Heisenberg Correspondence Principle Research Articles

Communication: Wigner functions in action-angle variables, Bohr-Sommerfeld quantization, the Heisenberg correspondence principle, and a symmetrical quasi-classical approach to the full electronic density matrix.

It is pointed out that the classical phase space distribution in action-angle (a-a) variables obtained from a Wigner function depends on how the calculation is carried out: if one computes the standard Wigner function in Cartesian variables (p, x), and then replaces p and x by their expressions in terms of a-a variables, one obtains a different result than if the Wigner function is computed directly in terms of the a-a variables. Furthermore, the latter procedure gives a result more consistent with classical and semiclassical theory-e.g., by incorporating the Bohr-Sommerfeld quantization condition (quantum states defined by integer values of the action variable) as well as the Heisenberg correspondence principle for matrix elements of an operator between such states-and has also been shown to be more accurate when applied to electronically non-adiabatic applications as implemented within the recently developed symmetrical quasi-classical (SQC) Meyer-Miller (MM) approach. Moreover, use of the Wigner function (obtained directly) in a-a variables shows how our standard SQC/MM approach can be used to obtain off-diagonal elements of the electronic density matrix by processing in a different way the same set of trajectories already used (in the SQC/MM methodology) to obtain the diagonal elements.

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.

New forms of wavefunctions for the isotropic harmonic oscillator in a time-dependent magnetic field

After a unitary transformation, exact wavefunctions of the time-dependent two-dimensional harmonic oscillator in a time-dependent magnetic field are written in the form of the product of two one-dimensional time-dependent harmonic oscillators in the (x, py) representation. Applying the Heisenberg correspondence principle (HCP) to the system, the exact classical solution is obtained from the quantum matrix elements. Meanwhile, the coherent state is discussed.

Quantum and Classical Exact Solutions ofHarmonic Oscillator in a Time-Dependent Magnetic Field

In cylindrical coordinate, exact wave functionsof the two-dimensional time-dependent harmonic oscillator ina time-dependent magnetic field are derived by using the trialfunction method. Meanwhile, the exact classical solution as wellas the classical phase is obtained too. Through the Heisenbergcorrespondence principle, the quantum solution and the classicalsolution are connected together.

Nondivergent classical response functions from uncertainty principle: Quasiperiodic systems

Time-divergence in linear and nonlinear classical response functions can be removed by taking a phase-space average within the quantized uncertainty volume O(hn) around the microcanonical energy surface. For a quasiperiodic system, the replacement of the microcanonical distribution density in the classical response function with the quantized uniform distribution density results in agreement of quantum and classical expressions through Heisenberg's correspondence principle: each matrix element (u/alpha(t)/v) corresponds to the (u-v)th Fourier component of alpha(t) evaluated along the classical trajectory with mean action (Ju+Jv)/2. Numerical calculations for one- and two-dimensional systems show good agreement between quantum and classical results. The generalization to the case of N degrees of freedom is made. Thus, phase-space averaging within the quantized uncertainty volume provides a useful way to establish the classical-quantum correspondence for the linear and nonlinear response functions of a quasiperiodic system.

Quantum-classical correspondence of the relativistic equations

According to the Heisenberg correspondence principle, in the classical limit, quantum matrix element of a Hermitian operator reduces to the coefficient of the Fourier expansion of the corresponding classical quantity. In this article, such a quantum-classical connection is generalized to the relativistic regime. For the relativistic free particle or the charged particle moving in a constant magnetic field, it is shown that matrix elements of quantum operators go to quantities in Einstein’s special relativity in the classical limit. Especially, matrix element of the standard velocity operator in the Dirac theory reduces to the classical velocity. Meanwhile, it is shown that the classical limit of quantum expectation value is the time average of the classical variable.

Quantum–Classical Correspondence of the Time-Dependent Linear Potential

Using the trial-function method, the general solution of the Schrodinger equation for the time-dependent linear potential is obtained. Based on the Heisenberg correspondence principle, the solution of the classical equation of motion is derived from the quantum matrix elements.

Forced Time-Dependent Harmonic Oscillator in a Static Magnetic Field: Exact Quantum and Classical Solutions

Exact wave functions of the forced time-dependent two-dimensional harmonic oscillator in a static magnetic field are derived by unitary transformation. The geometrical phase induced by the driving force is the phase of the de Broglie wave associated with the particle moving according to the classical equation. Extending the idea of the Heisenberg correspondence principle to the time-dependent system, the exact classical solution \({\dot \nu }\) obtained from quantum matrix elements.

