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Journal : progressive physics journal

Application of Physics-Informed Neural Networks (PINNs) for the Numerical Solution of the Time-Independent Schrödinger Equation Hamdi Akhsan; Khoirun Nisa; Putri Nurhikmah; Wailaina; Melly Ariska
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 Agung Mataram Agung Sugiarto Agustina, Atika Ahmad Fauzi Ismail Ahmad Fudholi Aisyah Veronika Aji, Dwi Purnomo Aldi Prayoga Aljuzza, Muhammad Ikhsan Almafie, Muhammad Rama Amalia, Wardah Amanda Kurnia Utami Amanda Rahmannisa Andriani, Nelly Anggelina, Putri Anggelina, Zanne Anisa Qoriq Atun Nasiroh Apit Fathurohman Apit Fathurohman Arini Rosa Sinensis Atika Agustina Ayu Sekar Sari Azizah Putri Berimah Berimah, Azizah Putri Cahyani, Delvina Putri Cahyati, Nilam Damayanti, Fitria Siska Dani, Rahma Dea Amalia Dea Rizky Oktavia Dendi Juliansyah Deni Sadly Desilawati Detrika Mawar Arrum Devi Ardiantini Devi Ardiantini Dia Oktarina Gumanti Dwi Purnomo Aji Dwi Purnomo Aji Dwicahyani, Rania Dwita Kartika Sari Edo Anugrah Edray, Auliya Einsrizkianne Effendi Effendi Eka Sutinah Eka Sutinah Elsie Sulastry Erazando Alfa Seira Ermalia, Widia Evelina Astra Patriot Fahreza, Muhammad Aditya Fajriah, Nurkholisa Fatmawati, Fatmawati Fatonah, Amelia Febrianti, Tanti Fena Siska Putriyani Fernando Eric Redondo Fika Nurul Hidayati Fina Maliya Fitri Nisa Fitria Siska Damayanti Fitria Siska damayanti Fitriyani Fitriyani Guruh Sukarno Putra Hayati, Suci Husnul Huriyatul Firdausi I. Ismet Ida Sriyanti Iful Amri Iin Seprina Ina Agustina Intan Mahdiah Intan Purnama Sari Irenta Irenta Ishkaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Iskhaq Iskandar Isma Sari Ismet Ismet Ismet Ismet Ismet, Ismet Ismet, I. Izzah Nurhidayah, Izzah Jamiatul Khairunnisa Putri Jesi Pebralia Juwita Febrianti Kartika, Nadia Dwi Kayla Azzakia Ketang Wiyono Ketang Wiyono Ketang Wiyono Ketang Wiyono Kheisya Nadiva Az-Zahra KHOIRUN NISA Kiki Ayu Winarni Kistiono Kistiono Kistiono Kistiono Kistiono Kistiono Kistiono Kistiono Komarudin Komarudin Kurnia, Mega Lamura, M. Danny Pratama Lara Sania Laras Sapitri, Cindy Leni Marlina Leni Marlina Leni Marlina Lestari, Yora Inda Lira Diska Wati Lira Lira Lira, Lira Lismayana, Pegi Putri M. Nicola Prajaska Maghfira Aulia Mahdiah, Intan Mahmudah, Indri Markos Siahaan, Sardianto Masduki Masduki Mega Kurnia Mega Kurnia Meliani Devina Melinda Maharani Melly Ariska Melly Ariska Melly Junita Melvany, Nanda Eva Michell Butarbutar Miftahul Jannah Mindia Vanessa Pratiwi, Sri Muhamad Yusup Muhamad Yusup Muhamad Yusup Muhammad Aditiyah Muhammad Afrizal 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 . Mutia Faradini Ramadhanty Mutiah Lestari Mutiara Putri Nadiah Alfania Nadiar Pertiwi Nadya Anggraini Nadya Anggraini Nasiroh, Anisa Qoriq Atun Nazhifah, Rizqia Nely Andriani Nely andriani Nely Andriani, Nely Nestity, Alifia Nia Nopeliza, Nia Nia Three Manurung Nia Three Manurung Nia Three Manurung Nilam Cahyati Nita Arrum Sari Noor Dayana Abd Halim Nor Farahwahidah Abdul Rahman Nor Farahwahidah Abdul Rahman Nora Herliza Noviana, RA Sania Nur Kayla Lekati Nurkholisa Fajriah Pertiwi, Nadiar Pratiwi Ineke Anwar Putra, Gede Mudita Edi Putra, Guruh Sukarno Putri Anggelina Putri Anisah Putri Maulida, Nabila Putri Nurhikmah Putri Wijayanti Putri, Arselly Rahmanda Putri, Jamiatul Khairunnisa Putri, Mutiara Juliana Putriyani, Fena Siska Qathrunnada, Sausan Rahma Hafitri Rahman, Nor Farahwahidah Abdul Rahmannisa, Amanda Ramadhani, Neysya Ditha Rara Rara Resta Ulis Muslimah Resta Ulis Muslimah Ribkoh, Ribkoh Risa Albeta Rita Inderawati Rizki Novianti, Rizki Romadoni, Muhammad Rosada, Mita Rosdiana Salmah Rianti Salmah Rianti Salsa bila Salsabila Endana Putri Salsabillah, Zahwa Nur Saparini Saparini Sardianto Markos Siahaan Sardianto Markos Siahaan Sardianto Markos Siahaan Sardianto Markos Siahaan, Sardianto Markos Sari, Dwita Kartika Sari, Mona Sekarani, Tresnane Suandti Selvia Mariska Syahputri Seprina, Iin Sherry Junira Sagita Sholeha, Nurlita Dwi Siti Kurdiah Siti Nur Azizah Sopia Sopia Sri Mindia Vanessa Pratiwi Sri Zakiyah Sudirman Sudirman Sudirman Sudirman Suhadi Suhadi Suhadi Suhadi Suhadi Sunarti, Arinda Rahma Supardi Supardi Supardi Supari Supari Supari Supari Supari Supari Supari, Supari Sutanto, Taufiq Syahputri, Selvia Mariska Syahrani, Revanola Gusti Syuhendri, Syuhendri Tamara Shelvy Maharani Tanti Febrianti Taufiq Sutanto Taufiq Taufiq Taufiq Taufiq Tina Oktasari Tine Aprianti Tita Ratna Wulan Dari Tresnane Suandti Sekarani Trista Meilanie Tsabita Husna Tyse Ramadhona Utami, Amanda Kurnia Vania, Amanda Asti VIOLANTI ANARKY, VIOLANTI Wahyuningsi, Wahyuningsi Wailaina Wardah Amalia Wati, Lira Diska Winarni, Kiki Ayu Yendri Tri Hartati Napitupulu Yenny Anwar Yesi Monalika Yuli Setianingsih Yuliana, Rahma Zahlia, Marsela Zahra Ayanna Firmansyah Zahrani, Ami Zulherman Zulherman Zulherman Zulherman, Zulherman