The green's functions of the Schrödinger equation for the simplest systems

The Green's functions for the simplest quantum mechanical systems the linear harmonic oscillator, the three-dimensional isotropic oscillator, the Morse oscillator, the Kratcer potential, and the double-minimum potential V_(x) = (mw²/2)(/x/?R)² are presented in closed analytical forms.     

Introducing the γ function in quantum theory

Through a series of postulates, we define a function γ(x) whose square acts as Dirac's δ(x) and exhibits several unusual properties. Though the square root of δ cannot be defined among distributions, it appears in quantum theory if one wants to associate a wave function to a (quasi)classical particle having charge distribution δ(x) . The newly defined function γ(x) serves to describe quasi-classical particles using part of the quantum formalism (eg, wave functions, operators, expectation values) but exhibiting classical properties. The function γ(x) appears to be useful to define model wave functions for simple (quasi)quantum systems. In a spherical coordinate system, γ(r − r₀) leads to a quasi-classical “bubble” model of the hydrogen atom, where the electron is distributed on the surface of a sphere with radius r₀, and it provides exact quantum mechanical energies of its total symmetric levels. For other simple quantum systems, it provides approximate but meaningful energies. In particular, exact energy differences for harmonic oscillator levels are obtained, with the zero-point energy missing.     

SWKB formalism for confined quantum systems

A formalism is developed to study spatially confined one‐dimensional quantum mechanical systems in the framework of the supersymmetric Wentzel–Kramers–Brillouin (SWKB) method. The approximation technique is applied to estimate the energy eigenvalues of two confined potentials—the harmonic oscillator V(x)=x² and the screened Coulomb potential V(x)=−V₀sech²x. The results thus obtained are found to be in better agreement with the exact numerical values than are those from the ordinary WKB approach. © 2000 John Wiley & Sons, Inc. Int J Quant Chem 79: 267–273, 2000     

New unitary transformation for the three coupled oscillators

A new unitary operator U, which can transform the Fock space of a three-dimensional isotropic harmonic oscillator into the space in which the Hamiltion of three coupled oscillators is diagonized, is found. The coordinate representation of U is presented and is used to directly derive the wave function of the energy eigenstate of the coupled oscillators.     

Degeneracy in one‐dimensional quantum mechanics: A case study

In this work we study the isotonic oscillator, V(x) = Ax² + Bx^?2, on the whole line ?∞ < x < + ∞ as an example of a one‐dimensional quantum system with energy level degeneracy. A symmetric double‐well potential with a finite barrier is introduced to study the behavior of energy pattern between both limit: the harmonic oscillator (i.e., a system without degeneracy) and the isotonic oscillator (i.e., a system with degeneracy). © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2010     

Application of the strong-coupling correspondence principle to atom–triatom collinear collisions

The strong-coupling correspondence principle has been used to calculate the T–V transition probabilities in the collinear collision of He and Kr with CO₂. For a harmonic CO₂ potential the results for He agree well with published quantum mechanical probabilities. For Kr the agreement is less satisfactory but at worst the ratio of quantum to semiclassical transition probability is approximately 0.2. Introduction of anharmonicity in the CO₂ potential was found to increase the semiclassical transition probabilities but this may just be due to the lowering of the vibration frequencies.     

Interpreting nonlinear vibrational spectroscopy with the classical mechanical analogs of double-sided Feynman diagrams

Observables in coherent, multiple-pulse infrared spectroscopy may be computed from a vibrational nonlinear response function. This response function is conventionally calculated quantum-mechanically, but the challenges in applying quantum mechanics to large, anharmonic systems motivate the examination of classical mechanical vibrational nonlinear response functions. We present an approximate formulation of the classical mechanical third-order vibrational response function for an anharmonic solute oscillator interacting with a harmonic solvent, which establishes a clear connection between classical and quantum mechanical treatments. This formalism permits the identification of the classical mechanical analog of the pure dephasing of a quantum mechanical degree of freedom, and suggests the construction of classical mechanical analogs of the double-sided Feynman diagrams of quantum mechanics, which are widely applied to nonlinear spectroscopy. Application of a rotating wave approximation permits the analytic extraction of signals obeying particular spatial phase matching conditions from a classical-mechanical response function. Calculations of the third-order response function for an anharmonic oscillator coupled to a harmonic solvent are compared to numerically correct classical mechanical results.     