Quantum and Classical Exact Solutions of the Time-Dependent Driven Generalized Harmonic Oscillator

Using the trial function method, exact wave functions of the time-dependent driven generalized harmonic oscillator (TDGHO) were derived. The wave functions are wave packets with the center moving according to the classical equation of motion. Correspondingly, the geometrical phase is divided into two parts with one part generated by the shape change of the wave packet and the other part by the motion of the wave packet center. The exact classical solution as well as the classical phase was obtained from a quantum description by the Heisenberg correspondence principle. Finally, the relationship between the quantum and classical phases was achieved.

The hydrogen atom's quantum-to-classical correspondence in Heisenberg's correspondence principle

The famous Gordon formula for matrix element n',l|r|n,l-1 is transformed into a new form which can be easily treated when n', n and l are all large. Then its asymptotic expression is derived which turns out to be that obtained from Heisenberg's correspondence principle. Finally, its classical limit form is used to construct classical quantities x, y and z representing a singe Keplerian orbit in terms of a Fourier series of time variable t.

Semiclassical mechanics in one dimension: II. Approximate matrix elements

The semiclassical formula for matrix elements, sometimes called the Heisenberg correspondence principle, relates matrix elements (operators in energy representation) to phase space functions (Weyl representatives). The formula does not make sense for arbitrary operators; when it is valid it implicitly fixes the relative phases of the eigenfunctions of the Hamiltonian. The conventions, which have to be used for the wavefunctions in position or momentum representation, are given here in explicit form, and we present a class of operators related to coherent states of high energy for which the Heisenberg correspondence principle holds.

Phase space structure of triatomic molecules

The bifurcation structure is investigated for a Hamiltonian for the three coupled nonlinear vibrations of a highly excited triatomic molecule. The starting point is a quantum Hamiltonian used to fit experimental spectra. This Hamiltonian includes 1:1 Darling–Dennison resonance coupling between the stretches, and 2:1 Fermi resonance coupling between the stretches and bend. A classical Hamiltonian is obtained using the Heisenberg correspondence principle. Surfaces of section show a pronounced degree of chaos at high energies, with a mixture of chaotic and regular dynamics. The large-scale bifurcation structure is found semianalytically, without recourse to numerical solution of Hamilton’s equations, by taking advantage of the fact that the spectroscopic Hamiltonian has a conserved polyad quantum number, corresponding to an approximate constant of the motion of the molecule. Bifurcation diagrams are analyzed for a number of molecules including H2O, D2O, NO2, ClO2, O3, and H2S.

Semiclassical integrable matrix elements.

A semiclassical expression for matrix elements of an arbitrary operator with respect to the eigenstates of an integrable Hamiltonian is derived. This is essentially the Heisenberg correspondence principle, and it is shown via the Weyl correspondence that the approximation is valid through the lowest two orders in \ensuremath{\Elzxh}. The result is used to prove that an asymptotic form of the Clebsch-Gordan coefficients for two large and one small angular momenta is valid through two orders. \textcopyright{} 1996 The American Physical Society.

Semiclassical approach to Rydberg-atom intercombination transitions in collisions with electrons.

The intercombination transitions between l-resolved Rydberg levels due to collisions with fast electrons are studied within the Ochkur approximation for principal quantum numbers n\ensuremath{\le}20, where l is the angular momentum. The use of the Heisenberg correspondence principle for radial integrals enables one to obtain analytic expressions for any term in the multipole expansion of the cross section for collisions with an arbitrary n and l change. For transitions involving angular momenta that are small compared to the principal quantum numbers, the semiclassical and the exact results are found to be fairly close, within the accuracy consistent with the Born approximation. As a result, the semiclassical approach furnishes a ready estimate for cross sections, and its validity range proves broad enough to cover the region of large momenta transferred to the atom. The numerical results also confirm that dipole transitions have no dominance over the collisions with other \ensuremath{\Delta}l values, in accord with the general model of intercombination scattering.

