⚛️ general relativity

Stationary states for a particle in a box with slanted walls

This paper investigates the quantum mechanical problem of a particle in an infinite potential well with slanted walls, deriving energy eigenvalues via Airy functions and demonstrating that the system exhibits unique asymptotic energy spacing and confinement behaviors that surpass standard models in fulfilling Bohr's correspondence principle.

Original authors: Nivaldo A. Lemos

Published 2026-09-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Nivaldo A. Lemos

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the rules of motion are dictated by the shape of the space a particle occupies. Imagine a tiny object, like an electron, trapped inside a container. If the walls of that container are perfectly flat and vertical, the object can only exist at specific, distinct energy levels, much like a ladder with rungs that are spaced further apart as you go up. This is the classic "particle in a box," a fundamental model used to understand everything from chemical bonds to the behavior of tiny electronic components. However, nature rarely offers such perfect geometric simplicity. When the walls of the container are not vertical but instead slope inward or outward, the rules change. The energy levels shift, the spacing between them alters, and the way the particle behaves near the edges becomes far more complex. Understanding these slanted environments is crucial because real-world traps for particles, such as those found in advanced semiconductor devices or nanostructures, often have sloped boundaries rather than sharp, vertical ones.

A physicist at the Federal Fluminense University in Brazil has recently explored exactly this scenario: a particle trapped in a box where the walls are not vertical but slanted. The study investigates how the particle's energy levels and its position change when the container's sides tilt. The researcher found that the mathematical description of this system relies on a special set of functions known as Airy functions, which are not part of the standard toolkit for most physics students but are essential for solving problems where forces change linearly with position. By using these functions, the study determined the precise energy levels the particle can hold and how likely it is to be found outside the region where classical physics says it should be.

The most striking discovery concerns the spacing of these energy levels. In the standard box with vertical walls, the gap between energy levels grows larger as the energy increases, following a pattern where the difference between consecutive levels increases linearly with the quantum number. In the harmonic oscillator, a model for a spring-like force, the gaps remain exactly the same size forever. But in this box with slanted walls, the behavior is different. As the particle's energy gets higher, the gaps between the allowed energy levels become smaller and smaller, eventually shrinking to the point where the levels are almost indistinguishable from one another. This means that at high energies, the quantum system begins to look and act almost exactly like a classical system, where energy can vary continuously. This result suggests that the famous correspondence principle, which states that quantum mechanics should match classical physics at large scales, is satisfied even more accurately in this slanted box than in the standard models taught in textbooks.

The study also looked at how tightly the particle is confined within the box. A state is considered more confined if the particle is less likely to be found in the "forbidden" regions outside the box, a phenomenon known as tunneling. When the walls are only slightly slanted, the ground state—the lowest energy state—is the least confined, meaning the particle leaks out the most. This is similar to what happens in a harmonic oscillator. However, the researcher found a surprising turning point. When the slope of the walls becomes steep enough, the behavior flips. Beyond a specific critical slope value, the ground state is no longer the least confined. Instead, the first excited state, which is the next energy level up, becomes the one that leaks out the most. This reversal was unexpected and does not happen in the standard harmonic oscillator model.

To reach these conclusions, the researcher solved the equations governing the particle's motion for different wall slopes. The solutions were found using graphical and numerical methods, as the equations are too complex for a simple algebraic solution. The results showed that as the walls become steeper, the particle is forced closer to the center of the box, and its energy levels rise. The study calculated the probability of the particle being found in the forbidden regions for several different states and wall angles. The data revealed that for very steep walls, the ground state becomes extremely well confined, while the first excited state remains more spread out. This counterintuitive finding highlights how sensitive quantum systems are to the precise shape of their environment.

The work serves a dual purpose. Physically, it provides a more realistic model for particles in nanostructures where potential energy does not change abruptly but rises gradually. Mathematically, it offers a practical way for students to learn about Airy functions, which are often overlooked in undergraduate courses. By working through this problem, students can see how these functions describe real physical situations, from the behavior of electrons in tilted fields to the vibration of strings under tension. The study confirms that while the mathematics can be intricate, the physical picture is clear: the shape of the trap dictates the rhythm of the particle's energy, and changing that shape can lead to surprising shifts in how the particle is confined.

