Zeeman Effect In Hydrogen Research Articles

Quadratic Zeeman effect in hydrogen at 2-3 MG magnetic fields.

The Zeeman effect is used for measurement of magnetic fields in astrophysical and laboratory plasmas. Magnetic fields in atmospheres of magnetic white dwarf stars are in the range 40 kG-1 GG. The quadratic Zeeman effect results in the additional split and shift of lines for magnetic fields >2 MG. Hydrogen Balmer lines were studied in magnetic fields delivered by a 1 MA pulse power generator. The magnetic field was generated by rod loads 0.8-1 mm in diameter. A droplet of CH oil on the load center was the source of hydrogen. A low ionized oil layer was backlit by blackbody emission from the rod with a temperature of 0.5-0.6 eV. Zeeman splitting of H-alpha and H-beta absorption lines were with a grating spectrometer. A spectral shift of the central component of the triplet indicated the quadratic Zeeman effect in hydrogen lines.

Recurrence Relations in the Rayleigh-Ritz Method

We report a novel approach for finding upper-bounds for energy levels in quantum systems. This method uses the Rayleigh-Ritz method in conjunction with an expansion series solution of the Schrodinger equation. In the traditional Rayleigh-Ritz method, a N × N matrix is diagonalized to find the N coefficients of a N -term series expansion solution. In a new approach, we directly substitute a series expansion into the Schrodinger equation, and find a recurrence relationship for the coefficients. Hence, the linear coefficients in the series expansion are not all independent. We use this dependency of the coefficients to show that it is possible to find all N coefficients by diagonilizing a matrix that finds only the independent coefficiencts. The dependent coefficients are then automatically found by the recursion relations. This approach significantly reduces the size of the matrix diagonalization. The resultant matrix equation is only R × R, where R is the number of independent series coefficients. By reducing the size of the matrix equation, this approach saves computational time and machine memory. We have applied this method to the one-dimensional quartic anharmonic oscillator and also the anomalous Zeeman effect in Hydrogen. The results are comparable to the results of the N × N diagonalization energies of the traditional Rayleigh-Ritz approach.

Perturbation theory without wave functions for the Zeeman effect in hydrogen.

We apply perturbation theory to the hydrogen atom in a magnetic field by means of a recurrence relation for properly chosen moments of the wave function. The method is suitable for both numerical and symbolic computation and allows the treatment of classes of states with common symmetry properties. We derive analytic expressions in terms of the principal quantum number for the perturbation corrections to the energy of the unperturbed states with quantum numbers n-1=\ensuremath{\Vert}m\ensuremath{\Vert}=l, n-2=\ensuremath{\Vert}m\ensuremath{\Vert}=l-1, n-2=\ensuremath{\Vert}m\ensuremath{\Vert}=l, n-3=\ensuremath{\Vert}m\ensuremath{\Vert}=l-1, and (n-3=\ensuremath{\Vert}m\ensuremath{\Vert}=l,n-3=\ensuremath{\Vert}m\ensuremath{\Vert}=l-2) some of which have not been reported before. Disagreements with some previous results are pointed out.

Action-angle variables for the diamagnetic Kepler problem.

The diamagnetic Kepler problem, or quadratic Zeeman effect in hydrogen, is converted into an asymmetric top via a transformation to Lissajous action-angle variables. These variables are especially suitable for uniform semiclassical quantization of the system for arbitrary values of the azimuthal quantum number m.

Dynamical symmetry of the quadratic Zeeman effect in hydrogen: Semiclassical quantization.

By exploiting the approximate SU(2) symmetry of the {ital m}=0 quadratic Zeeman Hamiltonian within multiplets of fixed principal quantum number, the dynamics is mapped onto a twofold hindered rotor. This allows a singularity-free uniform semiclassical quantization of Solov'ev's approximate integral of the motion to be performed. Excellent agreement with quantum results is obtained.

Bounds to the energy of quantum-mechanical models from variational functionals

We derive semiclassical energy functions in configuration space that provide upper or lower bounds to the energies of quantum-mechanical systems. The bounds are predicted by rigorous theorems about the local energy which we choose to depend on adjustable parameters in order to improve the accuracy of the results. We apply the method to one-dimensional models, central-field systems, and the Zeeman effect in hydrogen.