Angular momentum and Heisenberg's correspondence principle: III. Rotation matrix elements

Heisenberg's correspondence principle is applied to the matrix elements of the rotation operator; in this way, an approximation for the reduced rotation matrix elements dJM'M( theta ) in terms of Bessel functions is obtained. It is shown that two distinct approximate forms are necessary to give sufficient accuracy over the entire range 0 to pi of the angle theta if the approximation is to be of value. The two forms are most accurate for theta near 0 and pi respectively, deteriorating as 0= pi /2 is approached; however, they retain a surprising degree of accuracy over the full range, particularly when (M'-M) is small and J large, the case for which the exact expression is most complex. Taken in conjunction with the results of the previous papers in this series, the present work allows Bessel function approximations to be obtained for both Clebsch-Gordan and Racah coefficients; indications as to the likely accuracy of such approximations are given.

Semiclassical hyperspherical matrix elements for helium doubly excited states

A classical description of the two-electron atom, analogous to the quantum adiabatic hyperspherical channel approach, is presented. The classical problems, analogous to the quantum eigenvalue problem for the grand angular momentum operator, and the separated dynamical systems defined by each of the other constants of the motion of the non-interacting system are solved by using the Hamilton-Jacobi method. Some matrix elements of the Coulomb interaction terms of the Hamiltonian for doubly excited helium atoms using the Heisenberg correspondence principle are calculated.

Angular momentum and Heisenberg's correspondence principle

Heisenberg's form of the correspondence principle is applied to the matrix elements of tensor operators. A correspondence limit of the Wigner-Eckart theorem is obtained. It is shown that the Clebsch-Gordan coefficient tends in this limit to the product of a reduced rotation matrix element with a simple multiplicative factor. A similar, but not identical, expression has long been known as a formal limit obtaining when two of the three angular momenta involved become large. It is characteristic of correspondence principle expressions that the initial and final quantal states must be represented by some 'mean' classical orbit, the choice of which remains ambiguous. Here recursion relations are used to arrive at the optimal choice. In this way, a correspondence principle approximation to the Clebsch-Gordan coefficient is obtained which may be used with success far from the limit to which it nominally refers. This approximation is tested numerically against both exact and semiclassical computations.

Unified semiclassical dynamics for molecular resonance spectra

A method is presented to depict the intramolecular dynamics of resonantly coupled vibrations, starting from the experimental overtone and combination spectrum. The nonlinear least-squares fit of the spectrum is used to obtain a semiclassical phase space Hamiltonian via the Heisenberg correspondence principle. This integrable Hamiltonian, corresponding to quasiperiodic motion, is used to generate a classical trajectory in phase space for each energy level in a resonance polyad. Polyad phase space profiles are shown to have complete mutual consistency starting from a fit in either the local or normal representation. It is argued that the best way to depict the phase space profile is on a spherical surface called the polyad phase sphere. Represented in this way, the local and normal mode phase spaces are seen to be a single entity, manifestly equivalent by a 90° rotation. The phase space trajectories can be converted into a coordinate space representation. This gives an easily visualized picture of the semiclassical intramolecular dynamics corresponding to each energy level. The polyad phase spheres from the fits of the experimental stretching spectra of H2O, O3 and SO2 are displayed. H2O and O3 are seen to be molecules with a local to normal modes transition, while SO2 is seen to be very near the pure normal modes limit. The experimentally determined phase space dynamics of H2O seen on the phase sphere are compared with the dynamics determined by Lawton and Child from trajectory calculations on the Sorbie–Murrell potential surface. The coordinate space trajectories corresponding to the phase spheres are compared with wave functions from the fit of the spectrum.

Fermi resonance phase space structure from experimental spectra

The method of algebraic resonance dynamics is developed to extract semiclassical dynamics of molecules from the fit to the experimental spectrum of bend and stretch normal modes with 2:1 Fermi resonance. The Heisenberg correspondence principle is used to determine phase space profiles for resonance polyads. Each quantum level in a polyad corresponds to a trajectory in action-angle phase space. Profiles for experimental spectra of CO2 and a series of substituted methanes show a common phase space structure with four distinct types of semiclassical motion.

A correspondence principle for strongly coupled states

Heisenberg's correspondence principle for non-relativistic matrix elements has been generalized: quantal transition amplitudes between strongly coupled states are expressed as Fourier components of integrals over classical trajectories. The new theory reduces to Heisenberg's correspondence principle in particular cases. The theory is applicable whenever the change in quantum number is small compared with the initial quantum number. In these situations it is more comprehensive than both quantal first order perturbation theory and the sudden approximation. Furthermore, compared with standard quantal methods, the evaluation of amplitudes is particularly simple. The theory is worked out for one- and three-dimensional separable systems and the generalization to a system of arbitrary dimension is indicated.