Technical Summary: Stationary States for a Particle in a Box with Slanted Walls

Problem Statement
This paper investigates the quantum mechanical stationary states of a particle of mass $m$ confined within a one-dimensional infinite potential well featuring linearly rising (slanted) walls. The potential energy $V(x)$ is defined as zero within the interval $0 \le x \le a$ , rises linearly with slope $F_1$ for $x > a$ , and rises linearly with slope $F_2$ for $x < 0$ . Unlike the standard infinite square well or the harmonic oscillator, this hybrid potential requires solutions involving Airy functions rather than simple trigonometric or parabolic functions. The study aims to determine the energy eigenvalues, analyze the asymptotic behavior of the energy spectrum, and quantify the degree of spatial confinement (tunneling probability) for various stationary states, with a specific focus on the symmetric case where $F_1 = F_2$ .

Methodology
The solution is derived by solving the time-independent Schrödinger equation in three distinct regions:

Region $x < 0$ : The potential is linear ( $V = -F_2 x$ ). The solution is expressed in terms of the Airy function of the first kind, $Ai(\eta)$ , as the second kind, $Bi(\eta)$ , diverges at infinity.
Region $0 \le x \le a$ : The potential is zero. The solution is a linear combination of sine and cosine functions.
Region $x > a$ : The potential is linear ( $V = F_1(x-a)$ ). The solution is proportional to $Ai(\xi)$ .

The physical wave function and its first derivative are required to be continuous at the boundaries $x=0$ and $x=a$ . These continuity conditions yield a system of linear equations for the integration constants. Eliminating these constants results in a transcendental equation involving $Ai$ , its derivative $Ai'$ , and trigonometric functions of the dimensionless energy parameter $z$ . This equation determines the allowed energy eigenvalues.

The transcendental equation is solved using graphical and numerical methods (specifically utilizing Gnuplot) to find the positive zeros of a constructed function $f(z)$ . The paper also employs an intuitive semiclassical argument and asymptotic properties of Airy functions to analyze the behavior of the spectrum for large quantum numbers. Finally, the wave functions are normalized, and the tunneling probability (the probability of finding the particle in the classically forbidden region) is calculated for the symmetric case ( $F_1 = F_2$ ) and compared against the harmonic oscillator.

Key Results

Energy Spectrum: The allowed energies are determined by the roots of the derived transcendental equation. For large quantum numbers ( $n \to \infty$ ), the energy levels scale as $E_n \propto n^{2/3}$ . Consequently, the energy difference between consecutive levels, $\Delta E_n$ , tends to zero as $n \to \infty$ .
Bohr's Correspondence Principle: The paper argues that this system satisfies Bohr's correspondence principle more accurately than the standard particle in a box or the harmonic oscillator. In the standard box, $\Delta E$ increases with $n$ ; in the harmonic oscillator, $\Delta E$ is constant. In the slanted-wall box, $\Delta E$ vanishes as $n \to \infty$ , causing the spectrum to appear effectively continuous, which aligns more closely with classical expectations for a particle in a potential where the confining force remains constant rather than increasing indefinitely.
Confinement and Tunneling: The degree of confinement is defined inversely by the tunneling probability.
- For the harmonic oscillator, the ground state is the least confined (highest tunneling probability) of all stationary states.
- In the symmetric slanted-wall box, a critical slope exists (identified numerically at $z_1 = z_2 \approx 2.55581$ ). Below this slope, the ground state is the least confined. However, above this critical slope, the ground state becomes more confined than the first excited state. For slopes steeper than this critical value, the first excited state (and eventually higher excited states) exhibits a higher tunneling probability than the ground state, a counter-intuitive result not observed in the harmonic oscillator.

Significance and Claims
The paper positions this problem as a pedagogical tool that introduces undergraduate physics students to Airy functions, which are typically absent from standard mathematical repertoires in physics curricula. Beyond its mathematical utility, the study highlights unique physical features of linearly rising potentials:

The specific asymptotic behavior of the energy spectrum ( $E_n \propto n^{2/3}$ ) provides a distinct example of how the nature of the confining force (constant vs. linearly increasing) dictates the high-energy limit of quantum systems.
The discovery of the "inversion" in confinement order—where the ground state ceases to be the least confined eigenstate beyond a critical wall slope—offers a non-trivial physical insight into the interplay between potential shape and wave function localization.

The author explicitly avoids proposing new experimental applications or future implications, focusing instead on the theoretical characterization of the stationary states and the educational value of familiarizing students with non-standard special functions in quantum mechanics.

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