A semianalytical approach to the quantum dynamics of a plane pendulum is developed, based on Mathieu functions which appear as stationary wave functions. The time-dependent Schr\"odinger equation is solved for pendular analogs of coherent and squeezed states of a harmonic oscillator, induced by instantaneous changes of the periodic potential energy function. Coherent pendular states are discussed between the harmonic limit for small displacements and the inverted pendulum limit, while squeezed pendular states are shown to interpolate between vibrational and free rotational motion. In the latter case, full and fractional revivals as well as spatiotemporal structures in the time evolution of the probability densities (quantum carpets) are quantitatively analyzed. Corresponding expressions for the mean orientation are derived in terms of Mathieu functions in time. For periodic double well potentials, different revival schemes, and different quantum carpets are found for the even and odd initial states forming the ground tunneling doublet. Time evolution of the mean alignment allows the separation of states with different parity. Implications for external (rotational) and internal (torsional) motion of molecules induced by intense laser fields are discussed.
In this chapter, the authors solve the Schrodinger equation for a harmonic oscillator using two methods, analytical method and algebraic method. They discuss the quantum properties of the harmonic oscillator. A harmonic oscillator is a system that exhibits simple harmonic motion or periodic motion, a motion that repeats itself after equal intervals of time. Many systems such as spring, simple pendulum, vibrating string and molecular vibration can be approximated as simple harmonic oscillators. The authors consider a simple harmonic oscillator, a spring whose spring constant is "k" attached to a mass "m" as a simple harmonic oscillator. They describe the Schrodinger equation in spherical coordinates. The authors also solve the three-dimensional Schrodinger equation for the hydrogen atom. For the hydrogen atom, the potential energy function is dependent only on radial distance and is independent of angular variables. The hydrogen atom consists of a positively charged proton surrounded by an electron that orbits around it.
In physics, harmonic motion is among the most representative types of motion. A simple harmonic oscillator is often the source of any vibration with a restoring force proportional to Hooke’s law. Every minimum potential has a solution in the form of the harmonic oscillator potential. Little oscillations at the minimum are characteristic of almost all natural potentials and of many quanta mechanical systems. Harmonic motion is an essential building block for these more complex uses. The Schrödinger equation is a defining feature of the harmonic oscillator. Here, we demonstrate that the time-frequency plane is a useful tool for analyzing their dynamics. We numerically integrate several examples involving different input forces and demonstrate that the oscillations are clearly displayed and easily interpretable in the time-frequency plane.
The consideration of the resonant interaction between a linear harmonic oscillator (LO) and a (nonlinear) angular momentum oscillator (NO) is motivated by its relation to the interaction between a field mode and a number of atomic systems, its relation to the interaction between three field modes, and the appearance of qualitatively different results quantum mechanically and classically even in cases that can be regarded as macroscopic. The analysis is carried out both classically and quantum mechanically in the rotating-wave approximation, and the time dependence of the energies of the two oscillators is examined, mainly by means of a nonlinear differential equation for the NO energy. In the classical analysis, the problem turns out to be identical to that of a spherical pendulum, where the NO energy
L3 corresponds to the pendulum potential energy, the LO energy
n corresponds to the pendulum kinetic energy, and the coupling energy between the two oscillators $\frac{1}{2}\frac{\ensuremath{\gamma}K}{\ensuremath{\omega}}$ is proportional to the angular momentum
K of the pendulum about the vertical axis, all quantities being expressed in suitable units. A solution for
L3(t) is obtained, and is shown to be described uniquely, apart from a shift in the time origin, by the constants of motion $E(\ensuremath{\equiv}{L}_{3}+n)$ and
K.
L3 oscillates periodically, except for the "unstable-equilibrium" solution and the "conical-pendulum" solution. The quantum-mechanical problem is solved exactly for the cases in which the NO is a two-level system and a three-level system, and certain aspects of the exact solution are derived in the case of a four-level system. The constants
E and
K become a complete set of quantum numbers. It is found that unstable equilibrium, in the sense that
〈L3〉 is constant, does not exist, because of spontaneous emission; it is also found that
〈L3〉 oscillates sinusoidally for the two-level system, oscillates periodically but not sinusoidally for the three-level system, and oscillates aperiodically for a four- or higher-level system, conclusions about the higher-level systems coming from general considerations and numerical solutions found in the literature. Approximate solutions for NO' s with arbitrarily large numbers of levels are obtained. One method of approximation displays spontaneous emission but not aperiodicity. Another approximate method, which contains the statistical spread of the initial phase difference between the oscillations of the two oscillators inherent in the uncertainty principle, displays both spontaneous emission and aperiodicity. It is shown that the aperiodicity - a property which does not exist classically for precise initial conditions - does not disappear even in the macroscopic limit and is due to the averaging performed, in taking the quantum-mechanical expectation value, over an ensemble of periodic oscillations with a spread in frequencies.
Two of the most iconic systems of quantum physics are the particle in a box and the Coulomb potential (the third is, of course, the harmonic oscillator). In this expository paper, we consider the quantum solution to the problem of an electron confined between the grounded planes of an infinite capacitor. The potential arises from the image charges that form in the grounded planes, along with the added condition that at x = 0, L, where L is the distance between the planes, the wavefunction must be zero. This effectively couples a hydrogen like system to a particle-in-a-box (PIB) based on L, the distance between the planes. The problem of finding the electrostatic potential of this infinite series of image charges is an old one, going back to at least 1929. Here, we give a short derivation for one of the limiting cases that yields a compact expression and show how the Kellogg infinite summation formula converges to that value. We note here that this potential is a symmetric double well potential, so there will be many familiar properties of its solutions. Then using that potential, we solve Schrödinger's equation using a spectral technique. The limiting forms of a particle in a box for small L (and high E), and that of a (degenerate) bound image charge at large L and small energy are recovered. We also discuss the tunneling level splitting that occurs in the transition from the large L to the small L regime.
The rigid pendulum, both as a classical and as a quantum problem, is an interesting system as it has the exactly soluble harmonic oscillator and the rigid rotor systems as limiting cases in the low- and high-energy limits respectively. The energy variation of the classical periodicity (
tau) is also dramatic, having the special limiting case of
tautoinfty at the 'top' of the classical motion (i.e. the separatrix.) We study the time-dependence of the quantum pendulum problem, focusing on the behavior of both the (approximate) classical periodicity and especially the quantum revival and superrevival times, as encoded in the energy eigenvalue spectrum of the system. We provide approximate expressions for the energy eigenvalues in both the small and large quantum number limits, up to 4th order in perturbation theory, comparing these to existing handbook expansions for the characteristic values of the related Mathieu equation, obtained by other methods. We then use these approximations to probe the classical periodicity, as well as to extract information on the quantum revival and superrevival times. We find that while both the classical and quantum periodicities increase monotonically as one approaches the 'top' in energy, from either above or below, the revival times decrease from their low- and high-energy values until very near the separatrix where they increase to a large, but finite value.