Semiclassical quantisation of the hydrogen atom in crossed electric and magnetic fields (original) (raw)
1982, Journal of Physics B: Atomic and Molecular Physics
https://doi.org/10.1088/0022-3700/15/8/012
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
This method, based on the adiabatic invariance of semiclassical quantisation conditions , is used to calculate the energy levels of a three-dimensional, non-integrable system: a hydrogen atom in crossed electric and magnetic fields. The results are presented for a ground state and two excited states, for various ratios of field strengths and angles between the fields. The limitations of the method are discussed.
Related papers
2022
The Hamiltonian of a pure hydrogen atom possesses the SO(4) symmetry group generated by the integrals of motion: the angular momentum and the Runge-Lenz vector. The pure hydrogen atom is a supersymmetric and superintegrable system, since the Hamilton-Jacobi and the Schrödinger equations are separable in several different coordinate systems and has an exact analytical solution. The Schrödinger equation for a hydrogen atom in a uniform electric field (Stark effect) is separable in parabolic coordinates. The system has two conserved quantities: z-projections of the generalized Runge-Lenz vector and of the angular momentum. The problem is integrable and has the symmetry group SO(2)xSO(2). The ion of the hydrogen molecule (problem of two Coulomb centers) has similar symmetry group SO(2)xSO(2) generated by two conserved z-projections of the generalized Runge-Lenz and of the angular momentum on the internuclear axis. The corresponding Schrödinger equation is separable in the elliptical coordinates. For the hydrogen atom in a uniform magnetic field, the respective Schrödinger equation is not separable. The problem is non-separable and non-integrable and is considered as a representative example of quantum chaos that cannot be solved by any analytical method. Nevertheless, an exact analytical solution describing the quantum states of a hydrogen atom in a uniform magnetic field can be obtained as a convergent power series in two variables, the radius and the sine of the polar angle. The energy levels and wave functions for the ground and excited states can be calculated exactly, with any desired accuracy, for an arbitrary strength of the magnetic field. Therefore, the problem can be considered superintegrable, although it does not possess supersymmetry and additional integrals of motion.
The semiclassical resonance spectrum of hydrogen in a constant magnetic field
Nonlinearity, 1996
We present the first purely semiclassical calculation of the resonance spectrum in the diamagnetic Kepler problem (DKP), a hydrogen atom in a constant magnetic field with L z = 0. The classical system is unbound and completely chaotic for a scaled energy ∼ EB −2/3 larger than a critical value c > 0. The quantum mechanical resonances can in semiclassical approximation be expressed as the zeros of the semiclassical zeta function, a product over all the periodic orbits of the underlying classical dynamics. Intermittency originating from the asymptotically separable limit of the potential at large electron-nucleus distance causes divergences in the periodic orbit formula. Using a regularization technique introduced in (Tanner G and Wintgen D 1995 Phys. Rev. Lett. 75 2928) together with a modified cycle expansion, we calculate semiclassical resonances, both position and width, which are in good agreement with quantum mechanical results obtained by the method of complex rotation. The method also provides good estimates for the bound state spectrum obtained here from the classical dynamics of a scattering system. A quasi-Einstein-Brillouin-Keller (QEBK) quantization is derived that allows for a description of the spectrum in terms of approximate quantum numbers and yields the correct asymptotic behaviour of the Rydberg-like series converging towards the different Landau thresholds.
