Motion in Stillness: An Open-Source Python Toolkit for Visualizing Time-Dependent Hydrogen Wavefunctions
Year: 2026
Type: preprint
Abstract: Orbital shapes and nodes are core learning objectives in chemistry and physics. Many tools can render a wavefunc- tion’s phase as a color map. Yet rendering it is not teaching it: students often cannot explain what phase means, nor where Ψ’s time dependence lies. Since the density |Ψ|2 of a stationary state is constant, many infer Ψ is static. The opposite is true. For a complex stationary state Ψ(t) = ψe−iEnt/ℏ, the phase argΨ = argψ −Ent/ℏ is time-dependent, advancing uniformly everywhere except at the nodes; because all points advance in step, phase differences remain fixed, and the nodal structure is unchanged. At a node (ψ = 0) the amplitude vanishes and the phase is undefined, changing abruptly across the nodal surface. Ψ’s time dependence thus resides in its phase, sharply distinguished at nodal and non-nodal points. We present Time-HWV, an open-source Python toolkit that decouples and animates the real part, imaginary part, and density in synchronized 2D cross-section and 3D views. Students observe the quadrature oscillation of Re[Ψ(t)] and Im[Ψ(t)] while confirming that |Ψ(t)|2 = |ψ|2 is invariant. Time-HWV offers 2D cross-sections (slices), 3D surface plots of those cross-sections (height fields), and interactive Plotly volume renderings in a lightweight NumPy/SciPy/Matplotlib package, with standalone executables requiring no programming environment. Code and manual are included as Supporting Information.
Source: ChemRxiv
Author: Yiming Ma
Institution: Northwest University
Language: English
Subfield: Education
Field: Social Sciences
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7225782034
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