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Hamdi Akhsan
Department Of Physics Education Faculty Of Teaching And Education, Sriwijaya University, Indralaya, 30139, Indonesia
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Journal : Progressive Physics Journal

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Application of Physics-Informed Neural Networks (PINNs) for the Numerical Solution of the Time-Independent Schrödinger Equation Akhsan, Hamdi; Khoirun Nisa; Nurhikmah, Putri; Wailaina; Ariska, Melly
Progressive Physics Journal Vol. 6 No. 2 (2025): Progressive Physics Journal
Publisher : Program Studi Fisika, Jurusan Fisika, FMIPA, Universitas Mulawarman

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30872/6881zx10

Abstract

This work investigates the application of Physics-Informed Neural Networks (PINNs) for numerical solutions to the time-independent Schrödinger equation of the quantum harmonic oscillator in one, two, and three spatial dimensions. Fully connected neural architectures are constructed to approximate wavefunctions over finite symmetric domains, while the corresponding energy eigenvalues are treated as trainable parameters. The training strategy utilizes randomly sampled interior points to enforce the Schrödinger operator residual and boundary points to impose vanishing wavefunction constraints. For the 1D quantum harmonic oscillator, the learned ground-state wavefunction yields an energy of E = 1.2939 after 12,000 iterations. In the 2D configuration, convergence is achieved at E = 2.1352 within 14,000 iterations, whereas the 3D model attains E = 2.6377 after 12,000 iterations. These values agree with the expected trend of increasing ground-state energy with dimensionality, although deviations from exact analytical values indicate that PINNs may experience optimization challenges and sensitivity to sampling density and boundary enforcement. Despite these limitations, the trained models successfully capture the characteristic spatial symmetries and Gaussian-like envelope of harmonic oscillator eigenstates across all dimensions. These findings demonstrate that PINNs offer a flexible, mesh-free alternative for solving stationary quantum systems, particularly when analytical or conventional numerical approaches become impractical. The method shows strong potential for higher-dimensional quantum applications, even though further refinement such as improved sampling, loss balancing, and network depth remains necessary to suppress residual error and enhance eigenvalue accuracy.
Co-Authors Abidin Pasaribu Abidin Pasaribu Abidin Pasaribu, Abidin Adam Darmawan Ade Kurniawan Ade Kurniawan Agustina, Atika Aji, Dwi Purnomo Anggelina, Putri Apit Fathurohman Ardiantini, Devi Arini Rosa Sinensis Atika Agustina Aulia, Maghfira Ayu Sekar Sari Azizah Putri Berimah Berimah, Azizah Putri Cahyati, Nilam Damayanti, Fitria Siska Dwi Purnomo Aji Dwicahyani, Rania Edray, Auliya Einsrizkianne Elsie Sulastry Fahreza, Muhammad Aditya Fajriah, Nurkholisa Fatonah, Amelia Febrianti, Tanti Fena Siska Putriyani Fika Nurul Hidayati Fitria Siska damayanti Fitriyani Fitriyani Guruh Sukarno Putra Halim, Noor Dayana Abd Huriyatul Firdausi Husna, Tsabita Iful Amri Iin Seprina Ina Agustina Intan Purnama Sari Ishkaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Ismet Ismet Ismet Ismet, Ismet Ismet, I. Izzah Nurhidayah, Izzah Jesi Pebralia Juliansyah, Dendi Juwita Febrianti Kartika, Nadia Dwi Ketang Wiyono Ketang Wiyono Ketang Wiyono KHOIRUN NISA Kistiono Kistiono Kistiono Kistiono Kistiono Komarudin Komarudin Kurnia, Mega Lara Sania Laras Sapitri, Cindy Leni Marlina Leni Marlina Lestari, Yora Inda Lira, Lira Lismayana, Pegi Putri M. FURQON Mahdiah, Intan Mahmudah, Indri Maliya, Fina Manurung, Nia Three Markos Siahaan, Sardianto Melly Ariska Melvany, Nanda Eva Miftahul Jannah Mindia Vanessa Pratiwi, Sri Muhamad Yusup Muhammad afrizal Muhammad Aufa Riyaldo Muhammad Irfan Muhammad Irfan Muhammad Irfan Muhammad Irfan Muhammad Irfan Muhammad Muslim muhammad muslim Muhammad Muslim Muhammad Muslim, Muhammad Muhammad Romadoni Muhammad Romadoni Muhammad Yusuf Muhammad Yusup Murnia Murniati Murniati . Muslimah, Resta Ulis Mutia Faradini Ramadhanty Napitupulu, Yendri Tri Hartati Nasiroh, Anisa Qoriq Atun Nazhifah, Rizqia Nely Andriani Nely Andriani, Nely Nestity, Alifia Nia Nopeliza, Nia Nia Three Manurung Nilam Cahyati Nora Herliza Noviana, RA Sania Nurhikmah, Putri Oktasari, Tina Patriot, Evelina Astra Pertiwi, Nadiar Pratiwi Ineke Anwar Putra, Gede Mudita Edi Putra, Guruh Sukarno Putri Maulida, Nabila Putri Wijayanti Putri, Jamiatul Khairunnisa Putri, Mutiara Juliana Putriyani, Fena Siska Qathrunnada, Sausan Rahman, Nor Farahwahidah Abdul Rahmannisa, Amanda Ramadhani, Neysya Ditha Rara, Rara Redondo, Fernando Eric Ribkoh, Ribkoh Risa Albeta Rizki Novianti, Rizki Romadoni, Muhammad Sadly, Deni Salmah Rianti Salmah Rianti Saparini Saparini Saparini Sardianto Markos Siahaan Sardianto Markos Siahaan, Sardianto Markos Sari, Dwita Kartika Sari, Isma Sari, Mona Seprina, Iin Sholeha, Nurlita Dwi Siti Nur Azizah Sri Mindia Vanessa Pratiwi Sri Zakiyah Sudirman Sudirman Sudirman Sudirman Suhadi Suhadi Suhadi Suhadi Suhadi Suhadi Supardi Supardi Supardi Supari Supari Supari Supari Supari Supari Supari, Supari Sutanto, Taufiq Sutinah, Eka Syahputri, Selvia Mariska Syuhendri, Syuhendri Taufiq Taufiq Tine Aprianti Utami, Amanda Kurnia VIOLANTI ANARKY, VIOLANTI Wailaina Wati, Lira Diska Winarni, Kiki Ayu Wulandari, Dwi Noviana Yenny Anwar Yuli Setianingsih Yuliana, Rahma Zulherman Zulherman Zulherman Zulherman, Zulherman