Chemical Physics, 2018
The mapping of an electronic state on a real-space support lattice may offer advantages over a basis set ansatz in cases where there are linear dependences due to basis set overcompleteness or when strong internal or external fields are present. Such discretization methods are also of interest because they allow for the convenient numerical integration of matrix elements of local operators. We have developed a pseudo-spectral approach to the numerical solution of the time-dependent and time-independent Schrödinger equations describing electronic motion in atoms and atomic ions in terms of a spherical coordinate system. A key feature of this scheme is the construction of a Variational Basis Representation (VBR) for the non-local component and of a Generalized Finite Basis Representation (GFBR) for the local component of the electronic Hamiltonian operator. Radial Hamiltonian eigenfunctions r () nl; of the H atom-like system and spherical harmonics form the basis set. Two special cases of this approach are explored: In one case, the functions of the field-free H atom are used as the elements of the basis set, and in the second case, each radial basis function has been obtained by variationally optimizing a shielding parameter β to yield a minimum energy for a particular eigenstate of the H atom in a uniform magnetic field. We derive a new quadrature rule of nearly Gaussian accuracy for the computation of matrix elements of local operators between radial basis functions and perform numerical evaluation of local operator matrix elements involving spherical harmonics. With this combination of radial and angular quadrature prescriptions we ensure to a good approximation the discrete orthogonality of Hamiltonian eigenfunctions of H atom-like systems for summation over the grid points. We further show that sets of r () nl; functions are linearly independent, whereas sets of the polar-angle-dependent components of the spherical harmonics, i.e., the associated Legendre functions, are not and provide a physical interpretation of this mathematical observation. The pseudo-spectral approach presented here is applied to two model systems: the field-free H atom and the H atom in a uniform magnetic field. The results demonstrate the potential of this method for the description of challenging systems such as highly charged atomic ions.
Semiclassical quantization of atomic systems through their normal form: the hydrogen atom
Theoretical Chemistry Accounts, 2014
The hydrogen atom and Rydberg systems continue to be a source of discovery to this day. This is surprising as last year was the centennial of the publication of Niels Bohr landmark paper [1] on the quantization of the hydrogen atom. This remarkable paper represents the height of what has become known as the old quantum theory . It was an extraordinary time. The question of the constitution of atoms and molecules was one of the major issues of the day. While the atomic theory of matter had been widely accepted, such basic properties as the size and mass of atoms were only just being measured.
Physical Review A, 2000
Excited states of the hydrogen molecule subject to a homogeneous magnetic field are investigated for the parallel configuration in the complete regime of field strengths B = 0 − 100a.u.. Up to seven excitations are studied for gerade as well as ungerade spin singlet states of Σ symmetry with a high accuracy. The evolution of the potential energy curves for the individual states with increasing field strength as well as the overall behaviour of the spectrum are discussed in detail. A variety of phenomena like for example the sequence of changes for the dissociation channels of excited states and the resulting formation of outer wells are encountered. Possible applications of the obtained data to the analysis of magnetic white dwarfs are outlined.
Hydrogen negative ion: Semiclassical quantization and weak-magnetic-field effect
Physical Review A, 1993
We report the results of semiclassical quantization of the hydrogen negative ion using the Gutzwiller trace formula and the unstable periodic orbits of the classically chaotic repeller of the collinear configuration with the two electrons on opposite sides of the nucleus. The energies of the 'S' and 'S' au- toionizing states have been calculated for principal quantum numbers up to X = 16 and 12, respectively. We also report the results of a semiclassical study of the quadratic effect of a weak magnetic field on the autoionizing states of H
Exact solution for a hydrogen atom in a magnetic field of arbitrary strength
An exact solution describing the quantum states of a hydrogen atom in a homogeneous magnetic field of arbitrary strength is obtained in the form of a power series in the radial variable with coefficients being polynomials in the sine of the polar angle. Energy levels and wave functions for the ground state and for several excited states are calculated exactly for the magnetic field varying in the range 0ϽB/(m 2 e 3 c/ប 3 )р4000.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.
References (11)
- Arnol'd V I 1962 Sou. Math. Dokl. 3 136 -1963 Russ. Math. Surveys 18 86 -1979 Matematicheskie Metody Klassicheskoi Mekhaniki (Moskva: Nauka)
- Bethe H A and Saltpeter E A 1957 Quantum Mechanics of One-and Two-Electron Atoms (New York: Born M 1960 Mechanics of the Atom (London: Bell)
- Cabib D, Fabri E and Fiorio G 1972 Nuovo Cim. 10B 185
- Demkov Yu N, Monozon B S and Ostrovskii V N 1970 Sou. Phys.-JETP 30 775
- Landau L D and Lifshitz E M 1976 Mechanics (Oxford: Pergamon)
- Noid D W and Marcus R A 1977 J. Chem. Phys. 67 559
- Percival I C 1977 Adv. Chem. Phys. 36 1-61
- Silverstone H J 1978 Phys. Rev. A 18 1853
- Solov'ev E A 1978 Sou. Phys.-JETP 48 635
- Stiefel E L and Schiefele G 1971 Linear and Regular Celestial Mechanics (Berlin: Springer)
- Strandt M P and Reinhard W P 1979 J. Chem. Phys. 70 3812 Academic)
Related papers
Semiclassical quantization of the hydrogen atom in crossed electric and magnetic fields
Physical Review A, 2003
The S-matrix theory formulation of closed-orbit theory recently proposed by Granger and Greene is extended to atoms in crossed electric and magnetic fields. We then present a semiclassical quantization of the hydrogen atom in crossed fields, which succeeds in resolving individual lines in the spectrum, but is restricted to the strongest lines of each n-manifold. By means of a detailed semiclassical analysis of the quantum spectrum, we demonstrate that it is the abundance of bifurcations of closed orbits that precludes the resolution of finer details. They necessitate the inclusion of uniform semiclassical approximations into the quantization process. Uniform approximations for the generic types of closed-orbit bifurcation are derived, and a general method for including them in a high-resolution semiclassical quantization is devised.
