Uniformly Valid Asymptotic Approximation for the Quantized Anharmonic Oscillator

A uniformly valid asymptotic expansion is obtained for the solution of the one-dimensional time-independent Schrödinger equation which describes an anharmonic oscillator. As is well known this leads to a two-turning-point problem in asymptotic analysis. The expansion itself is determined for large values of an appropriately constructed parameter λ which can be specialized to 4J(E) / 2πℏ. It is shown to be valid for high energies which, in turn, implies that the turning points are well separated. The technique employed is an extension of that used previously by the authors for low energies, i.e., for nearby turning points. The energy levels are obtained implicitly to all orders in λ−1 by a relation which generalizes the traditional Bohr–Wilson–Sommerfeld quantum condition. For analytic potentials the leading terms in the asymptotic expansion of this relation are obtained in terms of contour integrals and are found to agree with the corresponding higher-order JWKB results of Dunham. Finally, the normalization of the eigen-solutions is discussed and a procedure for obtaining the asymptotic expansion of the normalization constant is indicated.

We consider the asymptotic behaviour of the second discrete Painlevé equation in the limit as the independent variable becomes large. Using asymptotic power series, we find solutions that are asymptotically pole-free within some region of the complex plane. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. We subsequently apply exponential asymptotic techniques to investigate such phenomena, and obtain mathematical descriptions of the rapid switching behaviour associated with Stokes curves. Through this analysis, we determine the regions of the complex plane in which the asymptotic behaviour is described by a power series expression, and find that the behaviour of these asymptotic solutions shares a number of features with the tronquée and tri-tronquée solutions of the second continuous Painlevé equation.

We consider the asymptotic behavior of solutions of the first -difference Painlevé equation in the limits and . Using asymptotic power series, we describe four families of solutions that contain free parameters hidden beyond-all-orders. These asymptotic solutions exhibit Stokes phenomena, which is typically invisible to classical power series methods. In order to investigate such phenomena we apply exponential asymptotic techniques to obtain mathematical descriptions of the rapid switching behavior associated with Stokes curves. Through this analysis, we also determine the regions of the complex plane in which the asymptotic behavior is described by a power series expression, and find that the Stokes curves are described by curves known as -spirals.

Asymptotic power series

This technical paper of mathematical physics arose as an aftermath of the 2002 Cassini experiment (Bertotti et al 2003 Nature 425 374–6), in which the PPN parameter γ was measured with an accuracy σγ = 2.3 × 10−5 and found consistent with the prediction γ = 1 of general relativity. The Orbit Determination Program (ODP) of NASA's Jet Propulsion Laboratory, which was used in the data analysis, is based on an expression (8) for the gravitational delay Δt that differs from the standard formula (2); this difference is of second order in powers of m—the gravitational radius of the Sun—but in Cassini's case it was much larger than the expected order of magnitude m2/b, where b is the distance of the closest approach of the ray. Since the ODP does not take into account any other second-order terms, it is necessary, also in view of future more accurate experiments, to revisit the whole problem, to systematically evaluate higher order corrections and to determine which terms, and why, are larger than the expected value. We note that light propagation in a static spacetime is equivalent to a problem in ordinary geometrical optics; Fermat's action functional at its minimum is just the light-time between the two end points A and B. A new and powerful formulation is thus obtained. This method is closely connected with the much more general approach of Le Poncin-Lafitte et al (2004 Class. Quantum Grav. 21 4463–83), which is based on Synge's world function. Asymptotic power series are necessary to provide a safe and automatic way of selecting which terms to keep at each order. Higher order approximations to the required quantities, in particular the delay and the deflection, are easily obtained. We also show that in a close superior conjunction, when b is much smaller than the distances of A and B from the Sun, say of order R, the second-order correction has an enhanced part of order m2R/b2, which corresponds just to the second-order terms introduced in the ODP. Gravitational deflection of the image of a far away source when observed from a finite distance from the mass is obtained up to O(m2).

