Project 1, 2013

In this project we consider an infinite square well of length a in the interval [0,a]. We know the energy spectrum and the eigenfunctions of the single-particle Schrödinger equation for this system

E_n=E₁·n²,  n=1,2,3...,   E₁=(ħπ)²/(2ma²),

ψ_n(x)=A_nsin(k_nx),  k_n=nπ/a,  A_n=(2/a)^1/2.

We will now place a potential V(x)=αδ(x-a/2) into the center of the well. This problem can be solved analytically, but the solution for the energy spectrum is of an implicit type, see the second example in Problem set 4, (ItQM).

• Solve the Schrödinger equation for the problem in a truncated basis made from the complete set of eigenfunctions for a particle in an infinite well without the delta-potential.
• Graph the energy spectrum as a function of the strength of the delta-potential.
• Explore the eigenfunctions of the problem with the delta-potential present.

Please explore the convergence of the solution. The delta-potential is known to be difficult with respect to convergence. Allow α to be negative and positive in your calculations. How is this approach different from traditional perturbation methods? Scaling of all varibles in terms of the natural length a and energy E₁ is essential. Check the dimension of α.

We will talk about the method in the lecture August 29. Important links with relevant information are:

• Fallagrunnar, (Líkön og mælingar)
• Mean-fields, technical details
• Small FORTRAN example program that finds eigenvalues and vectors