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Student Class

2029

Student Affiliation

WISP Intern

First Advisor

Rufus Boyack

First Advisor Department

Department of Physics and Astronomy

Abstract

Quantum wave packets are localized, time-dependent solutions of the Schrödinger equation that mimic classical particle motion. In his 1926 paper, Schrödinger famously showed that displacing the ground state of the harmonic oscillator produces a wave packet whose probability density maintains its shape and saturates the Heisenberg uncertainty bound. Motivated by this result, we construct and analyze wave-packet solutions for the Airy potential, the simple harmonic oscillator (SHO), and the pseudoharmonic oscillator (PHO). Our general approach is to find expansion coefficients and take a superposition of the energy eigenstates of the time-independent Schrödinger equation. We examine spatially displaced eigenstates alongside alternative constructions, computing probability densities and uncertainty products to determine whether each wave packet maintains shape, is periodic in time, and whether it approaches the minimum uncertainty limit. For the Airy potential, we build exact, non-dispersive solutions from Airy functions, illustrating how a gravitational field shapes quantum motion. For the SHO, we generalize Schrödinger's original construction, showing that infinitely many displaced eigenstates preserve their shape while oscillating at the classical frequency. For the PHO, we extend this construction to a potential with a singular repulsive core. These results clarify when localized wave packets behave classically and remain coherent. Open questions include identifying the symmetry underlying the SHO's shape-preserving behavior and determining which PHO wave packets minimize the uncertainty product.

Publication Date

Spring 2026

Keywords

Quantum wave packets, Schrödinger equation, Coherent states, Heisenberg uncertainty principle, Simple harmonic oscillator, Pseudoharmonic oscillator, Airy functions, Displaced eigenstates, Energy eigenstate superposition, Inverse square potential

Disciplines

Quantum Physics

Dynamics of Localized Wave Packets in Quantum Mechanics

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