Asymptotic approximations ($n \to \infty$) to the truncation errors $r_n = - \sum_{\nu=0}^{\infty} a_{\nu}$ of infinite series $\sum_{\nu=0}^{\infty} a_{\nu}$ for special functions are constructed by solving a system of linear equations. The linear equations follow from an approximative solution of the inhomogeneous difference equation $\Delta r_n = a_{n+1}$. In the case of the remainder of the Dirichlet series for the Riemann zeta function, the linear equations can be solved in closed form, reproducing the corresponding Euler-Maclaurin formula. In the case of the other series considered -- the Gaussian hypergeometric series ${}_2 F_1 (a, b; c; z)$ and the divergent asymptotic inverse power series for the exponential integral $E_1 (z)$ -- the corresponding linear equations are solved symbolically with the help of Maple. The practical usefulness of the new formalism is demonstrated by some numerical examples.

An asymptotic expansion for energy eigenvalues of anharmonic oscillators

We introduce a generalization of Wick-ordering which maps the anharmonic oscillator (AO) Hamiltonian for mass m and coupling λ exactly into a “Wick-ordered” Hamiltonian with an effective mass M which is a simple analytic function of λ and m. The effective coupling Λ = λ M 3 is bounded. We transform the AO perturbation series in λ into one in Λ. This series may then be summed using Borel summation methods. We also introduce a new summation method for the AO series (which is a practical necessity to obtain accurate energy levels of the excited states). We obtain a numerical accuracy for (E PT − E exact) E exact of at least 10 −7 (using 20 orders of perturbation theory) and 10 −3 (using only 2 orders of perturbation theory) for all couplings and all energy levels of the anharmonic oscillator. The methods are applicable also to the double-well potential (DWP, the AO with a negative mass-squared). The only change is that now the effective coupling is unbounded as λ → 0. The series in Λ is, however, still summable. The relative accuracy in the energy levels for 20 orders of perturbation theory varies from 10 −7 for large coupling to 1% at λ = 0.1 and to 10% at λ = .05. We also present results for the sextic oscillator.

More matter, with less art. Hamlet II ii The quantum mechanics of a single electron in a central potential leads to classification of the one-electron states in terms of four quantum numbers. The individual electrons' spin and orbital angular momenta are coupled in an isolated manyelectron ion to give total spin and orbital quantum numbers S and L . Spin-orbit coupling then operates to split the energy levels into a series of J -multiplets, the lowest of which is specified by Hund's rules. Curie-law susceptibility, χ = C /T , is calculated for a general value of J . When placed in a solid, the ion experiences a crystal field due to the charge environment which disturbs the spin-orbit coupling, leaving either S or J as the appropriate quantum number. The crystal field modifies the structure of the lowest M S or M J magnetic sublevels which are split by the Zeeman interaction and it introduces single-ion anisotropy. Atomic physics is concerned with the energy levels of an atom or ion and the possible transitions between them, which are usually in the optical or ultraviolet energy range (1–10 eV). Magnetism is concerned with the energy levels that are occupied at ambient temperature, which usually means only the ground state , and its sublevels resulting from interactions with magnetic or electric fields, which produce a splitting of less than 0.1 eV. At ambient temperature k B T is about 25 meV.

We present an approximation of a non-Markovian solution for the time evolution of the mean number of quanta in an anharmonic oscillator linearly and stochastically coupled to a thermal bath. A colored Gaussian bath is explicitly considered, and a quantum treatment of the bath correlation functions, based on the use of a relation between their real and imaginary parts, is applied. The behavior of the oscillator is characterized by a reduced set of (five) parameters referring to the oscillator, the bath, and their mutual interaction. A comparison is made between the prediction of the approach for the mean stationary number of quanta and its exact value, which enables us to analyze its accuracy. When the Markovian limit holds, an explicit expression for a parameter measuring the time scale for the relaxation of the oscillator is obtained in terms of the characteristic parameters of the model. Except for very high temperatures, the anharmonicity has a small influence on this relaxation time. A numerical study of the time dependence of the mean number of quanta is presented on the basis of the reported analytical solution. We find that the anharmonicity is not a significant parameter in the appearance nor in the increase of non-Markovian effects.