Physical Review A, 2007
A hierarchical ordering is demonstrated for the periodic orbits in a strongly coupled multidimensional Hamiltonian system, namely the hydrogen atom in crossed electric and magnetic fields. It mirrors the hierarchy of broken resonant tori and thereby allows one to characterize the periodic orbits by a set of winding numbers. With this knowledge, we construct the action variables as functions of the frequency ratios and carry out a semiclassical torus quantization. The semiclassical energy levels thus obtained agree well with exact quantum calculations.
Adiabatic representation for a hydrogen atom photoionization in a uniform magnetic field
Physics of Atomic Nuclei, 2008
A new effective method of calculating wave functions of discrete and continuous spectra of a hydrogen atom in a strong magnetic field is developed on the basis of the adiabatic approach to parametric eigenvalue problems in spherical coordinates. The two-dimensional spectral problem for the Schr¨odinger equation at a fixed magnetic quantum number and parity is reduced to a spectral parametric problem for a one-dimensional angular equation and a finite set of ordinary second-order radial differential equations. The results are in good agreement with the photoionization calculations by other authors and have a true threshold behavior.
THE ENERGY EIGENVALUES OF THE TWO DIMENSIONAL HYDROGEN ATOM IN A MAGNETIC FIELD
International Journal of Modern Physics E, 2006
In this paper, the energy eigenvalues of the two dimensional hydrogen atom are presented for the arbitrary Larmor frequencies by using the asymptotic iteration method. We first show the energy eigenvalues for the no magnetic field case analytically, and then we obtain the energy eigenvalues for the strong and weak magnetic field cases within an iterative approach for n = 2 − 10 and m = 0 − 1 states for several different arbitrary Larmor frequencies. The effect of the magnetic field on the energy eigenvalues is determined precisely. The results are in excellent agreement with the findings of the other methods and our method works for the cases where the others fail.
Physical Review A, 2001
Excited states of the hydrogen molecule subject to a homogeneous magnetic field are investigated for the parallel configuration in the complete regime of field strengths B = 0 − 100a.u.. Up to seven excitations are studied for gerade as well as ungerade spin singlet states of Σ symmetry with a high accuracy. The evolution of the potential energy curves for the individual states with increasing field strength as well as the overall behaviour of the spectrum are discussed in detail. A variety of phenomena like for example the sequence of changes for the dissociation channels of excited states and the resulting formation of outer wells are encountered. Possible applications of the obtained data to the analysis of magnetic white dwarfs are outlined.
Hydrogen atom in a magnetic field: electromagnetic transitions of the lowest states
Revista Mexicana De Fisica - REV MEX FIS, 2008
A detailed study of the lowest states 1s0,2p−1,2p01s_0, 2p_{-1}, 2p_01s0,2p−1,2p0 of the hydrogen atom placed in a magnetic field Bin(0−4.414times1013rmG)B\in(0-4.414\times 10^{13} {\rm G})Bin(0−4.414times1013rmG) and their electromagnetic transitions ($1s_{0} \leftrightarrow 2p_{-1}$ and $ 1s_{0} \leftrightarrow 2p_{0}$) is carried out in the Born Oppenheimer approximation. The variational method is used with a physically motivated recipe to design simple trial functions applicable to the whole domain of magnetic fields. We show that the proposed functions yield very accurate results for the ionization (binding) energies. Dipole and oscillator strengths are in good agreement with results by Ruder {\em et al.} \cite{Ruderbook} although we observe deviations up to sim30\sim 30%sim30 for the oscillator strength of the (linearly polarized) electromagnetic transition 1s0leftrightarrow2p01s_{0} \leftrightarrow 2p_{0}1s0leftrightarrow2p0 at strong magnetic fields Bgtrsim1000B\gtrsim 1000Bgtrsim1000 a.u. Bibtex entry for this abstract Preferred format for this abstract (see Preferences) Find Similar Abstracts: Use: Autho...
