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Playing Quantum Physics Jeopardy with zero-energy eigenstates

Gilbert, L P, Belloni, M, Doncheski, M A, Robinett, R W

Published Jun 23, 2006• 8 citations
OpenAlex

Abstract

We describe an example of an exact, quantitative Jeopardy-type quantum mechanics problem. This problem type is based on the conditions in one-dimensional quantum systems that allow an energy eigenstate for the infinite square well to have zero curvature and zero energy when suitable Dirac delta functions are added. This condition and its solution are not often discussed in quantum mechanics texts and have interesting pedagogical consequences.

Related papers

Playing Quantum Physics Jeopardy with zero-energy eigenstates

L. P. Gilbert, M. Belloni, M. A. Doncheski +1 more

We describe an example of an exact, quantitative Jeopardy-type quantum mechanics problem. This problem type is based on the conditions in one-dimensional quantum systems that allow an energy eigenstate for the infinite square well to have zero curvature and zero energy when suitable Dirac delta functions are added. This condition and its solution are not often discussed in quantum mechanics texts and have interesting pedagogical consequences.

arXiv:quant-ph/0606196•American Journal of Physics•Oct 18, 2006•9 citations

More on the asymmetric infinite square well: energy eigenstates with zero curvature

L. P. Gilbert, Mario Belloni, M. A. Doncheski +1 more

We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wavefunction to have zero curvature over an extended region of space and still be a valid wavefunction. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space.

arXiv:quant-ph/0512156•European Journal of Physics•Jul 8, 2005•19 citations

Piecewise zero-curvature energy eigenfunctions in one dimension

L. P. Gilbert, Mario Belloni, M. A. Doncheski +1 more

We discuss the general mathematical condition in one-dimensional time-independent quantum mechanics for an energy eigenfunction to have zero curvature over an extended region of space and still be valid. This condition and its solution are not often discussed as part of quantum mechanics texts at any level, yet have interesting consequences for experimental, pedagogical and theoretical investigations of systems with piecewise-constant potential energy functions. We present the solutions for several standard cases in which zero-curvature energy eigenfunctions are allowed as bound states, scattering states and threshold states. These states are of interest to supersymmetric quantum mechanics and the procedures we outline can be used to 'design' quantum wells for studies of wave packet dynamics.

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A Quantum Physics Jeopardy Problem of a One-Bound-State Double Dirac Delta Potential with a Centrifugal, Angular-Momentum-Like Tail

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Infinite Square Well, Self-Adjointness, and the Dirac Delta Function

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Abstract One hundred years ago, Dirac presented a groundbreaking method for quantizing classical Hamiltonians. Simplicity is one of the virtues of the procedure, but certain classes of problems require special care. Here, we discuss one such class, the quantization of confined systems, with emphasis on the one-dimensional square well. We examine two presentations of the problem. The first presentation restricts the analysis to the interval $$0\le x\le L$$ . The second presentation considers the entire x axis, but an infinite potential traps the particle in the region $$0\le x\le L$$ . In contrast with the typical treatment in quantum mechanics textbooks, the first presentation imposes nontrivial boundary conditions. Proper treatment of the problem in the second presentation demands the use of delta functions or distributions.

Brazilian Journal of Physics•Mar 19, 2026•0 citations

Chapter 8. The Infinite Square Well

The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend

Sciendo eBooks•Sep 1, 2018•0 citations
We describe an example of an exact, quantitative Jeopardy-type quantum mechanics problem. This problem type is based on the conditions in one-dimensional quantum systems that allow an energy eigenstate for the infinite square well to have zero curvature and zero energy when suitable Dirac delta functions are added. This condition and its solution are not often discussed in quantum mechanics texts and have interesting pedagogical consequences.
We extend the standard treatment of the asymmetric infinite square well to include solutions that have zero curvature over part of the well. This type of solution, both within the specific context of the asymmetric infinite square well and within the broader context of bound states of arbitrary piecewise-constant potential energy functions, is not often discussed as part of quantum mechanics texts at any level. We begin by outlining the general mathematical condition in one-dimensional time-independent quantum mechanics for a bound-state wavefunction to have zero curvature over an extended region of space and still be a valid wavefunction. We then briefly review the standard asymmetric infinite square well solutions, focusing on zero-curvature solutions as represented by energy eigenstates in position and momentum space.
We discuss the general mathematical condition in one-dimensional time-independent quantum mechanics for an energy eigenfunction to have zero curvature over an extended region of space and still be valid. This condition and its solution are not often discussed as part of quantum mechanics texts at any level, yet have interesting consequences for experimental, pedagogical and theoretical investigations of systems with piecewise-constant potential energy functions. We present the solutions for several standard cases in which zero-curvature energy eigenfunctions are allowed as bound states, scattering states and threshold states. These states are of interest to supersymmetric quantum mechanics and the procedures we outline can be used to 'design' quantum wells for studies of wave packet dynamics.
Abstract One hundred years ago, Dirac presented a groundbreaking method for quantizing classical Hamiltonians. Simplicity is one of the virtues of the procedure, but certain classes of problems require special care. Here, we discuss one such class, the quantization of confined systems, with emphasis on the one-dimensional square well. We examine two presentations of the problem. The first presentation restricts the analysis to the interval 0≤x≤L0\le x\le L0≤x≤L . The second presentation considers the entire x axis, but an infinite potential traps the particle in the region 0≤x≤L0\le x\le L0≤x≤L . In contrast with the typical treatment in quantum mechanics textbooks, the first presentation imposes nontrivial boundary conditions. Proper treatment of the problem in the second presentation demands the use of delta functions or distributions.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend