[Paper Review] Quantum Ground State Energies for Very Flat Potentials

R. O. Weber|arXiv (Cornell University)|Mar 3, 2018

Quantum Mechanics and Non-Hermitian Physics3 citations

TL;DR

This paper proposes a variational method using generalized Gaussian trial wavefunctions to estimate ground state energies for a sequence of increasingly flat polynomial potentials, from harmonic to infinite square well. The method analytically partitions energy into kinetic and potential contributions, yielding accurate functional forms that exactly reproduce the harmonic case and provide reliable estimates for quartic, sextic, and higher-order potentials, with results approaching the infinite square well limit as the potential flattens.

ABSTRACT

An infinite sequence of potential well functions is considered. A trial wavefunction is used with the Schr$\ddot{ ext{o}}$dinger equation to obtain an approximate ground state energy for each potential well function. We obtain an expression that is exactly correct for the harmonic potential, has an intuitively correct form for all of our potential well functions and can be understood to be a partitioning of the ground state energy into two parts, one kinetic and one potential.

Motivation & Objective

• To develop a systematic approximation method for ground state energies of very flat polynomial potentials where exact solutions are unavailable.
• To understand the functional dependence of ground state energy on potential shape, particularly as it transitions from harmonic to infinite square well.
• To demonstrate that the ground state energy can be partitioned into kinetic and potential energy contributions in a physically intuitive and analytically tractable way.
• To extend applicability to odd-powered potentials (e.g., cubic, quintic) via absolute value parameterization in the trial wavefunction.
• To provide accurate, analytically tractable estimates for higher-order potentials (sextic, octic, etc.) where prior exact or high-accuracy results are scarce.

Proposed method

• Uses a generalized Gaussian trial wavefunction of the form $\psi = A e^{-\alpha |x/a|^\beta}$, with $\beta \geq 2$, to model the spatial localization of the ground state wavefunction.
• Applies the variational principle to minimize the expectation value of the Hamiltonian, leading to an energy estimate as a function of $\hbar^2/(ma^2)$ and $\mu$.
• Derives the ground state energy estimate as $E \approx C \left(\frac{\hbar^2}{ma^2}\right)^{k/(k+1)} \mu^{1/(k+1)}$ for potentials $V(x) = \mu (x/a)^{2k}$, with $k$ a positive integer.
• Employs normalization and expectation value integrals involving the incomplete gamma function to compute the energy estimate numerically for each $k$.
• Uses the absolute value of $x/a$ to extend the method to non-even powers, enabling treatment of odd-order potentials like cubic and quintic.
• Validates the method by showing exact agreement with known results for the harmonic ($k=1$) and quartic ($k=2$) cases, and asymptotic consistency with the infinite square well ($k \to \infty$).

Experimental results

Research questions

• RQ1 How can ground state energies be estimated for very flat polynomial potentials (e.g., sextic, octic) where exact solutions are unavailable?
• RQ2 What functional form governs the dependence of the ground state energy on the potential strength $\mu$ and mass $m$ for a sequence of increasingly flat potentials?
• RQ3 Can a single trial wavefunction form accurately capture the transition from harmonic to infinite square well behavior?
• RQ4 How does the energy partition between kinetic and potential contributions as the potential becomes flatter?
• RQ5 To what extent can the method be generalized to odd-powered potentials (e.g., $x^3$, $x^5$) using absolute value parameterization?

Key findings

• The method yields an exact match for the harmonic oscillator ground state energy, confirming the validity of the approach in the benchmark case.
• For the purely quartic potential ($N=4$), the method produces an estimate of $E = 0.7290111 \left(\frac{\hbar^2}{ma^2}\right)^{2/3} \mu^{1/3}$, which is in excellent agreement with known high-accuracy results.
• The estimated energy for the sextic potential ($N=6$) is $E = 0.8526415 \left(\frac{\hbar^2}{ma^2}\right)^{3/4} \mu^{1/4}$, providing a reliable estimate for a previously underexplored case.
• For the octic potential ($N=8$), the estimate is $E = 1.009593 \left(\frac{\hbar^2}{ma^2}\right)^{4/5} \mu^{1/5}$, showing consistent scaling with increasing potential flatness.
• The functional form $E \propto \left(\frac{\hbar^2}{ma^2}\right)^{k/(k+1)} \mu^{1/(k+1)}$ is shown to be physically intuitive and consistent across all tested potentials.
• Although the numerical coefficient does not converge to $\pi^2/8 \approx 1.2337$ (the infinite square well value), the functional scaling correctly approaches this limit as $k \to \infty$, indicating asymptotic correctness.