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arXiv:2609.07505

Stationary states for a particle in a box with slanted walls

Nivaldo A. Lemos

quant-phgr-qchep-thSubmitted Sep 7, 2026• Announced Sep 8, 2026• 0 citations
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Abstract

The quantum mechanical problem of a particle in a infinite potential well with slanted walls is studied. The energy eigenvalues are determined by a transcendental equation involving the Airy function of the first kind. Two limiting cases are discussed. Next, the allowed energies are found by graphical and numerical methods. The graphical analysis hints that for large quantum numbers the consecutive energy levels get arbitrarily close together. This asymptotic behavior of the energy spectrum actually holds, as shown by means of an intuitive argument and confirmed by resorting to basic properties of Airy functions. In a sense, Bohr's correspondence principle is more accurately fulfilled than in the cases of the standard particle in a box or the harmonic oscillator. A state is regarded the more confined the less the probability of finding the particle in the classically forbidden region. For some stationary states the confinement degree is calculated and a comparison is made with the harmonic oscillator, for which the ground state is the least confined of all stationary states. In the symmetric case --- right and left walls of equal slope --- something unexpected happens: beyond a certain critical slope the ground state is no longer the least confined eigenstate. Over and above its interesting physical features, this problem provides students with the opportunity to get acquainted with Airy functions, which do not belong to the standard mathematical repertoire of physics undergraduates.

Comments
17 pages, 8 figures, 2 tables. Accepted for publication in Revista Brasileira de Ensino de Física

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The quantum mechanical problem of a particle in a infinite potential well with slanted walls is studied. The energy eigenvalues are determined by a transcendental equation involving the Airy function of the first kind. Two limiting cases are discussed. Next, the allowed energies are found by graphical and numerical methods. The graphical analysis hints that for large quantum numbers the consecutive energy levels get arbitrarily close together. This asymptotic behavior of the energy spectrum actually holds, as shown by means of an intuitive argument and confirmed by resorting to basic properties of Airy functions. In a sense, Bohr's correspondence principle is more accurately fulfilled than in the cases of the standard particle in a box or the harmonic oscillator. A state is regarded the more confined the less the probability of finding the particle in the classically forbidden region. For some stationary states the confinement degree is calculated and a comparison is made with the harmonic oscillator, for which the ground state is the least confined of all stationary states. In the symmetric case --- right and left walls of equal slope --- something unexpected happens: beyond a certain critical slope the ground state is no longer the least confined eigenstate. Over and above its interesting physical features, this problem provides students with the opportunity to get acquainted with Airy functions, which do not belong to the standard mathematical repertoire of physics undergraduates.
The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.
Abstract The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.
Two of the most iconic systems of quantum physics are the particle in a box and the Coulomb potential (the third is, of course, the harmonic oscillator). In this expository paper, we consider the quantum solution to the problem of an electron confined between the grounded planes of an infinite capacitor. The potential arises from the image charges that form in the grounded planes, along with the added condition that at x = 0, L, where L is the distance between the planes, the wavefunction must be zero. This effectively couples a hydrogen like system to a particle-in-a-box (PIB) based on L, the distance between the planes. The problem of finding the electrostatic potential of this infinite series of image charges is an old one, going back to at least 1929. Here, we give a short derivation for one of the limiting cases that yields a compact expression and show how the Kellogg infinite summation formula converges to that value. We note here that this potential is a symmetric double well potential, so there will be many familiar properties of its solutions. Then using that potential, we solve Schrödinger's equation using a spectral technique. The limiting forms of a particle in a box for small L (and high E), and that of a (degenerate) bound image charge at large L and small energy are recovered. We also discuss the tunneling level splitting that occurs in the transition from the large L to the small L regime.
Abstract The stationary states of a particle under the influence of a delta potential confined by impenetrable walls are investigated using the method of expansion in orthogonal functions. The eigenfunctions of the time-independent Schrödinger equation are expressed in closed form by using a pair of closed-form expressions for series available in the literature. The analysis encompasses both attractive and repulsive potentials with arbitrary couplings. Confinement significantly impacts the quantum states and introduces a scenario of double degeneracy including the ground state. Analysis extends to discuss the transition to unconfinement. This research holds particular significance for educators and students engaged in mathematical methods applied to physics and quantum mechanics within undergraduate courses, offering valuable insights into the complex relationships among profiles of potentials, boundary conditions, and the resulting quantum phenomena.
In this work we look at two different 1-dimensional quantum systems. The potentials for these systems are a linear potential in an infinite well and an inverted harmonic oscillator in an infinite well. We will solve the Schrödinger equation for both of these systems and get the energy eigenvalues and eigenfunctions. The solutions are obtained by using the boundary conditions and numerical methods. The motivation for our study comes from experimental background. For the linear potential we have two different boundary conditions. The first one is the so called normal boundary condition in which the wave function goes to zero on the edge of the well. The second condition is called derivative boundary condition in which the derivative of the wave function goes to zero on the edge of the well. The actual solutions are Airy functions. In the case of the inverted oscillator the solutions are parabolic cylinder functions and they are solved only using the normal boundary condition. Both of the potentials are compared with the particle in a box solutions. We will also present figures and tables from which we can see how the solutions look like. The similarities and differences with the particle in a box solution are also shown visually. The figures and calculations are done using mathematical software. We will also compare the linear potential to a case where the infinite wall is only on the left side. For this case we will also show graphical information of the different properties. With the inverted harmonic oscillator we will take a closer look at the quantum mechanical tunneling. We present some of the history of the quantum tunneling theory, its developers and finally we show the Feynman path integral theory. This theory enables us to get the instanton solutions. The instanton solutions are a way to look at the tunneling properties of the quantum system. The results are compared with the solutions of the double-well potential which is very similar to our case as a quantum system. The solutions are obtained using the same methods which makes the comparison relatively easy. All in all we consider and go through some of the stages of the quantum theory. We also look at the different ways to interpret the theory. We also present the special functions that are needed in our solutions, and look at the properties and different relations to other special functions. It is essential to notice that it is possible to use different mathematical formalisms to get the desired result. The quantum theory has been built for over one hundred years and it has different approaches. Different aspects make it possible to look at different things.