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arXiv:quant-ph/0512156 · European Journal of Physics

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

L. P. Gilbert, Mario Belloni, M. A. Doncheski, Richard W. Robinett

Published Jul 8, 2005• 19 citations
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Abstract

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.

DOI
10.1088/0143-0807/26/5/014

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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.
It is well known that quantum mechanics allows the penetration into classically forbidden regions (tunneling). Less well known seems to be the fact that in some sense the converse is true also. Potentials with classically allowed regions where a particle can move freely to infinity can nevertheless lead to bound states in quantum mechanics due to the stringent requirements of the boundary conditions, thus forbidding an escape to infinity. This effect is demonstrated by using an obvious generalization of the well-known one-dimensional (D=1) square well potential to arbitrary space dimensions.
Abstract We address an interesting problem in elementary quantum mechanics, namely how can one obtain the momentum state eigenfunctions for the infinite square well potential as a direct solution of the time-independent Schrödinger equation in momentum space. The conventional method for obtaining the momentum state eigenfunctions for the infinite square well potential is to take the Fourier transform of the coordinate space eigenfunctions. Unfortunately, the Schrödinger equation in momentum space for the infinite well potential is not well-defined. As such we must use a potential whose Fourier transform is well-defined and then take a limit in which the potential approaches that of the infinite well potential. We present three approaches to the problem: one based on dimensional analysis, one in which the infinite well potential is viewed as the limit of a finite well whose depth approaches infinity, and one in which the infinite well potential is viewed as the limit of a power law potential in the limit that the power approaches infinity.
Abstract One-dimensional potentials defined by V ( S ) ( x ) = S ( S + 1 ) ℏ 2 π 2 / [ 2 ma 2 sin 2 ( π x / a ) ] (for integer S ) arise in the repeated supersymmetrization of the infinite square well, here defined over the region (0, a ). We review the derivation of this hierarchy of potentials and then use the methods of supersymmetric quantum mechanics, as well as more familiar textbook techniques, to derive compact closed-form expressions for the normalised solutions, ψ n ( S ) ( x ) , for all V ( S ) ( x ) in terms of well-known special functions in a pedagogically accessible manner. We also note how the solutions can be obtained as a special case of a family of shape-invariant potentials, the trigonometric Pöschl–Teller potentials, which can be used to confirm our results. We then suggest additional avenues for research questions related to, and pedagogical applications of, these solutions, including the behaviour of the corresponding momentum-space wave functions ϕ n ( S ) ( p ) for large ∣ p ∣ and general questions about the supersymmetric hierarchies of potentials which include an infinite barrier.