摘要
本文基于劈裂算符方法,数值研究了描述微观粒子的高斯型波包在一维、二维势垒中的动态演化过程,计算了体系的坐标、动量、哈密顿算符的期望值,并验证了在整个势垒隧穿动力学过程中满足海森堡不确定性关系。通过将时间演化算符拆分为动能与势能的交替作用,利用快速傅里叶变换对波函数进行坐标与动量表象间的转换,实现了含时薛定谔方程的数值求解。结果表明,若势能函数不显含时间,体系的平均能量含时演化在大尺度范围保持守恒,在小尺度范围“不严格守恒”,当微观粒子隧穿势垒时能量将快速涨落,离开势垒后恢复至隧穿前的能量。在一维双势阱中,高斯波包在势阱内往复运动,频繁贯穿势垒,能量平均值随时间小幅度周期性快速振荡。对于二维势垒情况,二维高斯波包的反射方向可与入射方向不同,一段时间后粒子主要出现在势垒两侧。本文为量子力学教学的可视化研究,以及量子隧穿动力学的深入探索奠定了数值基础,加深对微观粒子波-粒二象性以及海森堡测不准原理的理解。
In this paper,the dynamic evolution process of Gaussian wave packet which describes a microscopic particle in one-dimensional and two-dimensional potential barriers are numerically studied by virtue of the split operator method.Moreover,the expected values of coordinates,momentum and Hamiltonian operators of the particle are calculated,and the Heisenberg uncertainty relationship is verified throughout the quantum tunneling process.Specifically,the time evolution operator is split into two parts which independently include the contributions of kinetic energy and potential energy.The wave function between coordinate and momentum representations is converted arbitrarily by utilizing the fast Fourier transform,and the numerical solution of the time-dependent Schrödinger equation is achieved.Accordingly,the results indicate that if the potential barrier is time independent,the time evolution of average total energy of the system is conserved on a large scale,but not strictly conserved on a small scale.When microscopic particle tunnels through a potential barrier,the total energy fluctuates rapidly.While leaving the potential barrier,it restores to the total energy value be-fore quantum tunneling.Furthermore,in a one-dimensional double well potential,Gaussian wave packet oscillates back and forth within the double potential well,frequently crossing the middle potential barrier.Then the evolution of average energy oscillates rapidly and periodi-cally with small amplitude.For the two-dimensional potential barrier,the reflection direction of the wave packet may not be the same as the incident direction,and after a period of time,particle mainly appears on both sides of the potential barrier.As a consequence,this paper presents a numerical foundation for the visualization research of quantum mechanics teaching and the in-depth exploration of quantum tunneling dynamics,deepening the understanding of wave-particle duality for the microscopic particle and Heisenberg's uncertainty principle.
作者
李海凤
黄思哲
刘喆
潘雷雷
LI Haifeng;HUANG Sizhe;LIU Zhe;PAN Leilei(School of Sciences,Xi'an Technology University,Xi'an,Shaanxi 710021)
出处
《物理与工程》
2026年第1期115-121,128,共8页
Physics and Engineering
基金
西安工业大学研究生教育教学改革研究项目(XAGDYJ250237
XAGDYJ250241)
西安工业大学本科教学改革研究项目(25JCY42)
2025年大创项目(创新训练,S202510702162)。
关键词
劈裂算符法
含时薛定谔方程
高斯波包
量子隧穿
split operator method
time-dependent Schrödinger equation
Gaussian wave packet
quantum tunneling
