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

TL;DR

The paper addresses the quantum-classical correspondence for a hydrogen atom by showing that a highly excited eigenfunction $\psi_{nlm}$ is effectively an equal-weight superposition of all classical elliptic orbits with energy $E_n$, angular momentum $L=\sqrt{l(l+1)}\hbar$, and $L_z=m\hbar$. The authors derive classical probability densities $p_c(r)$ and $p_c(\theta)$ for the ensemble of such orbits and compare them to the quantum densities $p_q(r)=|R_{nl}(r)|^2 r^2$ and $p_q(\theta)=|Y_l^m(\theta,\phi)|^2\sin\theta$, showing strong agreement in the semiclassical limit $n\to\infty$ when appropriate scaling is maintained (fixed $l/n$ and $m/l$). Radial densities converge with quantum oscillations around the classical envelope, while angular densities align in trend and improve with larger $l$, supporting the view that eigenstates reduce to invariant phase-space ensembles rather than single classical trajectories. The work highlights a general semiclassical principle, extends to other central potentials and scattering states, and underscores the role of Liouville/Moyal formalisms in connecting quantum states to classical phase-space distributions. Overall, it provides a concrete, quantitative bridge between quantum eigenfunctions and classical orbital ensembles in a fundamental atomic system, with implications for semiclassical analyses in strong-field and multi-degree-of-freedom contexts.

Abstract

We show that a highly-excited energy eigenfunction $ψ_{nlm}(\vec{r})$ 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 $|ψ_{nlm}(\vec{r})|^2$ 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.

Figures (9)

• Figure 1: Probability density of a harmonic oscillator. Blue line is the quantum probability density for the 10th energy eigenstate; the gray line is the corresponding classical probability density. The unit of $x$ is $\sqrt{\frac{m\omega}{\hbar}}$, where $m$ and $\omega$ are the mass and frequency of the oscillator, respectively.
• Figure 2: Comparison of the quantum and classical probability densities: (a) the surface where the quantum probability is $|\psi_{nlm}|^2=10^{-5} \text{m}^3$ with $n=6$, $l=4$, $m=2$; (b) the surface where the classical probability divided by $\sin\theta$ is $10^{-5} \text{m}^3$. In the figure, $a$ is the Bohr radius.
• Figure 3: Classical orbits of hydrogen atom. The origin of the coordinates is the position of proton. (a) In the special case of zero angular momentum $l=0$, the classical orbits are straight lines orientated in all possible directions. These orbits collectively fill a spherical region , reflecting that the corresponding eigenstates are $s$-waves. The radius of this sphere is determined by the eigen-energy $E_n$. (b) For non-zero angular momentum $l\neq 0$, the classical orbits for a quantum eigenstate $\psi_{nlm}(\vec{r})$ are ellipses, which can be divided into two groups: (1) the ellipses that can be transformed into each other rotating around the $z$ axis and reflecting off the $XY$ plane; (2) all the ellipses that rotate around the focus of proton in the orbital plane, which is perpendicular to the angular momentum $\vec{L}$. For convenience, the $X'Y'$ coordinate system is set up in the orbital plane. The shape and size of the ellipses are determined by quantum number $n$ and $l$, the orientation of the ellipses relative to the $Z$ axis is determined by $l$ and $m$.
• Figure 4: One elliptic orbit in a coordinate system where the origin is the position of proton. $\vec{L}$ is the angular momentum vector. A is a point on the elliptic orbit. All the blue lines lie in the plane of the elliptic orbit. (a) $\alpha$ is the angle between $\vec{L}$ and the $Z$ axis, and $\theta$ is the angle between OA and the $Z$ axis. $X'Y'$ is a two-dimensional coordinate system in the plane of elliptic orbit. The $X'$ axis, $Z$ axis, and $\vec{L}$ lie in the same plane. $\gamma_0$ is the angle between the major axis of the ellipse and the $X'$-axis, and $\gamma$ is the angle between OA and the $X'$ axis. (b) B is a point on the $X'$ axis, and C is a point on the $Z$ axis. Their positions are chosen so that triangle ABC is perpendicular to the $Z$ axis.
• Figure 5: The radial probability density of the hydrogen atom for the case of zero angular momentum ($l=0$). The blue lines are quantum results $p_q({\tilde{r}})$, the black solid lines are classical results $p_c({\tilde{r}})$, and the dashed lines are doubled classical results $2p_c({\tilde{r}})$.