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How are quantum eigenfunctions of hydrogen atom related to its classical elliptic orbits? |
| Yi-Xuan Yin(殷艺轩)1, Tian-Tian Wang(王天天)1, and Biao Wu(吴飙)1,2,3,† |
1 International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China;
2 Wilczek Quantum Center, Shanghai Institute for Advanced Studies, Shanghai 201315, China;
3 Hefei National Laboratory, Hefei 230088, China |
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Abstract We show that a highly-excited energy eigenfunction $\psi_{nlm}$(r) of hydrogen atom can be approximated as an equal-weight superposition of classical elliptic orbits with energy $E_n$ and angular momentum $L=\sqrt{l(l+1)}\hbar$, and $z$ component of angular momentum $L_z=m\hbar$. This correspondence is established by comparing the quantum probability distribution |$\psi_{nlm}$(r)|2 and the classical probability distribution $p_{\rm c}$(r) of an ensemble of such orbits. This finding illustrates a general principle: in the semi-classical limit, an energy eigenstate of a quantum system is in general reduced to a collection of classical orbits, rather than a single classical orbit.
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Received: 20 October 2025
Revised: 23 December 2025
Accepted manuscript online: 04 January 2026
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PACS:
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03.65.-w
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(Quantum mechanics)
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03.65.Sq
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(Semiclassical theories and applications)
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| Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 92365202, 12475011, and 11921005), the National Key Research and Development Program of China (Grant No. 2024YFA1409002), the Shanghai Municipal Science and Technology Major Project (Grant No. 2019SHZDZX01), the Shanghai Municipal Science and Technology Project (Grant No. 25LZ2601100), and the Innovation Program for Quantum Science and Technology (Grant No. 2021ZD0302100). |
Corresponding Authors:
Biao Wu
E-mail: wubiao@pku.edu.cn
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Cite this article:
Yi-Xuan Yin(殷艺轩), Tian-Tian Wang(王天天), and Biao Wu(吴飙) How are quantum eigenfunctions of hydrogen atom related to its classical elliptic orbits? 2026 Chin. Phys. B 35 050301
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