Classical Representation for Quantum States of a Particle in $λz^{2m}$ potential

TL;DR

The paper develops a classical representation for quantum eigenstates in potentials $V(z)=\lambda z^{2m}$ by constructing energy distributions $f_n(\varepsilon)$ that, though quasiprobabilistic and sometimes negative, reproduce quantum position densities through the Abel transform and yield the quantum eigenenergies via their first moment. It provides both analytic results (notably the harmonic oscillator case with $f_n(\varepsilon)=(-1)^n e^{-\varepsilon} L_n(2\varepsilon)$) and numerical inversions of the Abel transform to obtain $f_n(\varepsilon)$ for general $m$, showing how the distributions diverge at $\varepsilon\to 0$ for $m>1$ but remain integrable in cumulative form; the mean energy constraint $\int_0^{\infty} \varepsilon f_n(\varepsilon) d\varepsilon = \varepsilon_n$ holds, connecting the classical representation to quantum spectra. The work also derives the Schrödinger equation in the classical representation, yielding a third-order equation for $\rho_n(x)$ and an integrodifferential equation for $\varphi_n(\varepsilon)$, with the harmonic-oscillator case providing a consistent check. In the large-$m$ limit, the energy distributions develop nonintegrable $\varepsilon^{-1}$-type singularities, indicating that the classical representation becomes ill-defined for the infinite square well, a key insight into the method's limitations and a motivation for extending the framework to more general or higher-dimensional systems.

Abstract

A classical representation for quantum eigenstates of a particle bound in $λz^{2m}$ $(λ>0, m=1,2,...)$ potentials is developed. It is represented by ensembles of classical trajectories with energy distributions that can take on negative values, for $m>1$ have integrable singularities at zero energy and whose mean energies coincide with quantum eigenenergies. The corresponding Schrödinger equation in classical representation is analyzed.

Figures (11)

• Figure 1: Ground-state position probability distributions $\rho_0(x)=\rho_0(-x)$ in various confining potentials $x^{2m}$. The values of distributions at classical turning points are marked by black points. Vertical thin line indicates the position of the infinite wall of the limiting ($m\to+\infty$) square well potential.
• Figure 2: Ground state eigenvalues $\varepsilon_0$ for various confining potentials $x^{2m}$. Numerical results are represented by black symbols (curve labeled exact). Zeroth and second order WKB approximations (curves labeled WKB0 and WKB2) are represented by red and blue symbols. Asymptotes corresponding to $m\to \infty$ are represented by dashed lines.
• Figure 3: Same as Fig.\ref{['Fig1']} but for excited $n=4$ state.
• Figure 4: Same as Fig\ref{['Fig2']} but for $n=4$ states.
• Figure 5: Ground-state energy distributions $f_0(\varepsilon)$ for various $x^{2m}$ potentials.