The relationship between effective potential, radial probability distribution, and radial nodes in hydrogenic systems via a computational approach

Bambang Supriadi (Universitas Jember)
Dyah Arum Arimurti (Universitas Jember)
Zilfiyatul Makida (Universitas Jember)
Kiki Dinda Octari (Universitas Jember)
Mah Citra Yunia Artika Putri (Universitas Jember)
Ulvia Khoirunisa Bisanti (Universitas Jember)
Nisrina Nafisah (Universitas Jember)

Gravity : Jurnal Ilmiah Penelitian dan Pembelajaran Fisika
Vol 12, No 2 (2026)

The hydrogen atom is a fundamental system in quantum mechanics, yet studies that integrate the effective potential structure, radial probability distribution, and radial node patterns remain limited. This study aims to characterize the interrelationship of these three aspects through a computational approach. The method used is a combination of analytical solutions to the radial Schrödinger equation and numerical evaluation to calculate and visualize the effective potential, radial wave functions, and probability distributions. Furthermore, radial nodes are identified using a combination of the principal quantum number n and the azimuthal quantum number l. The results show that variations in l modify the topology of the effective potential through centrifugal resistance, while an increase in n broadens the probability distribution according to the scaling law . The radial node pattern follows the relation , indicating an interaction between radial excitation and angular momentum. Overall, the probability distribution and radial nodes are direct consequences of the effective potential structure. This study provides a unified analytical framework for comprehensively understanding the radial structure of hydrogenic systems.