Limit Of Large Quantum Numbers Research Articles

Understanding the relation between classical and quantum mechanics: prospects for undergraduate teaching

Classical and quantum mechanics are two very different theories, each describing the world within its own range of validity. It is often stated that classical mechanics emerges from quantum mechanics in a certain limit. This is known as the correspondence principle. According to Planck’s version of the correspondence principle, classical mechanics is recovered when the limit in which a dimensionless parameter containing Planck’s constant h goes to zero is taken, while Bohr’s version entails taking the limit of large quantum numbers. However, despite what is usually stated in textbooks, the relation between the two theories is much more complex to state and understand. Here we deal with this issue by analysing some key examples, in some of which also the analogously subtle relation between wave and geometric optics is considered. Implications for quantum mechanics teaching at undergraduate level are carefully discussed.

Exact classical limit of the quantum bouncer

In this paper, we develop a systematic approach to determine the classical limit of periodic quantum systems and we applied it successfully to the problem of the quantum bouncer. It is well known that, for periodic systems, the classical probability density does not follow the quantum probability density. Instead, it follows the local average in the limit of large quantum numbers. Guided by this fact, and expressing both the classical and quantum probability densities as Fourier expansions, here we show that local averaging implies that the Fourier coefficients approach each other in the limit of large quantum numbers. The leading term in the quantum Fourier coefficient yields the exact classical limit, but subdominant terms also emerge, which we may interpret as quantum corrections at the macroscopic level. We apply this theory to the problem of a particle bouncing under the gravity field and show that the classical probability density is exactly recovered from the quantum distribution. We show that for realistic systems, the quantum corrections are strongly suppressed (by a factor of \(\sim 10^{-10}\)) with respect to the classical result.

Determination of classical behaviour of the Earth for large quantum numbers using quantum guiding equation

For quantum systems, we expect to see the classical behaviour at the limit of large quantum numbers. Hence, we apply Bohmian approach for describing the evolution of Earth around the Sun. We obtain possible trajectories of the Earth system with different initial conditions which converge to a certain stable orbit, known as the Kepler orbit, after a given time. The trajectories are resulted from the guiding equation $p=\nabla S$ in the Bohmian mechanics, which relates the momentum of the system to the phase part of the wave function. Except at some special situations, Bohmian trajectories are not Newtonian in character. We show that the classic behaviour of the Earth can be interpreted as the consequence of the guiding equation at the limit of large quantum numbers.

Revealing a quantum feature of dimensionless uncertainty in linear and quadratic potentials by changing potential intervals

As an undergraduate exercise, in an article (2012 Am. J. Phys. 80 780–14), quantum and classical uncertainties for dimensionless variables of position and momentum were evaluated in three potentials: infinite well, bouncing ball, and harmonic oscillator. While original quantum uncertainty products depend on and the number of states (n), a dimensionless approach makes the comparison between quantum uncertainty and classical dispersion possible by excluding . But the question is whether the uncertainty still remains dependent on quantum number n. In the above-mentioned article, there lies this contrast; on the one hand, the dimensionless quantum uncertainty of the potential box approaches classical dispersion only in the limit of large quantum numbers ()—consistent with the correspondence principle. On the other hand, similar evaluations for bouncing ball and harmonic oscillator potentials are equal to their classical counterparts independent of n. This equality may hide the quantum feature of low energy levels. In the current study, we change the potential intervals in order to make them symmetric for the linear potential and non-symmetric for the quadratic potential. As a result, it is shown in this paper that the dimensionless quantum uncertainty of these potentials in the new potential intervals is expressed in terms of quantum number n. In other words, the uncertainty requires the correspondence principle in order to approach the classical limit. Therefore, it can be concluded that the dimensionless analysis, as a useful pedagogical method, does not take away the quantum feature of the n-dependence of quantum uncertainty in general. Moreover, our numerical calculations include the higher powers of the position for the potentials.

On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity

Building on a recent proposal for a quantum reduction to spherical symmetry from full loop quantum gravity, we investigate the relation between a quantisation of spherically symmetric general relativity and a reduction at the quantum level. To this end, we generalise the previously proposed quantum reduction by dropping the gauge fixing condition on the radial diffeomorphisms, thus allowing us to make direct contact with previous work on reduced quantisation. A dictionary between spherically symmetric variables and observables with respect to the reduction constraints in the full theory is discussed, as well as an embedding of reduced quantum states to a subsector of the quantum symmetry reduced full theory states. On this full theory subsector, the quantum algebra of the mentioned observables is computed and shown to qualitatively reproduce the quantum algebra of the reduced variables in the large quantum number limit for a specific choice of regularisation. Insufficiencies in recovering the reduced algebra quantitatively from the full theory are attributed to the oversimplified full theory quantum states we use.

