Semiclassical quantisation of the hydrogen atom in crossed electric and magnetic fields (original) (raw)
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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.
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.
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.
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.
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.