state problem.

Two-electron atom with a screened interaction

We present analytical solutions to a quantum-mechanical three-body problem in three dimensions, which describes a helium-like two-electron atom. Similarly to Hooke's atom, the Coulombic electron-nucleus interaction potentials are replaced by harmonic potentials. The electron-electron interaction potential is taken to be both screened (decaying faster than the inverse of the interparticle separation) and regularized (in the limit of zero separation). We reveal the exactly solvable few-electron ground state, which explicitly includes electron correlation, for certain values of the harmonic containment.

Universal description of three two-component fermions

A quantum-mechanical three-body problem for two identical fermions of mass m and a distinct particle of mass m1 in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval ( and ) an additional parameter b determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. The dependence of the bound-state energies on m/m1 and b in the sector of angular momentum and parity is calculated and analysed with the aid of a simple model.

Confinement in the q-state Potts model: an RG-TCSA study

In the ferromagnetic phase of the q-state Potts model, switching on an external magnetic field induces confinement of the domain wall excitations. For the Ising model (q = 2) the spectrum consists of kink-antikink states which are the analogues of mesonic states in QCD, while for q = 3, depending on the sign of the field, the spectrum may also contain three-kink bound states which are the analogues of the baryons. In recent years the resulting "hadron" spectrum was described using several different approaches, such as quantum mechanics in the confining linear potential, WKB methods and also the Bethe-Salpeter equation. Here we compare the available predictions to numerical results from renormalization group improved truncated conformal space approach (RG-TCSA). While mesonic states in the Ising model have already been considered in a different truncated Hamiltonian approach, this is the first time that a precision numerical study is performed for the 3-state Potts model. We find that the semiclassical approach provides a very accurate description for the mesonic spectrum in all the parameter regime for weak magnetic field, while the low-energy expansion from the Bethe-Salpeter equation is only valid for very weak fields where it gives a slight improvement over the semiclassical results. In addition, we confirm the validity of the recent predictions for the baryon spectrum obtained from solving the quantum mechanical three-body problem.

Efimov three-body states on top of a Fermi sea

The stabilization of Cooper pairs of bound electrons in the background of a Fermi sea is the origin of superconductivity and the paradigmatic example of the striking influence of many-body physics on few-body properties. In the quantum-mechanical three-body problem the famous Efimov effect yields unexpected scaling relations among a tower of universal states. These seemingly unrelated problems can now be studied in the same setup thanks to the success of ultracold atomic gas experiments. In light of the tremendous effect of a background Fermi sea on two-body properties, a natural question is whether a background can modify or even destroy the Efimov effect. Here we demonstrate how the generic problem of three interacting particles changes when one particle is embedded in a background Fermi sea, and show that Efimov scaling persists. It is found in a scaling that relates the three-body physics to the background density of fermionic particles.

Three-body calculations of the triple-αreaction

Recently, the triple-\alpha (3\alpha) process, by which three 4He nuclei are fused into a 12C nucleus in stars, was studied by different methods in solving the quantum mechanical three-body problem. Their results of the thermonuclear reaction rate for the process differ by several orders at low stellar temperatures of 10^7 - 10^8 K. In this paper, we will present calculations of the 3\alpha process by a modified Faddeev three-body formalism in which the long-range effects of Coulomb interactions are accommodated. The reaction rate of the process is calculated via an inverse process, three-alpha (3-\alpha) photodisintegration of a 12C nucleus. Calculated reaction rate is about 10 times larger than that of the Nuclear Astrophysics Compilation of Reaction Rates (NACRE) at 10^7 K, and is remarkably smaller than the results of the recent three-body calculations. We will discuss a possible reason of the difference.

Photodetachment of Positronium Negative Ions

Photodetachment of the positronium negative ion, a bound state of one positron and two electrons, has been observed. Development of a method to produce the ions efficiently using a Na coated tungsten surface has enabled the first observation of the photodetachment. The obtained lower limit of the photodetachment cross section for the wavelength of 1064 nm is consistent with the theoretical calculations reported so far. The experimental field developed in the present work gives new opportunities to explore the quantum mechanical three-body problem and to develop energy-tunable positronium beams.

An integral transform method for solving a quantum mechanical three-body problem with exponential-type potentials

An integral transform or generator coordinate method is developed for the variational solution of the quantum mechanical three-body problem with exponentially screened (e.g., Yukawa or Haken) potentials. The method is applied to the specific example of calculation of the binding energy of the ionized donor-Wannier exciton complex in CdS. Convergence properties are tested and compared with results by other methods. The inherent error of Elkomoss' approach involving self-consistent averaging of the exponential factors is estimated. Some prospects for future application of the present method are pointed out.

The negative ion of positronium: measurement of the decay rate and prospects for further experiments

The negative positronium (Ps–) ion consisting of two electrons and a positron (e+e–e–), represents the simplest three-body system with a bound state. Its constituents are stable, point-like particles, and it is essentially free from perturbations by strong interaction effects. Together with the rather unique mass ratio, these properties make the positronium ion an interesting object for studying the quantum-mechanical three-body problem. We present a new determination of the decay rate of Ps–, using a beam-foil method and a stripping-based detection technique. The measured value of Γ = 2.089(15) ns–1 is a factor of six times more precise than the previous experimental result, and there is excellent agreement both with the latter and with the theoretical value. With the new high-intensity positron source NEPOMUC at the FRM-II research reactor in Munich being available, a further improvement in precision seems possible. Moreover, the high flux of low-energy positrons at this facility brings other properties of this exotic system within reach of experiments. The prospects for such investigations are discussed.PACS No.: 36.10Dr

