中国物理B ›› 2026, Vol. 35 ›› Issue (5): 50301-050301.doi: 10.1088/1674-1056/ae3303

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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. 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
  • 收稿日期:2025-10-20 修回日期:2025-12-23 接受日期:2026-01-04 发布日期:2026-05-07
  • 通讯作者: Biao Wu E-mail:wubiao@pku.edu.cn
  • 基金资助:
    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).

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. 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
  • Received:2025-10-20 Revised:2025-12-23 Accepted:2026-01-04 Published:2026-05-07
  • Contact: Biao Wu E-mail:wubiao@pku.edu.cn
  • Supported by:
    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).

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摘要： 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.

关键词: hydrogen, energy eigenfunction, invariant distribution, semiclassical limit

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.

Key words: hydrogen, energy eigenfunction, invariant distribution, semiclassical limit

中图分类号:  (Quantum mechanics)

  • 03.65.-w
03.65.Sq (Semiclassical theories and applications)