Transformation of the spectrum of atomic hydrogen in crossed electric and magnetic fields
Journal of Physics B Atomic and Molecular Physics
The spectrum of highly excited atomic hydrogen in weak mutually orthogonal electric and magnetic fields is investigated. An analytical description is given for the splitting of levels remaining degenerate in the first order of perturbation theory. Approximate expressions are obtained for the energy eigenvalues and for the exponentially small tunnelling splitting of doublet states. It is shown that the spectrum is radically transformed when the relative strength of the two fields is changed.
Semiclassically modeling Hydrogen at Rydberg states immersed in electromagnetic fields
2019
Originally, closed-orbit theory was developed in order to analyze oscillations in the near ionization threshold (Rydberg) densities of states for atoms in strong external electric and magnetic fields. Oscillations in the density of states were ascribed to classical orbits that began and ended near the atom. In essence, observed outgoing waves following the classical path return and interfere with original outgoing waves, giving rise to oscillations. Elastic scattering from one closed orbit to another gives additional oscillations in the cross-section. This study examines how quantum theory can be properly used in combination with classical orbit theory in order to study inelastic scattering for atoms in an external field. At Rydberg states, an electron wave function can be modeled numerically through semiclassical means, using the Coulombic interaction from the atom, but as it approaches lower states, it must be modeled quantum mechanically, using a ‘Modified Coulombic’ potential.
Le Journal de Physique Colloques
La quantification semi-classique de systèmes à plusieurs dimensions est discutée à l'aide de la méthode d'Einstein-Brillouin-Keller (EBK) sur les tores invariants de l'espace des phases, puis par la méthode des familles infinies de trajectoires périodiques. Les notions de séparabilitë, de systèmes classiques intégrables et non-intégrables sont introduites. L'intégrabilitë approchée existant dans le cas de l'effet Zeeman quadratique est utilisée pour la quantification de ce problème à l'aide de la forme normale de Birkhoff-Gustavson.
High Rydberg states of an atom in parallel electric and magnetic fields
Physical Review A, 1987
We have calculated the energy spectrum of a highly excited atom in parallel electric and magnetic fields. The eigenvalues were obtained by semiclassical quantization of action variables calculated from first-order classical perturbation theory. For the field strengths studied, the electron moves on a Kepler ellipse whose orbital parameters evolve slowly in time, and first-order perturbation theory reduces the problem to just one degree of freedom. Action variables were calculated from perturbation theory and the eigenvalues were obtained by semiclassical quantization of the action. The semiclassical analysis leads directly to a correlation diagram which connects the eigenstates of the Stark effect to those of the diamagnetic effect. A classification scheme for the eigenstates is proposed. Comparison with first-order degenerate quantum perturbation theory verifies the accuracy of the semiclassical treatment.
Related topics
Cited by
Classical approach in atomic physics
European Physical Journal D, 2011
The application of a classical approach to various quantum problems -the secular perturbation approach to quantization of a hydrogen atom in external fields and a helium atom, the adiabatic switching method for calculation of a semiclassical spectrum of a hydrogen atom in crossed electric and magnetic fields, a spontaneous decay of excited states of a hydrogen atom, Gutzwiller's approach to Stark problem, long-lived excited states of a helium atom discovered with the help of Poincaré section, inelastic transitions in slow and fast electron-atom and ion-atom collisions -is reviewed. Further, a classical representation in quantum theory is discussed. In this representation the quantum states are treated as an ensemble of classical states. This approach opens the way to an accurate description of the initial and final states in classical trajectory Monte Carlo (CTMC) method and a purely classical explanation of tunneling phenomenon. The general aspects of the structure of the semiclassical series such as renormgroup symmetry, criterion of accuracy and so on are reviewed as well.