Author ORCID Identifier
Degree Year
2024
Document Type
Thesis - Oberlin Community Only
Degree Name
Bachelor of Arts
Department
Physics and Astronomy
Advisor(s)
Daniel F. Styer
Committee Member(s)
Jason Stalnaker
Robert Owen
Yumi Ijiri
Dan Stinebring
Daniel F. Styer
Keywords
Mathematical physics, Analytic solutions, Non-relativistic quantum mechanics, Integral transform, Airy functions, Airy transform, Asymptotic expansions, Saddle point method, Oscillatory integrals, Constant-force potential, Gaussian wavepacket, Simple harmonic oscillator, Rididly-moving wavepackets
Abstract
This paper presents analytic solutions to quantal time-evolution problems in the simple harmonic oscillator and constant-force potential. We prove that in the simple harmonic oscillator potential, the only family of wave packets with rigidly moving probability density is the spatially translated energy eigenfunctions. In addition, we completely solve the time evolution of Gauss-Hermite waveforms, namely spatially translated and stretched energy eigenfunctions.
We also completely solve the time-evolution problem of an initially Gaussian wave packet in the constant-force potential. This solution involves using a convolutional integral transform ๐๐ผ over basis functions {Ai[๐ผ(๐ฅ โ ๐)] โถ ๐ โ R} (๐ผ โ 0 and ๐ฅ โ R), and we prove that the integral transform ๐๐ผ is invertible and satisfies the Parseval-Plancherel identity over the Schwartz space ๐ฎ(โ; โ). We suggest future work on generalizing the transformation ๐๐ผ to admit piece-wise continuous functions and (tempered) distributions, which can be used to solve time-evolution problems in the 1D Stark potentials, that is, 1D finite rectangular well with constant force field.
This paper also presents an overview of asymptotic methods in mathematical physics. We present asymptotic methods on Laplace-type integrals ranging from Laplaceโs method to the saddle point method. Large-๐ฅ asymptotic expressions for the Airy function Ai(๐ฅ) are derived explicitly using the saddle point method.
Repository Citation
Zhu, Shuran, "Quantal Time Evolution in the Simple Harmonic Oscillator and the Constant-force Potentials: Analytic Solutions" (2024). Honors Papers. 912.
https://digitalcommons.oberlin.edu/honors/912
