M. Belloni

Wigner quasi-probability distribution for the infinite square well: Energy eigenstates and time-dependent wave packets

We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, PW(n)(x,p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, PW(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrodinger-cat-like “mini-packets” which appear at fractional multiples of the exact revival time.

Quantum mechanical sum rules for two model systems

Sum rules have played an important role in the development of many branches of physics since the earliest days of quantum mechanics. We present examples of one-dimensional quantum mechanical sum rules and apply them to the infinite well and the single δ-function potential. These examples illustrate the different ways in which these sum rules can be realized and the varying techniques by which they can be confirmed. We use the same methods to evaluate the second-order energy shifts arising from the introduction of a constant external field, namely the Stark effect.

Exact Results for 'Bouncing' Gaussian Wave Packets

We consider time-dependent Gaussian wave packet solutions of the Schrodinger equation, with arbitrary initial central position, x0, and momentum, p0, for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets. We show how difference or mirror solutions of the form ψ(x,t) − ψ(−x,t) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate p2t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, xt, and momentum, pt, and their relation to the impulsive force during the `collision with the wall. Finally, the x0, p0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.

More on the asymmetric infinite square well: energy eigenstates with zero curvature

We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wavefunction to have zero curvature over an extended region of space and still be a valid wavefunction. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space.

Constraints on Airy function zeros from quantum-mechanical sum rules

We derive new constraints on the zeros of Airy functions by using the so-called quantum bouncer system to evaluate quantum-mechanical sum rules and perform perturbation theory calculations for the Stark effect. Using commutation and completeness relations, we show how to systematically evaluate sums of the form

Playing Quantum Physics Jeopardy with zero-energy eigenstates

S_{p}(n) = sum_{k neq n} 1/(zeta_k - zeta_n)^p

Less than perfect quantum wavefunctions in momentum-space: How ϕ(p) senses disturbances in the force

, for natural

New identities from quantum-mechanical sum rules of parity-related potentials

p > 1

The Biharmonic Oscillator and Asymmetric Linear Potentials: From Classical Trajectories to Momentum-Space Probability Densities in the Extreme Quantum Limit.

, where

Piecewise zero-curvature energy eigenfunctions in one dimension

zeta_n