Large-order perturbation theory of the Zeeman effect in hydrogen from a four-dimensional anisotropic anharmonic oscillator.

The Zeeman Hamiltonian for (spinless) hydrogen in a constant magnetic field is shown to be equivalent to a four-dimensional anisotropic anharmonic oscillator. Using this relation, Rayleigh-Schr\"odinger perturbation series expansions of both systems can be related to each other and analyzed in a unified way. Special emphasis is laid upon analytical estimates of their behavior in large orders of perturbation theory. Employing the path-integral approach, a new large-order formula is derived for the expansion of the ground-state energy of the oscillator system. With use of known Bender-Wu formulas for isotropic anharmonic oscillators, the major part of this calculation becomes straightforward. Combined with the new equivalence, this calculation represents the simplest path-integral derivation of large-order formulas for the Zeeman system.

On the Ricatti equation for eigenvalue problems

AbstractThe Riccati equation is shown to be suitable for obtaining implicit approximate analytic expressions for the eigenvalues of quantum‐mechanical systems. The Hamiltonian operator H = (1/2)p2 – (Z/r) + λr2 is used as a test example, and the resulting formulae are modified to deal with the Zeeman effect in hydrogen.

Perturbation theory from moment recurrence relations

Moment recurrence relations are shown to be useful in perturbation calculations. General equations are developed and different cases are discussed. The anharmonic oscillator and Zeeman effect in hydrogen are used as illustrative examples.

Implicit analytical expressions for eigenvalues: the Zeeman effect in hydrogen

A simple way of obtaining implicit analytical expressions for the eigenvalues of parameter-dependent systems is developed. The method takes into account the scaling laws satisfied by the eigenvalues and is suitable for matching weak- and strong-coupling series. Results are shown for the Zeeman effect in hydrogen.

Upper and lower bounds to eigenvalues from variational functionals.

Upper and lower bounds to the energy of quantum-mechanical parameter-dependent systems are obtained from variational functionals. Results are shown for the Zeeman effect in hydrogen and the anharmonic oscillator.

Relativistic Zeeman effect hydrogen and positronium

Relativistic corrections to the Zeeman effect in hydrogen are calculated for the lowest-energy states. The use of (approximate) relativistic centre-of-mass variables renders the problem much easier to handle than in previous treatments and with comparable precision. Positronium is also examined using the same method.

An approximate analytical expression for the energies of the tight-bound states of the Zeeman effect in hydrogen

Abstract A simple approximate analytical expression for the energies of the tight-bound states of the Zeeman effect in hydrogen is developed that incorporates both the perturbation series in powers of the field strength and the asymptotic logarithmic expansion for the superstrong-field case. Results for the lowest tight-bound states are compared with very accurate non-perturbative calculations.

Improved renormalized perturbation series

An improved version of a recently developed method for summing strongly divergent pertubation series is presented. The perturbation series is changed into a sequence of polynomials by means of a properly chosen order-dependent mapping. The limit of the resulting sequence is obtained approximately through the ϵ-algorithm of Shanks and Wynn. The Rayleigh-Schrödinger perturbation series for the eigenvalues of the anharmonic oscillator, the linear-confining potential and the Zeeman effect in hydrogen are discussed. Results for the strong-coupling limit show that the ϵ-algorithm accelerates the convergence of the renormalized perturbation series markedly.

A method for summing strongly divergent perturbation series: The Zeeman effect in hydrogen

Abstract A new method for summing strongly divergent power series is presented. It consists of an order-dependent mapping that transforms the divergent series into a convergent sequence. The procedure is applied to the perturbation series of the Zeeman effect in hydrogen obtaining accurate energy values for the states 1s and 2p ±1 over a wide range of field strengths. Difficulties in handling the 2s state are pointed out.