A semiclassical study of wave packet dynamics in anharmonic potentials

Classical and semiclassical methods are developed to calculate and invert the wave packet motion measured in pump-probe experiments. With classical propagation of the Wigner distribution of the initial wave packet created by the pump pulse, we predict the approximate probe signal with slightly displaced recurrence peaks, and derive a set of first-order canonical perturbation expressions to relate the temporal features of the signal to the characteristics of the potential surface. A reduced dynamics scheme based on the Gaussian assumption leads to the correct center of mass motion but does not describe the evolution of the shape of the wave packet accurately. To incorporate the quantum interference into classical trajectories, we propose a final-value representation semiclassical method, specifically designed for the purpose of computing pump-probe signals, and demonstrate its efficiency and accuracy with a Morse oscillator and two kinetically coupled Morse oscillators. For the case of one-color pump probe, a simple phase-space quantization scheme is devised to reproduce the temporal profile at the left-turning point without actual wave packet propagation, revealing a quantum mechanical perspective of the nearly classical pump-probe signal.     

Note on calculations by hypervirial relations and sufficient conditions for these relations: Box potential and harmonic oscillator models

The calculations by the diagonal and the off-diagonal hypervirial relations and by the sufficient conditions for these relations have been performed for the one-dimensional box potential and the harmonic oscillator models. In the case of the box potential model, the variation method gave the unsymmetrical wave functions for trial functions though it gave the best energies, but some of these hypervirial calculations gave the symmetrical functions and therefore they showed the exact expectation values of x, though they did not give the best energies. On the contrary, in the case of the harmonic oscillator model such a discrepancy for wave function and excitation energy did not happen. For the box potential model the expectation values of some operators were evaluated and compared with the results by the exact solution and the variation method. A weakness of the off-diagonal hypervirial calculation was pointed out and the removal of this weakness was tried. The effectiveness of these calculations were investigated.     

Rotational-vibrational rainbows in impulsive electron — diatomic molecule collisions: I. state-to-state transitions

Collisional induced combined rotational-vibrational excitation of diatomic molecules is discussed in a simple quantum mechanical spectator model with applications to electron-molecule collisions at intermediate collision energies (≈ 10² eV) within the rigid rotor/harmonic oscillator approximation. Quantum mechanical transition probabilities of the rotational-vibrational excitation, which show typical vibrational and rotational rainbow patterns, are calculated and compared with the structure of classical rainbow singularities.     

Time-dependent perturbation: a recursive generation of the transition amplitudes between bound states

We present a new expansion of the solution to the time-dependent Schrödinger equation it ∂U/∂t = {H₀ + V(t)}U. A complete set of eigenvectors of H₀ spanning the Hilbert space in which H₀ and V operate is partitioned in two subsets. Transition amplitudes from the first subspace to the second one are calculated by building an adequate series of intermediate representations with respect to the couplings which produce the transition from the model space into its orthogonal complement. These expansions yield a coupling matrix series V⁽ⁿ⁾ for which an iterative solution V⁽ⁿ⁾ → V⁽ⁿ⁺¹⁾ is derived. This solution leads to a recursive numerical treatment of the calculation of transition amplitudes. A simple example, concerning a harmonic oscillator under an intense laser field, is considered using a Fourier analysis of the perturbation.     

The semiclassical and the quasiclassical partition function of a quartic anharmonic oscillator

After a substitution a known Laplace-type integral is used to derive quantum corrections to the classical partition function of a quartic anharmonic oscillator in the framework of the Wigner—Kirkwood perturbation expansion. By straightforward calculations results are given in a closed form allowing the analytical formulation of the thermodynamic functions H, E, S, C_υ. The numerical results agree for arbitrary anharmonicity and for high and intermediate temperatures with the numerical partition function calculated from the Hioe—Montroll eigenvalues. Furthermore, the same integral type is used for the analytical calculation of a “quasiclassical” partition function and of “quasiclassical” moments. In the trace formulation of the partition function all commutators are neglected. The harmonic oscillator density matrix is applied to the evaluation of the truncated trace expressions. The “quasiclassical” partition function is an exact upper bound and lies always below the classical partition function.     

On the dynamics of a linear and a nonlinear quantum oscillator with randomly changing harmonic frequency

Numerical experiments with a nonlinear (λχ⁴) oscillator which has its harmonic frequency changing randomly with time reveal certain interesting features of its dynamics of quantum evolution. When λ = 0, the level populations are seen to oscillate. But, as the nonlinear coupling is switched on (λ > 0), a threshold is reached at λ = λ_c when the evolution is seen to be characterized by an abrupt transition dominantly to the highest available state of the unperturbed (initial) oscillator. It is shown that this transition probability is maximized at a particular value of λ. The time threshold for this transition decreases with increasing nonlinear coupling strength. The numerically obtained structures of the underlying quantum-phase spaces of the linear and nonlinear random oscillators are examined. Possible use of these results in a problem of chemical origin is explored. © 1997 John Wiley & Sons, Inc.     

