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arXiv:1001.0311 · American Journal of Physics

Exact solutions of a particle in a box with a delta function potential: The factorization method

Pouria Pedram, Mohammad Vahabi

Published Jul 14, 2010• 13 citations
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Abstract

We use the factorization method to find the exact eigenvalues and eigenfunctions for a particle in a box with the delta function potential V(x)=λδ(x−x0). We show that the presence of the potential results in the discontinuity of the corresponding ladder operators. The presence of the delta function potential allows us to obtain the full spectrum in the first step of the factorization procedure even in the weak coupling limit λ→0.

DOI
10.1119/1.3373925

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We use the factorization method to find the exact eigenvalues and eigenfunctions for a particle in a box with the delta function potential V(x)=λδ(x−x0). We show that the presence of the potential results in the discontinuity of the corresponding ladder operators. The presence of the delta function potential allows us to obtain the full spectrum in the first step of the factorization procedure even in the weak coupling limit λ→0.
A particle in a box with the δ-function potential Vλ(x)=λδ(x−x0) has been recently explored. We show that this example is solvable in the weak (λ→0±) and the strong (1∕λ→0±) coupling limits. In either limit the attractive and repulsive potentials lead to identical spectra, with the possible exception of a single negative-energy state that is present when 1∕λ→0−. We numerically obtain the spectra near the strong-coupling limit and discuss the consequences of the degeneracy that arises when 1∕λ→0±.
This paper presents several cases in which the effects of the addition of a delta function potential on bound states can be computed exactly. In the case of the one dimensional Schrödinger equation, a Green’s function technique is used to compute an exact implicit expression that gives the effect of the one−dimensional delta function potential on the bound states eigenvalues. It is also shown that the weak coupling limit of the implicit expression agrees with the perturbative treatment of the delta function. It is pointed out that this technique cannot be applied to obtain an exact treatment of the delta function potential δ3(r) in the three−dimensional Schrödinger equation. Instead, an alternate definition of the delta function in three dimensions as a limit of a sequence of square well potentials is used. Unlike the one dimensional case, the exact effect on the eigenvalue does not agree in the limit of weak coupling with the perturbative results. This indicates that the three−dimensional delta function potential should not be used to approximate strongly coupled short range potentials for which perturbation theory is inadequate.
The problem of bound states in delta potentials is revisited by means of Fourier transform approach. The problem in a simple delta potential sums up to solve an algebraic equation of degree one for the Fourier transform of the eigenfunction and the problem for more than one delta function also reveals itself to be a simple matter. Quite differently from direct methods, no knowledge about the jump discontinuity of the first derivative of the eigenfunction is required to determine the solution of the problem.
Abstract: In this work, the problem of the quantum harmonic oscillator with a delta-derivative potential λδ^' (x), where λ is a coupling constant, is solved using the Green’s function technique. A transcendental equation that governs the energy eigenvalues of the problems for each coupling constant is obtained. The eigenfunctions and their first derivative are proved to be discontinuous at the origin. The values of the discontinuity jumps are found to agree with the requirements of having a self-adjoint extension Hamiltonian. In the large coupling limit, the even energy eigenvalues and eigenfunctions for the quantum harmonic oscillator are annihilated, and only the odd parts survive. The dependence of the energy eigenvalues and eigenfunctions on the sign of λ was made clear. A mapping between the sign of λ and the positions of the particle was used to explain the discontinuity of the solution. In the large-λ regime, an educated guess for the wave functions was proposed. The proposed solutions led to the correct energy eigenvalues and obeyed the required conditions to have a self-adjoint extension Hamiltonian.
The problem of bound states in delta potentials is revisited by means of Fourier transform approach. The problem in a simple delta potential sums up to solve an algebraic equation of degree one for the Fourier transform of the eigenfunction and the problem for more than one delta function also reveals itself to be a simple matter. Quite differently from direct methods, no knowledge about the jump discontinuity of the first derivative of the eigenfunction is required to determine the solution of the problem.