Perturbation theory without wave function for multidimensional systems

AbstractA new method to obtain perturbation corrections to the eigenvalues of multidimensional quantummechanical models is developed. It consists of rearranging the Rayleigh‐Schrodinger perturbation theory so that any coefficient of the perturbation series is obtained from a simple and compact recursion relationship. The Zeeman effect in hydrogen and the hydrogen molecule‐ion are used to illustrate the procedure.

Rayleigh-Schrodinger perturbation theory with a strong perturbation: the quadratic Zeeman effect in hydrogen

The ground state energy of a hydrogen atom in a uniform magnetic field has been computed by means of low-order variational-perturbation theory. High accuracy is obtained for arbitrary strengths up to 1012 G by the simple expedient of including the leading effect of the field in the zero-order model.

Approximate constants of motion for classically chaotic vibrational dynamics: Vague tori, semiclassical quantization, and classical intramolecular energy flow

Coupled nonlinear Hamiltonian systems are known to exhibit regular (quasiperiodic) and chaotic classical motions. In this and the preceding paper by Jaffé and Reinhardt, we find substantial short time regularity even in the chaotic regions of phase space for what appears to be a large class of systems. This regularity is demonstrated by the behavior of approximate constants of motion calculated by Padé summation of the Birkhoff–Gustavson normal form expansion and is attributed to remnants of destroyed invariant tori in phase space. The remnant toruslike manifold structures are used to suggest justification for use of Einstein–Brillouin–Keller semiclassical quantization procedures for obtaining quantum energy levels even in the absence of complete tori and to form a theoretical basis for the calculation of rate constants for intramolecular mode–mode energy transfer. These results are illustrated in a thorough analysis of the Hénon–Heiles oscillator problem. Possible generality of the analysis is shown by brief consideration of classical dynamics for the Barbanis Hamiltonian, Zeeman effect in hydrogen, and recent results of Wolf and Hase for the H–C–C fragment.

Practical recursive solution of degenerate Rayleigh-Schrödinger perturbation theory and application to high-order calculations of the Zeeman effect in hydrogen

With the aim of high-order calculations, a new recursive solution for the degenerate Rayleigh-Schr\"odinger perturbation-theory wave function and energy has been derived. The final formulas, ${\ensuremath{\chi}}_{\ensuremath{\sigma}}^{(N)}={R}^{(\ensuremath{-}\ensuremath{\sigma})}{\ensuremath{\Sigma}}_{k=0}^{N\ensuremath{-}1}{\overline{\mathcal{H}}}_{\ensuremath{\sigma}+1}^{(\ensuremath{\sigma}+1+k)}{\ensuremath{\chi}}^{(N\ensuremath{-}1\ensuremath{-}k)}$ ${E}^{(N+\ensuremath{\sigma})}=〈0|{\mathcal{H}}_{\ensuremath{\sigma}+1}^{(N+\ensuremath{\sigma})}|0〉+〈0|{\ensuremath{\Sigma}}_{k=0}^{N\ensuremath{-}2}{\mathcal{H}}_{\ensuremath{\sigma}+1}^{(\ensuremath{\sigma}+1+k)}|{\ensuremath{\chi}}^{(N\ensuremath{-}1\ensuremath{-}k)}〉$ which involve new Hamiltonian-related operators ${\mathcal{H}}_{\ensuremath{\sigma}}^{(\ensuremath{\sigma}+k)}$ and ${\overline{\mathcal{H}}}_{\ensuremath{\sigma}}^{(\ensuremath{\sigma}+k)}$, strongly resemble the standard nondegenerate recursive formulas. As an illustration, the perturbed energy coefficients for the $3s\ensuremath{-}3{d}_{0}$ states of hydrogen in the Zeeman effect have been calculated recursively through 87th order in the square of the magnetic field. Our treatment is compared with that of Hirschfelder and Certain [J. Chem. Phys. 60, 1118 (1974)], and some relative advantages of each are pointed out.

Bender-Wu formulas for the Zeeman effect in hydrogen

We review the Bender-Wu theory of the Rayleigh-Schrödinger series in large order and derive the leading term in the Bender-Wu formulas for the perturbation coefficients for the hydrogen Zeeman Hamiltonian.