A Newtonian approach to quantum gravity is studied. At least for weak gravitational fields it should be a valid approximation. Such an approach could be used to point out problems and prospects inherent in a more exact theory of quantum gravity, yet to be discovered. Newtonian quantum gravity, e.g., shows promise for prohibiting black holes altogether (which would eliminate singularities and also solve the black hole information paradox), breaks the equivalence principle of general relativity, and supports non-local interactions (quantum entanglement). Its predictions should also be testable at length scales well above the "Planck scale", by high-precision experiments feasible even with existing technology. As an illustration of the theory, it turns out that the solar system, superficially, perfectly well can be described as a quantum gravitational system, provided that the $l$ quantum number has its maximum value, $n-1$. This results exactly in Kepler's third law. If also the $m$ quantum number has its maximum value ($\pm l$) the probability density has a very narrow torus-like form, centered around the classical planetary orbits. However, as the probability density is independent of the azimuthal angle $\phi$ there is, from quantum gravity arguments, no reason for planets to be located in any unique place along the orbit (or even \textit{in} an orbit for $m \neq \pm l$). This is, in essence, a reflection of the "measurement problem" inherent in all quantum descriptions.

This study investigated the conceptual challenges faced by Grade 12 STEM learners in mastering the electronic structure of atoms, encompassing the quantum mechanical model, energy levels, orbitals, sublevels, electron configuration, and quantum numbers. The research also examines learners’ self-efficacy in understanding these topics. A needs assessment questionnaire, validated by both subject-matter and methods experts, was administered to evaluate learners’ comprehension, alongside a self-efficacy survey to measure confidence in mastering the content. Results indicated generally low confidence among learners, with slightly higher self-assurance reported for determining magnetic properties from electron configurations (40% expressing moderate confidence). Analysis of mastery levels revealed that describing the quantum mechanical model of the atom was the least understood competency, averaging 55% mastery, while other topics—including energy levels, orbitals, sublevels, electron configuration, and quantum numbers—scored below 50%, with quantum numbers being the most poorly grasped concept. These findings underscore the need for targeted instructional interventions addressing the least understood topics and strategies to enhance learners’ confidence, thereby improving conceptual understanding in electronic structure of atoms.

In this study, we analyze the energy level of the diatomic molecules in the case of a non-relativistic quantum system influenced by an external magnetic field and exponential potential. The energy eigenvalue as the solution of the Schrödinger equation in the cosmic-string spacetime was simulated numerically. The energy spectrum was calculated for the three selected diatomic molecules involving RbH, NI, ScI. The influence of the quantum number, potential parameter, and the external magnetic field is considered in this computation to recover the behavior of the particle. The results showed that the energy value depends on the quantum number, potential parameter, mass of the molecule, and magnetic field strength. The magnetic field strength and the potential parameter most apparently affect the change in energy level, followed by the quantum number which also affects the increase in the energy level of the particle considered in this system. In addition, these results may be useful in studying the thermodynamic, optic, and magnetic properties of the atomic interaction of a system particle. Keywords: Non-relativistic system; Diatomic molecule; Position-dependent mass; External magnetic field; Cosmic string

The molecular energy level can be represented by: eE = eEe + eE v +eEJ, where eEe is the electronic energy level eE v is the vibrational energy level = \( hv\left( {v + {1 \over 2}} \right) \) eEj = eEr is the rotational energy level = \( J\left( {J + 1} \right){{{h^2}} \over {8{\pi ^2}I}} \) Note that v is the vibrational quantum number, and J is the rotational quantum number.

<p><em>Hamiltonian a known system in many matters does not guarantee that the equation can be resolved, e.g. due to minor disturbances such as electric fields or magnetic fields that can lead to slight changes in energy and its particle functions, for such issues it should be used disorder theory (perturbation theory). The perturbation theory can determine how much the result of the presence of disruption to energy levels and Eigen's functions. The energy possessed by particles in addition to being influenced by quantum numbers (n<sub>x</sub>, n<sub>y</sub> and n<sub>z</sub>) and the size of the potential box are also influenced by interference in the form of magnetic energy produced due to the motion of particles (electrons) in the core intrinsic magnetic field (proton) which affects the energy value does not appear due to the value of this magnetic energy disorder is very small, while for interference that is an extrinsic magnetic energy has shown a 10 (Tesla). The wider the size of the box then the energy that the particles have is smaller so that the energy level and the particle energy spectrum in the potential box will appear continuous. It can be concluded that the particle energy level relies on the length of the potential box, the quantum number of particles, and also the interference that particles have. </em></p>