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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 Aanisah Nadya Shafwah Abdul Fattah Nasution Addina Tafaquh. F Adib Ahmada Adinda Ramawati Aditya Dwi Prananda Adzkia Rizkianisa Awira Siregar Agisca Arifien Ahmad Syihabuddin A.F Ahmad Zainuri Aji Pramudya Akbar Rafsanjani Alip Laila Bintar Wati Alwiyah Nasution Alya Mardatilla Amalia Wahyuningtyas Amaluddin Tanjung Anastasia Wheni Indrianingsih Andra Abel Guvinda Angel Vionika Saharani Anika Ardila Anna Mukhayaroh Annastasya Nabila Elsa Wulandari Annisa Luthfiyyah Anzilly Ifadatun Ummah Arif Setia Sandi A. Arimbi Syahkila Simangunsong Aslikhatun Khoiriyah Asqia Viani Auffah Yumni Aulia Utami Aripin Ayu Lucy Larassaty Ayu Salsabila Ayu Shaumina Mubarokah Ayuda Dyah Nurekawati Baihakil Aziz carmidah carmidah Catur Febriyan Chusnul Chotimah Cici Darsih Deberina Syurfi Yunina Dede Cahyati Sahrir Dedi Masri Deny Tri Prasetia Devy Kusuma Dian Andani DEWI LEYLA RAHMAH Diah Arum Sari Nawang Ulan Dian Sulistyorini Wulandari Diardana Prima Sintasari Dina Oktapia Dina Oktavia Dina Tri Mulyani Djatnicka, Erlina Een Juhriah Elsa Amelia. A Elva Meilani Sagita Erlin Maulidatus. S Eva Ulfiana Dewi Eveline Stephani Fadhylah Zaini Pardede Fajarwati, Salsabila Fani Laffanillah Fara Amalia Dewi Fatimatus Zahro Fiki Khoirotussiha Firdatul Izza Fithriyani, Dina Frida Aprilia Frika Fatimah Zahra Galuh Sandi Gentio Harsono H. Moeljadi Hamdi Akhsan Haq, Furqonul Hendri Stiawan Ignatia Esti Sumarah Ika Ika Iksan Adi Nugroho Ilvi Mawaddah Inom Nasution Irma Tussa’diah Hasibuan Ismatul Izzah Iva Nur Zakiya Ivan Ilham Riyandi Izzatun Nafis Kale Khairi Putik Nurikhwani Kenconojati, Hapsari Khairun A Roni Khotimatul Husna Lestari Yusuf Lulu Nazma Lailatul Muchlis M. Mun’im M. Yunus Abu Bakar Machful Indra Kurniawan Manik, Santini Marista Febria Safutri Mauliana, Metatia Intan Maulidayani Maulidayani Mega Dewi Ambarwati Mela Safitri Situmorang Melly Ariska Meysah Wanda Sari Mia Indriyani Minsih Mochamad Rizal Yulianto Mochamad Wahyudi Moh. Hafidh Ubaidillah Al Hasan Moh. Mahmud Moh. Rofiki Mohammad Ahyar Ma'arif Mohammad Falahuddin Muhammad Aidil Fitri Muhammad Alfiansyah Muhammad Ispan Muhammad Kasran Munawaroh Munirul Abidin MUZAYANAH Nafia Ilhama Qurratu’aini Narwikah Narwikah Nashihin Naswa Ika Febrianty Nihayatus Sakhiyah Niken Eriana Azzahra Fauzi Noly Shifiyah Nor Anisa Novelya Aleena Rofani Novi Arilia Safitri Novia Dwi Rahmawati Novrina Feronita Br.Sibuea NUR HASANAH Nur Izzah Nur Subekti Nurheny Rahmawati Nurhikmah, Putri NURUL AZIZAH Nurul Ismaiyah Nurul Lathifatul Fithriya Prayoga, Estu Purwono Purwono, Purwono PUTRI HANDAYANI Putri Syafaatun Qoni'ah Nur Wijayani Qurrotul A’yuni Rachman Hidayat Randi Purnama Putra Reni Endrawat Retno Septiwulan Riani Nur Azizah Ridlotun Nuvi Setyan Rifky Adha Rina Devianty Riska Nur Aini Rismadefi - Woferst Rizal Fadly Ardiansyah Riziq Syehab Rizka Mufidah Rizki Aulianita Sabrienda Najwa. A Salwa Nabila Damanik Septi Nurhayati Shalvena Aura Azzura Sholihah Roekhana Silma Zulfaika Sibli Siti Anisah Siti Fatimah, Eva Sofyan Syamsuddin Soomro, Suhaib Ahmed Subhi Indriyani Subi anto Sukendra Martha Supriyanta Supriyanta Susan Widianingsih Tengku Darmansyah Tina Novi Lestari Trisnawati Hutagalung Valent Milenio Boas Santosa Vanda Rezania Viraesti Damayati Vita Taufika Rosyida Wahyu Eko Pujianto Wahyu Ningsih Wailaina Waluyo, Dangan Winih Budiarti Wisnu Panggah Setyono Wiwin Eka Kristina Wuri Apriyana Yayuk Sri Kasiani Yesi Hasneli Yesi Novia Pitriani Yudhi Novriansyah Yumi Kholifah Yunianita Rahmawati Zahro Ulfaturrohmatiririn Zakiyatun Nabila zulfatun anisah Zuyyina Fihayati