Entropic Kullback-Leibler type distance measures for quantum distributions

Entropic distance measures for quantum mechanical probability distributions, which are characterized by nodal structure and symmetry holes, are considered. We illustrate how the Kullback-Leibler (KL) distance is not well defined in some instances and propose instead the use of the cumulative residual Kullback-Leibler (CRKL) distance. The KL and CRKL measures are compared and contrasted for some representative quantum mechanical systems: The particle in an infinite well, the harmonic oscillator, and hydrogenic systems. We present cases where CRKL can be used to obtain distances whereas KL cannot be used, and also highlight examples where the KL and CRKL measures yield different behaviors and interpretations. An extension of the CRKL definition is obtained for application to harmonic oscillator systems defined over [−∞, ∞]. Distance measures for two-variable (particle) distributions are also considered to address generalizations of the mutual information correlation measure. The use of CRKL in measuring distances between orbitals is also discussed.     

Rotational-vibrational rainbows in impulsive electron-diatomic molecule collisions

Rotational and vibrational rainbow effects in electron-diatomic molecule scattering at intermediate impact energies (≈10² eV) are discussed in a simple quantum mechanical spectator model within the rigid rotor/harmonic oscillator approximation. The total vibrational (summed over all final rotational quantum numbers) and rotational (vibrationally summed) transition probabilities show vibrational or rotational rainbow patterns, characteristic steps, and rainbow singularities, which are analyzed and interpreted in terms of classical cross sections.     

Doubly excited3Se,3De and3Ge states of two-electron atomic systems

Summary Time-dependent perturbation theory has been applied to calculate the doubly excited triplet statesNsns:³S^e,Npnp:³D^e andNdnd:³G^e (N=2, 3, 4,n=N+1, ... ,5) for He, Li⁺, Be²⁺ and B³⁺. A time-dependent harmonic perturbation causes simulataneous excitation of both the electrons with a change of spin state. The doubly excited energy levels have been identified as the poles of an appropriately constructed linearized variational functional with respect to the driving frequency. In addition to the transition energies, effective quantum numbers of these doubly excited states have been calculated and analytic representations of their wave functions are obtained. These are utilized to estimate the Coulomb repulsion term for these states which checks the consistency of the wave functions. These wave functions may also be used for calculating other physical properties of the systems.     

Tests of semiclassical treatments of vibrational-translational energy transfer collinear collisions of helium with hydrogen molecules

Three kinds of semiclassical theory are tested against quantum mechanical results for vibrational transition probabilities and average vibrational energy transfers in collinear collisions of atoms with harmonic and Morse vibrators for the He-H₂ mass combination. The interaction potential is assumed to be a repulsive exponential function with an exponential parameter which is realistic for He-H₂ collisions. The energy range studied is total energies of 2–8 in units of ?ω_e. The uniform semiclassical approximations of classical S matrix theory are tested only for classically allowed transitions, i.e., for transition probabilities greater than about 0.2. They are accurate quantitatively for both harmonic and Morse vibrators. The integral expressions of classical S matrix theory are found to be quantitatively accurate for classically allowed and weakly classically forbidden transitions, i.e., for transition probabilities greater than about 0.01–0.05, and to be unreliable for strongly classically forbidden transitions. Quasiclassical trajectory methods yield qualitatively accurate results only for classically allowed transitions but the phase-averaged energy transfer in quasiclassical collisions may be accurate even when classically forbidden transition probabilities are important for the calculation of the average energy transfer. Forced quantum oscillator methods using a classical path whose initial velocity is the average of the initial and final velocities corresponding to the transition of interest are accurate for transition probabilities as small as 4 × 10^?8 for harmonic vibrators but do not seem to accurately account for the effect of anharmonicity.     

Preparation and resolution of molecular states by coherent sequences of phase-locked ultrashort laser pulses

We study the application of nonlinear wave packet interferometry to the preparation and resolution of the overlaps of nonstationary nuclear wave functions evolving in an excited electronic state of a diatomic molecule. It is shown that possible experiments with two phase-locked ultrashort pulsepairs can be used to determine a specific vibrational wave packet state in terms of coherent states of the ground electronic state. We apply this scheme to an idealized molecule with harmonic potential energy surfaces and to the X <-- B transition states of the iodine molecule. Our results indicate that this scheme is very promising as a potential tool to quantum control.