Hydrogen-Like Atom in Spaces of Lower Dimensions

The problem of finding discrete energy values of a particle with negative charge equal in absolute value to the elementary charge in one-dimensional space (e → e 1), located in the field of a nucleus with charge (Ze 1) with potential corresponding to the space of this dimension (N = 1) and different from the potential of the nucleus in three-dimensional space (N = 3) is solved in the quasiclassical approximation. For the one-dimensional case, the corresponding Schrodinger equation is solved and exact energy values are obtained, coincident with the quasiclassical approximation in the limit of large quantum numbers, and the wave function, expressed in terms of the Airy function, is found. In this latter approach, the energy values depend on the zeros of the Airy function. Considerations are discussed, touching on the possibility of solution of the Schrodinger equation for a hydrogen-like atom in two-dimensional space (N = 2).

The physical boundary Hilbert space and volume operator in the Lorentzian new spin-foam theory

A covariant spin-foam formulation of quantum gravity has been recently developed, characterized by a kinematics which appears to match well the one of canonical loop quantum gravity. In this paper we reconsider the implementation of the constraints that defines the model. We define in a simple way the boundary Hilbert space of the theory, introducing a slight modification of the embedding of the SU(2) representations into the ones. We then show directly that all constraints vanish on this space in a weak sense. The vanishing is exact (and not just in the large quantum number limit). We also generalize the definition of the volume operator in the spin-foam model to the Lorentzian signature and show that it matches the one of loop quantum gravity, as in the Euclidean case.

Energy-spin trajectories in AdS 5 × S 5 from semiclassical vertex operators

We study the relation between vertex operators in AdS 5 × S 5 and classical spinning string solutions. In the limit of large quantum numbers the treatment of vertex operators becomes semiclassical. In this regime, a given vertex operator carrying a certain set of quantum numbers defines a singular solution. We show in a number of examples that this solution coincides with the classical string solution with the same quantum numbers but written in a different two-dimensional coordinate system. The marginality condition imposed on an operator yields a relation between the energy and the other quantum numbers which is shown to coincide with that of the corresponding classical string solution. We also argue that in some cases vertex operators in AdS 5×S 5 cannot be given by expressions similar to the ones in flat space and a more involved consideration is required.

Quantum-classical correspondence of the Dirac equation with a scalar-like potential

Quantum matrix elements of the coordinate, momentum and the velocity operator for a spin-1/2 particle moving in a scalar-like potential are calculated. In the large quantum number limit, these matrix elements give classical quantities for a relativistic system with a position-dependent mass. Meanwhile, the Klein-Gordon equation for the spin-0 particle is discussed too. Though the Heisenberg equations for both the spin-0 and spin-1/2 particles are unlike the classical equations of motion, they go to the classical equations in the classical limit.

Long-range interaction between ground and excited state hydrogen atoms

The asymptotic expansion at large distances is obtained for the interaction between a ground state hydrogen atom and an excited hydrogen atom with principal quantum number n = 2, …, 10. A degenerate perturbation theory up to the second order is employed to obtain accurate results. The asymptotic representation for several special cases is found in the limit of large quantum number n.

Large transient states in quantum field theory

A study is made of the scattering of two large composite projectiles, such as heavy ions, which are initially prepared in a pure quantum state. It is shown that the quantum field theoretic evolution equation for this system, under certain conditions, goes over in form to the master equation of classical statistical mechanics. Thus, the statistical mechanical description of heavy ion collision is viewed as an implied outcome of the Correspondence Principle, which states that in the limit of large quantum number, quantum dynamics goes over to classical dynamics. This hypothesis is explored within the master equation transcription with particular focus on the quark–gluon formation scenario.

Quantum-mechanical vs. semi-classical spectral-line widths and shifts from the line core in the non-impact region for the Ar-perturbed/ K-radiator system

New quantum-mechanical (QM) and semi-classical (SC) shifts (d's) and widths (HWHM's, w's) were measured from the line core of computed full spectral-line shapes for the Ar-perturbed/K-radiator system (K/Ar). The initial state of our model was based on a 4 p 2 P 3/2,1/2 pseudo-potential for the K/Ar system, and the final state on a zero potential. The Fourier transform of the line shape formed the basis for the computations. Excellent agreement was found between the QM and SC values of d and of w in a high-pressure ( P) non-impact region, which was characterized by a √ P dependence of w and a P dependence of d. These agreements were shown to be another example of a correspondence between classical (SC) quantities and QM quantities in the limit of large quantum numbers. Typically at P=1×10 6 Torr and T=400 K, w QM= 448 cm −1 and w SC =479 cm −1, where the deviation from the mean is ±3.3%. Also, d QM =−3815 cm −1 and d SC =−3716 cm −1, where the deviation from the mean is ±1.3%. A new general method was formulated which yielded a definite pressure P 0, which was defined as an upper limit to the low-pressure impact approximation and a lower limit to the non-impact region.

Matrix elements and classical limit of relativistic particles in infinitely deep potential well

For relativistic particles with spin-0 (satisfying the Klein-Gordon equation) and spin-1/2 (satisfying the Dirac equation) in infinitely deep potential well, matrix elements for the coordinate, momentum and the velocity operators are calculated. In the limit of large quantum numbers, these matrix elements give the corresponding classical quantities (nowbeing related quantities in special relativity) and satisfy exact classical relations. These results show that the Heisenberg correspondence principle is applicable to such relativistic systems.

