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For a nonrelativistic particle acted on by a spatially uniform but possibly time-dependent force, or for a harmonic oscillator also subject to such a uniform force, the wave function differs from that without the force only by a spatial displacement and a phase factor. The shifts in position and phase are simply derived from a solution of the classical equations of motion. [c] 2010 American Association of Physics Teachers. [DOI: 10.1119/1.3481703]

We discuss four general features of the force-free evolution of wave packets: (1) The spatial spread of a packet changes with time in a simple way. (2) For sufficiently short durations (related to the spread in the momentum of the packet) the probability distribution will move with uniform speed and little change in shape. (3) After a sufficiently long time (related to the initial spatial spread) the wave function converges to a simple form that is simply related to the momentum distribution of the packet. In this asymptotic regime the shape of the probability distribution no longer changes, and its scale increases linearly with the time. (4) There is an infinite denumerable set of simple wave packets (the Hermite-Gauss packets) that do not change shape at any time during their evolution.

The coherent states of the simple harmonic oscillator with an impenetrable barrier at its center are studied. This half oscillator is the equivalent of a pendulum that bounces elastically off a vertical wall directly below the point of suspension with the angle of swing sufficiently small. The system can also be considered as a paddle ball, where the paddle is fixed and the ball is constrained by a spring attached to the paddle. The coherent states are almost the same as the familiar Gaussian coherent states of the full oscillator, except when they overlap the barrier. The solutions can be easily extended to two and three dimensions and gravity can be included if the impenetrable barrier is vertical. To better understand the form of the expectation values of the position and momentum, we investigate some general aspects of the effect of impenetrable barriers on the dynamics of wavepackets. [DOI: 10.1119/1.2825388]

The use of a total time derivative of operators, that depends on the time evolution of the wave function as well as on any intrinsic time dependence in the operators, simplifies the formal development of quantum mechanics and allows its development to more closely follow the corresponding development of classical mechanics. We illustrate the use of the total time derivative for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square potential. In these cases, operators whose total time derivative is zero can be found and yield general properties of wave packets and several useful time-dependent solutions of Schrodinger's equation, including the propagator. [DOI: 10.1119/1.1531579]

The motion of wave packets can be easily determined for any Hamiltonian that is quadratic in position and momentum, even if the coefficients of the terms in the Hamiltonian vary with time. The method is based on the existence of an invariant operator, linear in position and momentum, the coefficients of these operators being solutions of the corresponding classical system. This immediately yields a set of very simple wave packets whose evolution is easily determined, and whose magnitude has the same form as the energy eigenfunctions of the harmonic oscillator (i.e., Gaussian or Hermite-Gaussian). This set provides a complete basis for finding the evolution of any state and the exact propagator is readily determined. For the harmonic oscillator, this set includes coherent states, squeezed states, displaced number states, and squeezed number states as special cases. The following important properties hold for every quadratic Hamiltonian, no matter what time dependence is present in the Hamiltonian: (1) The motion of the centroid of any wave packet separates from that of any moments relative to the centroid. (2) Every detail of the evolution of the quantum system can be calculated from the solutions of the corresponding classical system. (3) Wave functions of Gaussian (or Hermite-Gaussian) form will retain that form as they evolve.

The concept of an equilibrium continuous charge density on a conducting line, although in some ways unphysical, does provide useful physical insight into needlelike conductors. It is shown that the charge density on such a line must be uniform. A simple method is developed for calculating the charge density on long, thin conductors with circular cross section. Also, the equilibrium distribution of N equal charges on a line is shown to approach equal spacing (except near the ends) as the number of charges grows.

An analytic solution for wave packets of different shapes bouncing off walls is discussed. The solution explains why a wave packet originally bouncing off a wall seems to be travelling away from the wall after a time. It is shown that if the wave packet lies entirely to the left of the wall at time=0, then the wave packet is a solution of the free Schrodinger equation. The solution is also applied to the evolution of a Gaussian wave packet.