[Paper Review] Infinite square well and periodic trajectories in classical mechanics

Bijan Bagchi, S. Basu Mallik+1 more|ArXiv.org|Jul 24, 2002

Dynamics and Control of Mechanical Systems4 references3 citations

TL;DR

This paper provides a detailed classical mechanics analysis of the infinite square well, demonstrating that particle motion is periodic via Hamilton's equations in 1D and the Hamilton-Jacobi equation in 2D. It derives exact periodic trajectories and shows that the period depends on mass, initial momentum, and well size, with explicit examples confirming periodicity through repeated return to initial conditions after finite time intervals.

ABSTRACT

We examine the classical problem of an infinite square well by considering Hamilton's equations in one dimension and Hamilton-Jacobi equation for motion in two dimensions. We illustrate, by means of suitable examples, the nature of the periodic motion of a particle trapped inside the well.

Motivation & Objective

• To provide a rigorous classical mechanics treatment of the infinite square well, often underdeveloped in graduate textbooks.
• To demonstrate the periodic nature of particle trajectories in one and two dimensions using Hamiltonian formalism.
• To construct explicit, nontrivial examples of periodic motion in 2D infinite square wells with non-uniform initial momenta.
• To clarify the relationship between classical periodicity and quantum mechanical quantization in the same system.

Proposed method

• Solving Hamilton's equations of motion in one dimension to derive piecewise linear trajectories with momentum reversals at infinite walls.
• Applying the Hamilton-Jacobi equation with separation of variables to analyze two-dimensional motion and identify periodic solutions.
• Using canonical transformations and generating functions to map the system into action-angle variables, revealing periodic behavior.
• Deriving the period as $ T = 4ma/p_0 $ in 1D and $ T = 12ma/p_{x0} $ in a 2D example with rational frequency ratios.
• Tracking position and momentum evolution through multiple reflections using piecewise-defined equations of motion.
• Verifying periodicity by showing the particle returns to the initial position and momentum after a finite time $ T $.

Experimental results

Research questions

• RQ1 How can the periodic nature of classical motion in an infinite square well be rigorously derived from Hamilton's equations in one dimension?
• RQ2 What conditions lead to periodic trajectories in two-dimensional infinite square wells, and how do they differ from the 1D case?
• RQ3 How does the Hamilton-Jacobi formalism reveal periodicity in multi-dimensional billiard systems?
• RQ4 What is the exact period of motion for a particle with arbitrary initial momentum in a 2D infinite square well?
• RQ5 How do the classical periodic frequencies relate to the quantum mechanical energy levels in the same system?

Key findings

• In one dimension, the motion is periodic with period $ T = 4ma/p_0 $, where $ m $ is mass, $ a $ is half the well width, and $ p_0 $ is initial momentum.
• In two dimensions, for a specific example with $ p_{x0} $ and $ p_{y0} $, the period is $ T = 12ma/p_{x0} $, with $ T_x = 4ma/p_{x0} $ and $ T_y = 6ma/p_{x0} $, satisfying $ T = 3T_x = 2T_y $.
• The system exhibits periodicity only when the ratio of the x- and y-motion periods is rational, as seen in the $ T_x : T_y = 2:3 $ ratio.
• The particle returns to its initial position and momentum $ (x_0, y_0, p_{x0}, p_{y0}) $ at $ t = T = 12t_1 $, confirming full periodicity.
• The classical characteristic frequencies are $ u_x = p_{x0}/(4ma) $ and $ u_y = p_{y0}/(4ma) $, consistent with action-angle variable analysis.
• The analysis confirms that classical periodicity arises from rational frequency ratios and reflection symmetry at infinite walls.