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European Journal of Physics

On the placement of scaled delta barriers in the infinite square well

Todd Keene Timberlake, Benjamin Roush

Published Feb 9, 2026• 0 citations
Publisher

Abstract

Abstract In the quantum infinite square well with hard walls at x = 0 and L the energy eigenvalues are proportional to ϵ n 2 , where ϵ n = n is a dimensionless reduced wave number. We examine how the values of ϵ n are modified by the addition of one or more Dirac delta barriers, scaled so that the probability T for a particle to transmit through the barrier is not dependent on the particle’s energy. If the barriers are placed at rational locations x = iL / D , where D > 1 is an integer common denominator and i is any integer such that 0 < i < D , then ϵ n follows a pattern that repeats every D values. Once the first D values of ϵ n are found, all other values follow from this pattern. Analytical solutions for ϵ n are derived for D = 2 and 3. Numerical solutions for D = 7 are shown to illustrate phenomena such as band structure, defects, and avoided crossings. When the barriers are placed at irrational locations there is no repeating pattern, but a finite sequence of ϵ values can be well-approximated by using results with the barriers placed at rational approximations of the irrational locations. Because this system admits analytical solutions, or numerical solutions obtained with minimal effort, it could serve as the basis for interesting exercises or projects in undergraduate or beginning graduate courses in quantum mechanics.

DOI
10.1088/1361-6404/ae43a1

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Abstract In the quantum infinite square well with hard walls at x = 0 and L the energy eigenvalues are proportional to ϵ n 2 , where ϵ n = n is a dimensionless reduced wave number. We examine how the values of ϵ n are modified by the addition of one or more Dirac delta barriers, scaled so that the probability T for a particle to transmit through the barrier is not dependent on the particle’s energy. If the barriers are placed at rational locations x = iL / D , where D > 1 is an integer common denominator and i is any integer such that 0 < i < D , then ϵ n follows a pattern that repeats every D values. Once the first D values of ϵ n are found, all other values follow from this pattern. Analytical solutions for ϵ n are derived for D = 2 and 3. Numerical solutions for D = 7 are shown to illustrate phenomena such as band structure, defects, and avoided crossings. When the barriers are placed at irrational locations there is no repeating pattern, but a finite sequence of ϵ values can be well-approximated by using results with the barriers placed at rational approximations of the irrational locations. Because this system admits analytical solutions, or numerical solutions obtained with minimal effort, it could serve as the basis for interesting exercises or projects in undergraduate or beginning graduate courses in quantum mechanics.
For the quantum particle in a finite and semi-infinite well, we scale space and energy to consider a dimensionless Schrdinger equation that can be solved for the allowed energy levels. In contrast to traditional approaches that determine the energies from dimensional transcendental equations associated with the boundary conditions, we solve for the square root of the dimensionless energies, which are also solutions of dimensionless transcendental equations. Our ability to define classes of wells depends on the value of the square root of the dimensionless well depth. We are also able to determine the maximum number of energy levels quite easily, and, besides deriving the relevant transcendental equations, we obtain asimple expression for the corresponding wave functions, which can be plotted using the numerical values of the allowed energies.
Certain superposition states of the 1-D infinite square well have transient zeros at locations other than the nodes of the eigenstates that comprise them. It is shown that if an infinite potential barrier is suddenly raised at some or all of these zeros, the well can be split into multiple adjacent infinite square wells without affecting the wavefunction. This effects a change of the energy eigenbasis of the state to a basis that does not commute with the original, and a subsequent measurement of the energy now reveals a completely different spectrum, which we call the interference energy spectrum of the state. This name is appropriate because the same splitting procedure applied at the stationary nodes of any eigenstate does not change the measurable energy of the state. Of particular interest, this procedure can result in measurable energies that are greater than the energy of the highest mode in the original superposition, raising questions about the conservation of energy akin to those that have been raised in the study of superoscillations. An analytic derivation is given for the interference spectrum of a given wavefunction Ψ ( x , t ) with N known zeros located at points s i = ( x i , t i ) . Numerical simulations were used to verify that a barrier can be rapidly raised at a zero of the wavefunction without significantly affecting it. The interpretation of this result with respect to the conservation of energy and the energy-time uncertainty relation is discussed, and the idea of alternate energy eigenbases is fleshed out. The question of whether or not a preferred discrete energy spectrum is an inherent feature of a particle’s quantum state is examined.
The computation of allowed energy levels for a particle bounded to a finite square well potential is ubiquitous in modern physics and introductory quantum mechanics textbooks. For stationary bound states, the matching conditions for the wave functions lead to a pair of transcendental equations whose roots correspond to the energy eigenvalues. However, the graphical solutions available do not make clear the dependence of the energies on the well potential parameters. In this note, I present a simpler graphical solution involving only one dimensionless parameter that determines a straight-line crossing identical sinusoidal curves. I then reduce this solution to a single cosine curve, and from a three-point interpolation, I derive an approximate formula for all energy levels valid for any square quantum well and that demands only a pocket calculator.
The time evolution of quantum mechanical states in a square well of infinite depth with a Dirac δ function at its center has been examined for cases where the initial state was localized in one of the wells. Let well A denote the well in which the initial state is localized, and let P(t) denote the integrated probability density Ψ*Ψ in well A. For opaque barriers the time-dependent system is adequately described by a two-state model in which only the pair of stationary states of even and odd parity are considered whose wave functions in well A, apart from an arbitrary phase factor, are largely identical with the wave function of the initial state. For P(t) a harmonic oscillation is observed whose frequency νt is well approximated by the well-known formula for the tunneling frequency νt≂ΔE/h, where ΔE represents the energy separation among the pair of states in the model. For the present model of a symmetric double well it has been shown that for highly transparent barriers a three-state model can describe the time evolution adequately. The three stationary states involved in this model are a state of odd parity whose wave function in well A is largely the same as the wave function of the initial state and a pair of stationary states of even parity which on the energy scale are immediately above and below the first state. In this three-state model the function P(t) is a superposition of two sinusoidal functions with nearly identical amplitudes and frequencies plus a constant. As a consequence, the amplitude of P(t) changes harmonically. In the present model a δ function has been used as a barrier in order to minimize the mathematical detail involved in the time-dependent treatment. It is to be expected that the beating in P(t) can be observed also in the time evolution of a state localized in one of the wells of a symmetric double-minimum potential with a more realistic ‘‘low’’ barrier if the density of energy levels near the energies of the levels to be considered in the corresponding three-state model varies slowly with energy.
Abstract: This work considered the infinite square well in one dimension with a contact potential. The Dirac delta derivative function potential λδ^' where λ is a coupling constant was used to represent the contact potential. Using Green’s function technique, exact implicit expressions of the energy eigenvalues and eigenfunctions were obtained. The energy eigenvalues were expressed using a transcendental equation. The energy eigenfunctions satisfy the Schrödinger equation and the infinite square well boundary conditions. Also, the eigenfunctions and their first derivative were shown to be discontinuous. The values of these discontinuity jumps agreed with the required conditions for a self-adjoint extension Hamiltonian. In the weak coupling region, the energy eigenvalues are close to that of the even parity solution before adding the contact potential. The energy eigenvalues in the strong coupling regime reveal the energy eigenvalues of the odd parity solution. Keywords: Point interactions, Infinite square well, Green’s function technique. PACS numbers: 03.65.-w, 03.65.Db, 03.65.Ge