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arXiv · 2411.18890

How Are Quantum Eigenfunctions of Hydrogen Atom Related To Its Classical Elliptic Orbits?

Yixuan Yin·Tiantian Wang·Biao Wu
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Abstract

We show that a highly-excited energy eigenfunction $\psi_{nlm}(\vec{r})$ 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 $|\psi_{nlm}(\vec{r})|^2$ and the classical probability distribution $p_c(\vec{r})$ 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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Yixuan Yin, Tiantian Wang, Biao Wu. 2024-11-28. How Are Quantum Eigenfunctions of Hydrogen Atom Related To Its Classical Elliptic Orbits?. https://doi.org/10.1088/1674-1056%2Fae3303

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