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European Journal of Physics

The simplest model of the zero-curvature eigenstate

Zafar Ahmed, Swayam Kesari

Published Dec 13, 2013• 7 citations
Publisher

Abstract

We consider the energy eigenvalue problem of the Dirac delta well potential, V ( x ) = − V 0 δ( x ), placed between two rigid walls at x = ± a . When the strength parameter is increased, the ground state eigenvalue, E 0 , passes from a positive to a negative value. We show that when s = 1, E 0 = 0, the eigenstate is forced to be of zero-curvature: ψ 0 ( x ) = A ( a − | x |), and it is orthogonal to all excited states. On the other hand, if V 0 is made negative then a zero-energy and zero-curvature eigenstate does not exist.

DOI
10.1088/0143-0807/35/1/018002

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We consider the energy eigenvalue problem of the Dirac delta well potential, V ( x ) = − V 0 δ( x ), placed between two rigid walls at x = ± a . When the strength parameter is increased, the ground state eigenvalue, E 0 , passes from a positive to a negative value. We show that when s = 1, E 0 = 0, the eigenstate is forced to be of zero-curvature: ψ 0 ( x ) = A ( a − | x |), and it is orthogonal to all excited states. On the other hand, if V 0 is made negative then a zero-energy and zero-curvature eigenstate does not exist.
Abstract We study a general double Dirac delta potential to show that this is the simplest yet still versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ( T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of a Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission (or zero reflectivity, R ( E ) ) at energy E = 0 is non-intuitive. However, this has been found earlier and called the ‘threshold anomaly’. Here we show that it is a critical phenomenon and we can have 0 ≤ R ( 0 ) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke a zero-energy and zero curvature eigenstate ( ψ ( x ) = Ax + B ) of the delta well between two symmetric rigid walls for R ( 0 ) = 0 . We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.
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