Limit Of Large Quantum Numbers Research Articles

Resonant electron transfer in ion-metal-surface interactions: The case of large quantum numbers

Some theoretical aspects of resonant electron transfer processes occurring in the interaction of slow atoms and ions with metal surfaces are surveyed. The emphasis is on cases involving atomic states with large quantum numbers, i.e., on surface interactions of Rydberg atoms and of highly charged ions. Resonant transition rates calculated in first-order adiabatic approximation are examined with regard to their scaling properties, and universal behavior of the rates is disclosed by analyzing the limit of large quantum numbers. Quasiclassical aspects of these results are discussed, and comparison is made with purely classical estimates for the transition rates. Angular momentum effects in resonance ionization of Rydberg atoms are considered explicitly.

Asymptotic form of the penetration probability of the quantum harmonic oscillator into the classically forbidden region

The penetration probability of the quantum harmonic oscillator into the classically forbidden region is examined in the limit of large quantum numbers using analytical methods. An expression is derived for the asymptotic form of the penetration probability, and is shown to be in good agreement with an earlier result obtained by numerical methods. The asymptotic form of the probability density in the region between the classical turning points is also presented and found to have a simple physical interpretation.

Separatrix eigenfunctions.

Near-separatrix eigenfunctions for the double-well potential are analyzed. These functions are needed for the study of systems with perturbed or slowly varying double-well potentials. The probability density of separatrix eigenfunctions collapses to the classical result (a \ensuremath{\delta} function at the unstable fixed point) logarithmically with the number of quantum states. Matrix elements with respect to this basis are also studied. Unlike the wave functions, the unnormalized matrix elements are nonsingular in the limit of large quantum number.

Synthesis of the Planck and Bohr formulations of the correspondence principle

Planck formulated the correspondence principle between quantum and classical mechanics as the limit in which the Planck constant h goes to zero. Bohr formulated the correspondence principle to be the limit of large quantum numbers. For three common quantum mechanical systems it is shown that in order for eigenvalues of quantum mechanical observables to have meaningful classical limits, it is necessary to take the double limit as both the Planck constant goes to zero and the quantum number goes to infinity, subject to the constraint that their product is equal to an appropriate classical action. This synthesis of the Bohr and Planck formulations of the correspondence principle is also used to show that the quantum mechanical transition frequency between adjacent levels approaches the corresponding classical frequency. The features these systems have in common in their classical limit are explained by general considerations of the classical limit of the Schrödinger equation.

Non adiabatic theory of collisions in a radiative field. Semi-classical limit

The quantal theory of atomic collisions in a radiative field is now well established for weak fields which we consider here. Several calculations of radiative cross sections and redistribution of polarization compare extremely well with experimental results. Semi-classical approaches often allow a better physical interpretation. However, the above mentioned results show that trajectory and internuclear reorientation effects during the collision play an essential role so that sophisticated methods have to be developed. In this paper (paper 1), we derive the semi-classical close coupled equations in the limit of large quantum numbers of the quantal equations thereby avoiding the rectilinear trajectory assumption. An adequate choice of the axis frame of reference allows us to show the exact correspondence between this approach and the direct semi-classical derivation of the close coupled equations. This leads us to interpret physically the P, Q, R spectroscopic branches in terms of the rotation of the internuclear axis during the collision Obtention des equations couplees semi-classiques et des sections efficaces radiatives comme limites, aux grands moments cinetiques, des expressions quantiques correspondantes, ce qui permet de s'affranchir d'une hypothese de trajectoire rectiligne. Par un choix adequat des systemes d'axes, etablissement d'une correspondance exacte avec l'approche semi-classique directe. Interpretation physique des branches spectroscopiques P, Q, R en termes de rotation de l'axe internucleaire durant la collision

Ehrenfest theorem and the classical trajectory of quantum motion

Ehrenfest theorem asserts that the quantum mechanical motion of a particle when considered in the expectation value sense should agree with classical mechanics in the correspondence limit. An explicit verification of this result is presented in the one-dimensional case for motion in an infinite potential well (large quantum number limit) and a brief mention is made of the case of a smooth potential (ℏ→0 limit).

Semiclassical kinetic equation and the WKB approximation

We study the connection between the classical response function of a nucleus described by the Vlasov equation and the corresponding quantum response function. In the limit of large quantum numbers, the Fourier coefficients which appear in the Vlasov theory correspond to the quantum matrix elements evaluated in the WKB approximation. The classical frequencies of motion give the quantum excitation energies according to the correspondence principle. For a central potential, we identify the normal modes of the classical systems with the corresponding shell-model transitions. Thus we can improve the Vlasov theory by excluding excitation modes which correspond to forbidden quantum transitions. Also, we study the effect of a spin-orbit force in the context of the Vlasov theory. Finally, always by exploiting the close analogy between the classical and the quantum response functions, we introduce exchange (Fock) term in the semiclassical RPA equation given by the Vlasov theory.

A projection based approach to the Clebsch-Gordan multiplicity problem for compact semisimple Lie groups. III. The classical limit

For pt.II see ibid., vol.19, p.1531 (1986). This paper is the third in a series directed towards a projection-based solution to the Clebsch-Gordan multiplicity problem for semisimple (compact) Lie groups. The authors show that the project states for the Clebsch-Gordan problem approach orthogonality in the classical limit of large quantum numbers in a manner analogous to that of Elliot's (1962) well known solution to the U(3) contains/implies O(3) state labelling problem.