Negative hydrogenic ion orD−center in a quantum wire

The negative hydrogenic ion is not only of physical but also historic importance for it is the first quantum mechanical three-body problem that involves the intrinsic electron-electron correlation. Recent advances in nanotechnology have brought the study of this system in a quantum wire into the forefront. The nonanalytic behaviors of the binding energy ${ϵ}_{b}^{2\mathrm{nd}\phantom{\rule{0.3em}{0ex}}\mathrm{elec}}$ of the second electron in a negative hydrogenic ion in one dimension and in a quantum wire (QWR) as a function of the Coulombic repulsive strength $\ensuremath{\gamma}$ and the wire radius $b$ are first identified. They are then exploited, together with the recognition of the well-known near-infinite (\`a la Loudon) binding energy of the neutral hydrogenic atom, to set up a model that calculates ${ϵ}_{b}^{2\mathrm{nd}\phantom{\rule{0.3em}{0ex}}\mathrm{elec}}$ in a QWR directly, thereby avoiding the pitfalls of a variational approach to this problem. This ${ϵ}_{b}^{2\mathrm{nd}\phantom{\rule{0.3em}{0ex}}\mathrm{elec}}$, found to be $0.4\phantom{\rule{0.3em}{0ex}}\mathrm{Ryd}$, is an exact solution for a wire whose radius is nearly zero. For a finite $b$, it is shown that still ${ϵ}_{b}^{2\mathrm{nd}\phantom{\rule{0.3em}{0ex}}\mathrm{elec}}\ensuremath{\approx}0.4\phantom{\rule{0.3em}{0ex}}\mathrm{Ryd}$, which is independent of $b$ as long as $0<b⪡{a}_{\text{Bohr}}$ due to the inverse square nature of the Coulomb force. Its comparisons with the corresponding cases in three dimensions and two dimensions is discussed.

Three-body problem in quantum mechanics: hyperspherical elliptic coordinates and harmonic basis sets.

Elliptic coordinates within the hyperspherical formalism for three-body problems were proposed some time ago [V. Aquilanti, S. Cavalli, and G. Grossi, J. Chem. Phys. 85, 1362 (1986)] and recently have also found application, for example, in chemical reaction theory [see O. I. Tolstikhin and H. Nakamura, J. Chem. Phys. 108, 8899 (1998)]. Here we consider their role in providing a smooth transition between the known "symmetric" and "asymmetric" parametrizations, and focus on the corresponding hyperspherical harmonics. These harmonics, which will be called hyperspherical elliptic, involve products of two associated Lame polynomials. We will provide an expansion of these new sets in a finite series of standard hyperspherical harmonics, producing a powerful tool for future applications in the field of scattering and bound-state quantum-mechanical three-body problems.

A New Method for Solving an Eigenvalue Problem for a System of Three Coulomb Particles within the Hyperspherical Adiabatic Representation

The quantum mechanical three-body problem with Coulomb interaction is formulated within the adiabatic representation method using the hyperspherical coordinates. The Kantorovich method of reducing the multidimensional problem to the one-dimensional one is used. A new method for computing variable coefficients (potential matrix elements of radial coupling) of a resulting system of ordinary second-order differential equations is proposed. It allows the calculation of the coefficients with the same precision as the adiabatic functions obtained as solutions of an auxiliary parametric eigenvalue problem. In the method proposed, a new boundary parametric problem with respect to unknown derivatives of eigensolutions in the adiabatic variable (hyperradius) is formulated. An efficient, fast, and stable algorithm for solving the boundary problem with the same accuracy for the adiabatic eigenfunctions and their derivatives is proposed. The method developed is tested on a parametric eigenvalue problem for a hydrogen atom on a three-dimensional sphere that has an analytical solution. The accuracy, efficiency, and robustness of the algorithm are studied in detail. The method is also applied to the computation of the ground-state energy of the helium atom and negative hydrogen ion.

Statistics of energy levels in the quantum treatment of an elementary reaction

Statistical tests (NNLSD, Δ 3 of Dyson and Mehta, correlation coefficients) are applied to the adiabatic energy level distribution of the prototypical reaction F+H 2 → HF+H, depending parametrically on the three-body total inertia measured by the hyperradius ρ. As a function of ρ (proposed as a natural control variable for the quantum mechanical three-body problem), a transition can be identified. At high ρ, the situation of separate partners of the reaction shows a Poissonian behavior while, at low ρ, the transition state (where the triatom can be viewed as a system of coupled oscillators) exhibits some Wigner-like features as signatures of quantum chaos.

Structure and Occurrence of Three-Body Halos in Two Dimensions

The quantum-mechanical three-body problem is reformulated in two dimensions by use of hyperspherical coordinates and an adiabatic expansion of the Faddeev equations. The effective radial potentials are calculated and their large-distance asymptotic behavior is derived analytically for short-range two-body interactions. Energies and wave functions are computed numerically for various potentials. An infinite series of Efimov states does not exist in two dimensions. Borromean systems, i.e. bound three-body systems without bound binary subsystems, can only appear when a short-range repulsive barrier at finite distance is present in the two-body interaction. The corresponding Borromean state is never spatially extended. For a system of three weakly interacting identical bosons we find two bound states with both binding energies proportional to the two-body binding energy. In the limit of small binding these states are spatially located at the very large distances characterized by the scattering length. Their properties are universal and independent of the details of the potential. We compare throughout with the corresponding properties in three dimensions.

Erratum: Approximate analytical solution of the quantum-mechanical three-body Coulomb continuum problem [Phys. Rev. A53, 2314 (1996)

Erratum: Approximate analytical solution of the quantum-mechanical three-body Coulomb continuum problem [Phys. Rev. A<b>53</b>, 2314 (1996)