Exact solutions of time-dependent Dirac equations and the quantum–classical correspondence

Exact solutions to the Dirac equations with a time-dependent mass and a static magnetic field or a time-dependent linear potential are given. Matrix elements of the coordinate, momentum and velocity operator are calculated. In the large quantum number limit, these matrix elements give the classical solution.

Some properties of the five-dimensional surface harmonics

Ladder operators and a triangular relation are used to derive the five-dimensional surface harmonics with definite angular momentum, as used in studies of the dynamics of a quadrupole shape in the nuclear collective model. A new basis is used which leads to solutions in terms of associated Legendre functions. The role of the Octahedral symmetry group and the limit of large quantum numbers are discussed.

Comment on “Coherent states for the hydrogen atom”

Majumdar and Sharatchandra recently proposed a set of ``coherent states'' for the hydrogen atom [Phys. Rev. A 56, R3322 (1997)]. These states satisfy some of the typical properties of coherent states, such as, for example, continuity in the parameters. The time evolution of these states is given by the classical evolution of the angle variables, and the expectation values of the quantum observables behave quasiclassically. However, those authors also claimed that, although one does not obtain exactly the same state after a Kepler period, the gross features of their wave packets do not change. We show that this is not correct, and that these wave packets do not remain localized on the classical variables, but spread rapidly over the Keplerian orbit. The localization properties of these states do not improve in the limit of large quantum numbers.

The Classical Lamb Shift: Why Jackson is Wrong!

At the Edirne summer school, a discussion developed on how it seemed to be possible to use classical methods to calculate atomic spontaneous emission rates, but not the Lamb shift. This point of view has particularly been emphasized in Jackson’s book, Classical Electrodynamics, in which he flat-out states that a classical argument based on radiation-reaction theory and the Abraham-Lorentz equation will give the correct Einstein A coefficient for spontaneous decay, but gives the wrong value for the Lamb shift1. This bit of folklore is often touted as some sort of proof that classical methods can not reproduce elements of quantum electrodynamics, even in a limiting sense. Usually, as Jackson does, the blame is placed on electromagnetic vacuum fluctuations — a manifestly nonclassical phenomenon — that is claimed to be the “true” origin of the Lamb shift. But people claim that vacuum fluctuations are also the cause of atomic spontaneous emission — how come classical radiation-reaction theory is right there? “Just good luck,” is the usual reply. This seems an unsatisfactory state of affairs, for several reasons. To me, the correspondence principle presents the biggest objection to the notion that there is no classical analog to the Lamb shift. In the limit of large quantum numbers, one would expect the quantum electrodynamical Lamb shift to go over into a classical frequency pulling that an oscillating macroscopic charge experiences by interacting with its self field. An additional disturbing fact is that classical theory is quite often used to compute cavity corrections to the Lamb shift2, and at least for excited atoms the results agree exactly with full QED calculations3. Why then can the classical approach get the cavity-induced Lamb shift right, but not the free-space shift? As we shall see, there is nothing wrong with the classical approach to the free-space Lamb shift; it is Jackson who is wrong!

Interpretation of time-reparametrization-invariant quantum mechanics: An exactly soluble model.

The classical and quantum dynamics of simple time-reparametrization- invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast to a similar proposal of Rovelli, evolution is with respect to the geometrical proper time, and the Heisenberg equation of motion is exact. The possibility of a ``test clock'', which would reveal time evolution while contributing negligibly to the Hamiltonian constraint is examined, and found to be viable in the semiclassical limit of large quantum numbers.

Function-mixing hypothesis and quantum chaos

Billiards are studied whose boundaries comprise two or more segments which allow local (but not global) separation of the Helmholtz equation. Related local solutions are labeled “sep” functions. Such billiards comprise the Σ and Σ sets. A definition of quantum chaos for these sets of billiards is presented based on the ratio of the “fluctuation length” of the wave function nodal pattern to the “c-diameter” of the billiard. The “function-mixing hypothesis” states that a sufficient condition for a billiard to be chaotic is that the billiard be an element of one of these sets. It further ascribes such chaotic behavior to be due to mixing of dissimilar sep functions. Examples of the application of this hypothesis are described. A set-theoretic formalism is introduced to describe perturbation theory for infinite potentials and applied to the concave-sided square billiard of sufficiently small concavity. It is concluded that the adiabatic theorem of quantum mechanics does not apply to this configuration in the limit of large quantum numbers.

Asymptotic Clebsch–Gordan coefficients

We present derivations of su(2) and su(3)⊇so(3) Clebsch–Gordan coefficients in the limit of large quantum numbers. For su(3), we derive simple analytic expressions for the asymptotic coefficients needed for the decomposition of the product (λ1,0)⊗(λ2,0), in the limit of λ1→∞, in an so(3) basis.