The correspondence principle revisited

The correspondence principle addresses the connection between classical and quantum physics. The simple statement that quantum mechanics reduces to classical mechanics in the limit where the principal quantum number n approaches infinity, while found in many textbooks, is not true in general. In this article we will give special attention to the notion that the quantum frequency spectrum of a periodic system reduces to the classical spectrum in this limit. Two simple counter-examples—a particle in a cubical box, and a rigid rotator—will show us that the classical result is not always recovered in the limit of large quantum numbers.

Semiclassical quantization of the nonlinear Schrödinger equation

Using the functional integral technique of Dashen, Hasslacher, and Neveu, we perform a semiclassical quantization of the nonlinear Schrödinger equation, which reproduces McGuire's exact result for the energy levels of the theory's bound states. We show that the stability angle formalism leads to the one-loop normal ordering and self-energy renormalization expected from perturbation theory and demonstrate that taking into account center-of-mass motion gives the correct nonrelativistic energymomentum relation. We interpret the classical solution in the context of the quantum theory, relating it to the matrix element of the field operator between adjacent bound states in the limit of large quantum numbers. Finally, we quantize the NLSE as a theory of N component fermion fields and show that the semiclassical method yields the exact energy levels and correct degeneracies.

Semiclassical approximations to 3j- and 6j-coefficients for quantum-mechanical coupling of angular momenta

The coupling of angular momenta is studied using quantum mechanics in the limit of large quantum numbers (semiclassical limit). Uniformly valid semiclassical expressions are derived for the 3j (Wigner) coefficients coupling two angular momenta, and for the 6j (Racah) coefficients coupling three angular momenta. In three limiting cases our new expressions reduce to those conjectured by Ponzano and Regge. The derivation involves solving the recursion relations satisfied by these coefficients, by a discrete analog of the WKB method. Terms of the order of the inverse square of the quantum numbers are neglected in the derivation, so that the results should be increasingly accurate for larger angular momenta. Numerical results confirm this asymptotic convergence. Moreover, the results are of a useful accuracy even at small quantum numbers.

Bohr correspondence principle for large quantum numbers

Periodic systems are considered whose increments in quantum energy grow with quantum number. In the limit of large quantum number, systems are found to give correspondence in form between classical and quantum frequency-energy dependences. Solely passing to large quantum numbers, however, does not guarantee the classical spectrum. For the examples cited, successive quantum frequencies remain separated by the incrementhI−1, whereI is independent of quantum number. Frequency correspondence follows in Planck's limit,h → 0. The first example is that of a particle in a cubical box with impenetrable walls. The quantum emission spectrum is found to be uniformly discrete over the whole frequency range. This quality holds in the limitn → ∞. The discrete spectrum due to transitions in the high-quantum-number bound states of a particle in a box with penetrable walls is shown to grow uniformly discrete in the limit that the well becomes infinitely deep. For the infinitely deep spherical well, on the other hand, correspondence is found to be obeyed both in emission and configuration. In all cases studied the classical ensemble gives a continuum of frequencies.

Asymptotic evaluation of the Green's function for large quantum numbers

In view of extending the shell model to highly deformed states of heavy nuclei, we discuss the evaluation of the Green's function for the Schrödinger equation in three dimensions with a smooth potential, in the limit of large quantum numbers. Such an evaluation is possible only after smoothing over the energy, which results in the introduction of complex energies. The zeroth order (semiclassical) approximation requires the resolution of the Hamilton-Jacobi equation for complex energies. For a sufficiently large smoothing width, the Green's function is expressed as a local expansion based on the Taylor series expansion of the potential. Higher order corrections to the semiclassical approximation are derived from an integral equation for the Green's function. These corrections produce in particular the reflected waves from the caustic (which generalizes in three dimensions the turning points). The connection with the usual BKW method is discussed in the one-dimensional case by considering the limit of a small smoothing width. The quantum corrections to the Thomas-Fermi model are derived as a first application of these methods.

Two Notes on Phase-Integral Methods

I. The connection formulas are derived by a new method. The general approach is that of Zwaan's discussion of Stokes' phenomenon, but no use is made of any assumptions about the reality of the coefficients of the differential equation. Instead of this, the proof is based only on the fact that actual solutions of the differential equation must be single-valued. Both this manner of proof and the form in which the results are obtained are suited to the discussion of certain problems in which a boundary condition consists in the requirement that the field at a great distance contain an out-going wave only. The results serve to establish the validity of Eckersley's phase-integral method for the treatment of problems of wave propagation. II. General arguments used to establish phase-integral methods are asymptotic in nature, and lead to the expectation that the methods will be valid only in a certain limit; in the language of quantum mechanics this is the limit of large quantum numbers. Much of the methods' usefulness, however, comes from the fact that they give in practice surprisingly accurate results even for small quantum numbers. In the case of the energy levels of the anharmonic oscillator, special arguments have been devised by Kemble and by Birkhoff to establish the usual phase-integral formula without assuming large quantum numbers. In this note a special argument is given for calculating normalization factors for the approximate wave functions of the oscillator. It is shown that the usual asymptotic formula for the normalization factor holds for the lowest quantum states, with about the accuracy with which the phase-integral solution approximates the shape of the exact wave function at the point at which the potential energy function has its minimum.